Systems of partial differential equations: modeling, numerical analysis and simulation

A.H. Erhardt, K. Tsaneva-Atanasova, G.T. Lines, E.A. Martens, eds., Dynamical Systems, PDEs and Networks for Biomedical Applications: Mathematical Modeling, Analysis and Simulations, Special Edition, articles published in Frontiers of Physics, Frontiers in Applied Mathematics and Statistics, and Frontiers in Physiology, Frontiers Media SA, Lausanne, Switzerland, 2023, 207 pages, (Collection Published), DOI 10.3389/978-2-8325-1458-0 .

E. Franck, J. Fuhrmann, V. Michel-Dansac, L. Navoret, eds., Finite Volumes for Complex Applications X -- Volume 1, Elliptic and Parabolic Problems: FVCA10, Strasbourg, France, October 30, 2023 -- November 03, 2023, Invited Contributions, 432 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham, 2023, 396 pages, (Collection Published), DOI 10.1007/978-3-031-40864-9 .

E. Franck, J. Fuhrmann, V. Michel-Dansac, L. Navoret, eds., Finite Volumes for Complex Applications X -- Volume 2, Hyperbolic and Related Problems: FVCA10, Strasbourg, France, October 30, 2023 -- November 03, 2023, 433 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham, 2023, 308 pages, (Collection Published), DOI 10.1007/978-3-031-40860-1 .

Z. Erkoç, A. Aman, U. Güdükbay, H. Si, Out-of-core constrained Delaunay tetrahedralizations for large scenes, V.A. Garanzha, L. Kamenski, H. Si, eds., 143 of Lecture Notes in Computational Science and Engineering, Springer Nature Switzerland AG, Cham, 2021, pp. 113--124, (Chapter Published), DOI 10.1007/978-3-030-76798-3 .

H. Si, On decomposition of embedded prismatoids in $R^3$ without additional points, V.A. Garanzha, L. Kamenski, H. Si, eds., 143 of Lecture Notes in Computational Science and Engineering, Springer Nature Switzerland AG, Cham, 2021, pp. 95--112, (Chapter Published), DOI 10.1007/978-3-030-76798-3 .

V.A. Garanzha, L. Kamenski, H. Si, eds., Numerical Geometry, Grid Generation and Scientific Computing. Proceedings of the 10th International Conference, NUMGRID 2020 / Delaunay 130, Celebrating the 130th Anniversary of Boris Delaunay, Moscow, Russia, November 2020, 143 of Lecture Notes in Computational Science and Engineering, Springer Nature Switzerland AG, Cham, 2021, 417 pages, (Collection Published), DOI 10.1007/978-3-030-76798-3 .

R. Ahrens, Z. Lakdawala, A. Voigt, V. Wiedmeyer, V. John, S. Le Borne, K. Sundmacher, Chapter 14: Numerical Methods for Coupled Population Balance Systems Applied to the Dynamical Simulation of Crystallization Processes, in: Dynamic Flowsheet Simulation of Solids Processes, S. Heinrich, ed., Springer, Cham, 2020, pp. 475--518, (Chapter Published), DOI 10.1007/978-3-030-45168-4_14 .

U.W. Pohl, A. Strittmatter, A. Schliwa, M. Lehmann, T. Niermann, T. Heindel, S. Reitzenstein, M. Kantner, U. Bandelow, Th. Koprucki, H.-J. Wünsche, Chapter 3: Stressor-Induced Site Control of Quantum Dots for Single-Photon Sources, in Semiconductor Nanophotonics, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., 194 of Springer Series in Solid-State Sciences, Springer, Heidelberg, 2020, pp. 53--90, (Chapter Published), DOI 10.1007/978-3-030-35656-9_3 .
Abstract
The strain field of selectively oxidized A10x current apertures in an A1GaAs/GaAs mesa is utilized to define the nucleation site of InGaAs/GaAs quantum dots. A design is developed that allows for the self-aligned growth of single quantum dots in the center of a circular mesa. Measurements of the strain tensor applying transmission-electron holography yield excellent agreement with the calculated strain field. Single-dot spectroscopy of site-controlled dots proves narrow excitonic linewidth virtually free of spectral diffusion due to quantum-dot growth in a defect-free matrix. Implementation of such dots in an electrically driven pin structure yields single-dot electroluminescence. Single-photon emission with excellent purity is provided for this device using a Hanbury Brown and Twiss setup. The injection efficiency of the initian pin design is affected by a substantial lateral current spreading close to the oxide aperture. Allpying 3D carier-transport simulation a ppn doping profile is developed achieving a substantial improvement of the current injection.

S. Rodt, P.-I. Schneider, L. Zschiedrich, T. Heindel, S. Bounouar, M. Kantner, Th. Koprucki, U. Bandelow, S. Burger, S. Reitzenstein, Chapter 8: Deterministic Quantum Devices for Optical Quantum Communication, in: Semiconductor Nanophotonics, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., 194 of Springer Series in Solid-State Sciences, Springer, Heidelberg, 2020, pp. 285--359, (Chapter Published), DOI 10.1007/978-3-030-35656-9_8 .
Abstract
Photonic quantum technologies are based on the exchange of information via single photons. The information is typically encoded in the polarization of the photons and security is ensured intrinsically via principles of quantum mechanics such as the no-cloning theorem. Thus, all optical quantum communication networks rely crucially on the availability of suitable quantum-light sources. Such light sources with close to ideal optical and quantum optical properties can be realized by self-assembled semiconductor quantum dots. These high-quality nanocrystals are predestined single-photon emitters due to their quasi zero-dimensional carrier confinement. Still, the development of practical quantum-dot-based sources of single photons and entangled-photon pairs for applications in photonic quantum technology and especially for the quantum-repeater scheme is very demanding and requires highly advanced device concepts and deterministic fabrication technologies. This is mainly explained by their random position and emission energy as well as by the low photon-extraction efficiency in simple planar device configurations.

V. John, P. Knobloch, U. Wilbrandt, Chapter 6: Finite Element Pressure Stabilizations for Incompressible Flow Problems, in: Fluids under Pressure, T. Bodnár, G. Galdi, Š. Nečasová, eds., Advances in Mathematical Fluid Mechanics, Birkhäuser, Cham, 2020, pp. 483--573, (Chapter Published), DOI 10.1007/978-3-030-39639-8_6 .
Abstract
Discretizations of incompressible flow problems with pairs of finite element spaces that do not satisfy a discrete inf-sup condition require a so-called pressure stabilization. This paper gives an overview and systematic assessment of stabilized methods, including the respective error analysis.

M. Kantner, Th. Höhne, Th. Koprucki, S. Burger, H.-J. Wünsche, F. Schmidt, A. Mielke, U. Bandelow, Chapter 7: Multi-Dimensional Modeling and Simulation of Semiconductor Nanophotonic Devices, in: Semiconductor Nanophotonics, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., 194 of Springer Series in Solid-State Sciences, Springer, Heidelberg, 2020, pp. 241--283, (Chapter Published), DOI 10.1007/978-3-030-35656-9_7 .
Abstract
Self-consistent modeling and multi-dimensional simulation of semiconductor nanophotonicdevices is an important tool in the development of future integrated light sources and quantumdevices. Simulations can guide important technological decisions by revealing performance bottle-necks in new device concepts, contribute to their understanding and help to theoretically exploretheir optimization potential. The efficient implementation of multi-dimensional numerical simulationsfor computer-aided design tasks requires sophisticated numerical methods and modeling tech-niques. We review recent advances in device-scale modeling of quantum dot based single-photonsources and laser diodes by self-consistently coupling the optical Maxwell equations with semi-classical carrier transport models using semi-classical and fully quantum mechanical descriptionsof the optically active region, respectively. For the simulation of realistic devices with complex,multi-dimensional geometries, we have developed a novel hp-adaptive finite element approachfor the optical Maxwell equations, using mixed meshes adapted to the multi-scale properties ofthe photonic structures. For electrically driven devices, we introduced novel discretization andparameter-embedding techniques to solve the drift-diffusion system for strongly degenerate semi-conductors at cryogenic temperature. Our methodical advances are demonstrated on variousapplications, including vertical-cavity surface-emitting lasers, grating couplers and single-photonsources.

M. Kantner, Electrically Driven Quantum Dot Based Single-Photon Sources: Modeling and Simulation, Springer Theses, Springer, Cham, 2020, XVII, 180 pages, (Monograph Published), DOI 10.1007/978-3-030-39543-8 .
Abstract
Semiconductor quantum optics is on the verge of moving from the lab to real world applications. When stepping from basic research to new technologies, device engineers will need new simulation tools for the design and optimization of quantum light sources, which combine classical device physics with cavity quantum electrodynamics. This thesis aims to provide a holistic description of single-photon emitting diodes by bridging the gap between microscopic and macroscopic modeling approaches. The central result is a novel hybrid quantum-classical model system that self-consistently couples semi-classical carrier transport theory with open quantum many-body systems. This allows for a comprehensive description of quantum light emitting diodes on multiple scales: It enables the calculation of the quantum optical figures of merit together with the simulation of the spatially resolved current flow in complex, multi-dimensional semiconductor device geometries out of one box. The hybrid system is shown to be consistent with fundamental laws of (non-)equilibrium thermodynamics and is demonstrated by numerical simulations of realistic devices.

R. Klöfkorn, E. Keilegavlen, F.A. Radu , J. Fuhrmann, eds., Finite Volumes for Complex Applications IX -- Methods, Theoretical Aspects, Examples -- FVCA 9, Bergen, June 2020, 323 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2020, 775 pages, (Collection Published), DOI 10.1007/978-3-030-43651-3 .

V.A. Garanzha, L. Kamenski, H. Si, eds., Numerical Geometry, Grid Generation and Scientific Computing. Proceedings of the 9th International Conference, NUMGRID 2018 / Voronoi 150, Celebrating the 150th Anniversary of G.F. Voronoi, Moscow, Russia, December 2018, 131 of Lecture Notes in Computational Science and Engineering, Springer Nature Switzerland AG, Cham, 2019, 319 pages, (Collection Published), DOI 10.1007/978-3-030-23436-2 .

M. Hintermüller, J.F. Rodrigues, eds., Topics in Applied Analysis and Optimisation -- Partial Differential Equations, Stochastic and Numerical Analysis, CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, 396 pages, (Collection Published).

P. Farrell, N. Rotundo, D.H. Doan, M. Kantner, J. Fuhrmann, Th. Koprucki, Chapter 50: Drift-Diffusion Models, in: Vol. 2 of Handbook of Optoelectronic Device Modeling and Simulation: Lasers, Modulators, Photodetectors, Solar Cells, and Numerical Methods, J. Piprek, ed., Series in Optics and Optoelectronics, CRC Press, Taylor & Francis Group, Boca Raton, 2017, pp. 733--771, (Chapter Published).

H.-Chr. Kaiser, D. Knees, A. Mielke, J. Rehberg, E. Rocca, M. Thomas, E. Valdinoci, eds., PDE 2015: Theory and Applications of Partial Differential Equations, 10 of Discrete and Continuous Dynamical Systems -- Series S, American Institute of Mathematical Science, Springfield, 2017, iv+933 pages, (Collection Published).

P. Exner, W. König, H. Neidhardt, eds., Mathematical Results in Quantum Mechanics. Proceedings of the QMath12 Conference, World Scientific Publishing, Singapore, 2015, xii+383 pages, (Collection Published).

A. Zisowsky, A. Arnold, M. Ehrhardt, Th. Koprucki, Chapter 7: Transient Simulation of k$cdot$p-Schrödinger Systems Using Discrete Transparent Boundary Conditions, in: Multi-Band Effective Mass Approximations -- Advanced Mathematical Models and Numerical Techniques, M. Ehrhardt, Th. Koprucki, eds., 94 of Lecture Notes in Computational Science and Engineering, Springer, Cham et al., 2014, pp. 247--272, (Chapter Published).

D. Klindworth, M. Ehrhardt, Th. Koprucki, Chapter 8: Discrete Transparent Boundary Conditions for Multi-band Effective Mass Approximations, in: Multi-Band Effective Mass Approximations -- Advanced Mathematical Models and Numerical Techniques, M. Ehrhardt, Th. Koprucki, eds., 94 of Lecture Notes in Computational Science and Engineering, Springer, Cham et al., 2014, pp. 273--318, (Chapter Published).

M. Ehrhardt, Th. Koprucki, eds., Multi-Band Effective Mass Approximations --- Advanced Mathematical Models and Numerical Techniques, 94 of Lecture Notes in Computational Science and Engineering, Springer, Cham et al., 2014, xvi+318 pages, (Monograph Published).

I. Laukaityte, R. Čiegis, M. Lichtner, M. Radziunas, Parallel Numerical Algorithm for the Traveling Wave Model, in: Parallel Scientific Computing and Optimization: Advances and Applications, R. Čiegis, D. Henty, B. Kågström, J. Žilinskas, eds., 27 of Springer Optimization and Its Applications, Springer, New York, 2008, pp. 237-251, (Chapter Published).

M. Tlidi, R. Lefever, A.G. Vladimirov, On Vegetation Clustering, Localized Bare Soil Spots and Fairy Circles, in: Dissipative Solitons: From Optics to Biology and Medicine, N. Akhmediev, A. Ankiewicz, eds., 751 of Lecture Notes in Physics, Springer, Berlin, Heidelberg, 2008, pp. 381-402, (Chapter Published).

U. Bandelow, H. Gajewski, R. Hünlich, Chapter 3: Fabry--Perot Lasers: Thermodynamics-based Modeling, in: Optoelectronic Devices --- Advanced Simulation and Analysis, J. Piprek, ed., Springer, New York, 2005, pp. 63-85, (Chapter Published).

D. Abdel, A. Glitzky, M. Liero, Analysis of a drift-diffusion model for perovskite solar cells, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 30 (2025), pp. 99--131, DOI 10.3934/dcdsb.2024081 .
Abstract
This paper deals with the analysis of an instationary drift-diffusion model for perovskite solar cells including Fermi--Dirac statistics for electrons and holes and Blakemore statistics for the mobile ionic vacancies in the perovskite layer. The free energy functional is related to this choice of the statistical relations. Exemplary simulations varying the mobility of the ionic vacancy demonstrate the necessity to include the migration of ionic vacancies in the model frame. To prove the existence of weak solutions, first a problem with regularized state equations and reaction terms on any arbitrarily chosen finite time interval is considered. Its solvability follows from a time discretization argument and passage to the time-continuous limit. Applying Moser iteration techniques, a priori estimates for densities, chemical potentials and the electrostatic potential of its solutions are derived that are independent of the regularization level, which in turn ensure the existence of solutions to the original problem.

L. Schmeller, D. Peschka, Sharp-interface limits of Cahn--Hilliard models and mechanics with moving contact lines, , 22 (2024), pp. 869--890, DOI 10.1137/23M1546592 .
Abstract
We construct gradient structures for free boundary problems with moving capillary interfaces with nonlinear (hyper)elasticity and study the impact of moving contact lines. In this context, we numerically analyze how phase-field models converge to certain sharp-interface models when the interface thickness tends to zero. In particular, we study the scaling of the Cahn--Hilliard mobility with certain powers of the interfacial thickness. In the presence of interfaces, it is known that the intended sharp-interface limit holds only for a particular range of powers However, in the presence of moving contact lines we show that some scalings that are valid for interfaces produce significant errors and the effective range of valid powers of the interfacial thickness in the mobility reduces.

A. Erhardt, D. Peschka, Ch. Dazzi, L. Schmeller, A. Petersen, S. Checa, A. Münch, B. Wagner, Modeling cellular self-organization in strain-stiffening hydrogels, Computational Mechanics, published online on 31.08.2024, DOI 10.1007/s00466-024-02536-7 .
Abstract
We develop a three-dimensional mathematical model framework for the collective evolution of cell populations by an agent-based model (ABM) that mechanically interacts with the surrounding extracellular matrix (ECM) modeled as a hydrogel. We derive effective two-dimensional models for the geometrical set-up of a thin hydrogel sheet to study cell-cell and cell-hydrogel mechanical interactions for a range of external conditions and intrinsic material properties. We show that without any stretching of the hydrogel sheets, cells show the well-known tendency to form long chains with varying orientations. Our results further show that external stretching of the sheet produces the expected nonlinear strain-softening or stiffening response, with, however, little qualitative variation of the overall cell dynamics for all the materials considered. The behavior is remarkably different when solvent is entering or leaving from strain softening or stiffening hydrogels, respectively.

Y. Hadjimichael, Ch. Merdon, M. Liero, P. Farrell, An energy-based finite-strain model for 3D heterostructured materials and its validation by curvature analysis, International Journal for Numerical Methods in Engineering, e7508 (2024), pp. 7508/1--7508/28, DOI 10.1002/nme.7508 .
Abstract
This paper presents a comprehensive study of the intrinsic strain response of 3D het- erostructures arising from lattice mismatch. Combining materials with different lattice constants induces strain, leading to the bending of these heterostructures. We propose a model for nonlinear elastic heterostructures such as bimetallic beams or nanowires that takes into account local prestrain within each distinct material region. The resulting system of partial differential equations (PDEs) in Lagrangian coordinates incorporates a nonlinear strain and a linear stress-strain relationship governed by Hooke?s law. To validate our model, we apply it to bimetallic beams and hexagonal hetero-nanowires and perform numerical simulations using finite element methods (FEM). Our simulations ex- amine how these structures undergo bending under varying material compositions and cross-sectional geometries. In order to assess the fidelity of the model and the accuracy of simulations, we compare the calculated curvature with analytically derived formula- tions. We derive these analytical expressions through an energy-based approach as well as a kinetic framework, adeptly accounting for the lattice constant mismatch present at each compound material of the heterostructures. The outcomes of our study yield valuable insights into the behavior of strained bent heterostructures. This is particularly significant as the strain has the potential to influence the electronic band structure, piezoelectricity, and the dynamics of charge carriers.

M. Heida, M. Landstorfer, M. Liero, Homogenization of a porous intercalation electrode with phase separation, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 22 (2024), pp. 1068--1096, DOI 10.1137/21M1466189 .
Abstract
In this work, we derive a new model framework for a porous intercalation electrode with a phase separating active material upon lithium intercalation. We start from a microscopic model consisting of transport equations for lithium ions in an electrolyte phase and intercalated lithium in a solid active phase. Both are coupled through a Neumann--boundary condition modeling the lithium intercalation reaction. The active material phase is considered to be phase separating upon lithium intercalation. We assume that the porous material is a given periodic microstructure and perform analytical homogenization. Effectively, the microscopic model consists of a diffusion and a Cahn--Hilliard equation, whereas the limit model consists of a diffusion and an Allen--Cahn equation. Thus we observe a Cahn--Hilliard to Allen--Cahn transition during the upscaling process. In the sense of gradient flows, the transition goes in hand with a change in the underlying metric structure of the PDE system.

M. O'Donovan, P. Farrell, J. Moatti, T. Streckenbach, Th. Koprucki, S. Schulz, Impact of random alloy fluctuations on the carrier distribution in multi-color (In,Ga)N/GaN quantum well systems, Physical Review Applied, 21 (2024), pp. 024052/1--024052/12, DOI 10.1103/PhysRevApplied.21.024052 .
Abstract
In this work, we study the impact that random alloy fluctuations have on the distribution of electrons and holes across the active region of a (In,Ga)N/GaN multi-quantum well based light emitting diode (LED). To do so, an atomistic tight-binding model is employed to account for alloy fluctuations on a microscopic level and the resulting tight-binding energy landscape forms input to a drift-diffusion model. Here, quantum corrections are introduced via localization landscape theory and we show that when neglecting alloy disorder our theoretical framework yields results similar to commercial software packages that employ a self-consistent Schroedinger-Poisson-drift-diffusion solver. Similar to experimental studies in the literature, we have focused on a multi-quantum well system where two of the three wells have the same In content while the third well differs in In content. By changing the order of wells in this multicolor quantum well structure and looking at the relative radiative recombination rates of the different emitted wavelengths, we (i) gain insight into the distribution of carriers in such a system and (ii) can compare our findings to trends observed in experiment. Our results indicate that the distribution of carriers depends significantly on the treatment of the quantum well microstructure. When including random alloy fluctuations and quantum corrections in the simulations, the calculated trends in the relative radiative recombination rates as a function of the well ordering are consistent with previous experimental studies. The results from the widely employed virtual crystal approximation contradict the experimental data. Overall, our work highlights the importance of a careful and detailed theoretical description of the carrier transport in an (In,Ga)N/GaN multi-quantum well system to ultimately guide the design of the active region of III-N-based LED structures.

S. Haberland, P. Jaap, S. Neukamm, O. Sander, M. Varga, Representative volume element approximations in elastoplastic spring networks, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 22 (2024), pp. 588--638, DOI 10.1137/23M156656X .
Abstract
We study the large-scale behavior of a small-strain lattice model for a network composed of elastoplastic springs with random material properties. We formulate the model as an evolutionary rate independent system. In an earlier work we derived a homogenized continuum model, which has the form of linearized elastoplasticity, as an evolutionary ?-limit as the lattice parameter tends to zero. In the present paper we introduce a periodic representative volume element (RVE) approximation for the homogenized system. As a main result we prove convergence of the RVE approximation as the size of the RVE tends to infinity. We also show that the hysteretic stress-strain relation of the effective system can be described with the help of a generalized Prandtl?Ishlinskii operator, and we prove convergence of a periodic RVE approximation for that operator. We combine the RVE approximation with a numerical scheme for rate-independent systems and obtain a computational scheme that we use to numerically investigate the homogenized system in the specific case when the original network is given by a two-dimensional lattice model. We simulate the response of the system to cyclic and uniaxial, monotonic loading, and numerically investigate the convergence rate of the periodic RVE approximation. In particular, our simulations show that the RVE error decays with the same rate as the RVE error in the static case of linear elasticity.

N. Ahmed, V. John, X. Li, Ch. Merdon, Inf-sup stabilized Scott--Vogelius pairs on general shape-regular simplicial grids for Navier--Stokes equations, Computers & Mathematics with Applications. An International Journal, 168 (2024), pp. 148--161, DOI 10.1016/j.camwa.2024.05.034 .
Abstract
This paper considers the discretization of the time-dependent Navier--Stokes equations with the family of inf-sup stabilized Scott--Vogelius pairs recently introduced in [John/Li/Merdon/Rui, Math. Models Methods Appl. Sci., 2024] for the Stokes problem. Therein, the velocity space is obtained by enriching the H -conforming Lagrange element space with some H (div)-conforming Raviart--Thomas functions, such that the divergence constraint is satisfied exactly. In these methods arbitrary shape-regular simplicial grids can be used. In the present paper two alternatives for discretizing the convective terms are considered. One variant leads to a scheme that still only involves volume integrals, and the other variant employs upwinding known from DG schemes. Both variants ensure the conservation of linear momentum and angular momentum in some suitable sense. In addition, a pressure-robust and convection-robust velocity error estimate is derived, i.e., the velocity error bound does not depend on the pressure and the constant in the error bound for the kinetic energy does not blow up for small viscosity. After condensation of the enrichment unknowns and all non-constant pressure unknowns, the method can be reduced to a P - P -like system for arbitrary velocity polynomial degree k. Numerical studies verify the theoretical findings.

L. Araujo, C. Lasser, B. Schmidt, FSSH-2: Fewest Switches Surface Hopping with robust switching probability, Journal of Chemical Theory and Computation, 20 (2024), pp. 3413--3419, DOI 10.1021/acs.jctc.4c00089 .
Abstract
This study introduces the FSSH-2 scheme, a redefined and numerically stable adiabatic Fewest Switches Surface Hopping (FSSH) method. It reformulates the standard FSSH hopping probability without non-adiabatic coupling vectors and allows for numerical time integration with larger step sizes. The advantages of FSSH-2 are demonstrated by numerical experiments for five different model systems in one and two spatial dimensions with up to three electronic states.

R. Araya, A. Caiazzo, F. Chouly, Stokes problem with slip boundary conditions using stabilized finite elements combined with Nitsche, Computer Methods in Applied Mechanics and Engineering, 427 (2024), pp. 117037/1--117037/16, DOI 10.1016/j.cma.2024.117037 .
Abstract
We discuss how slip conditions for the Stokes equation can be handled using Nitsche method, for a stabilized finite element discretization. Emphasis is made on the interplay between stabilization and Nitsche terms. Well-posedness of the discrete problem and optimal convergence rates, in natural norm for the velocity and the pressure, are established, and illustrated with various numerical experiments. The proposed method fits naturally in the context of a finite element implementation while being accurate, and allows an increased flexibility in the choice of the finite element pairs.

R. Araya, C. Cárcamo, A.H. Poza, E. Vino, An adaptive stabilized finite element method for the Stokes--Darcy coupled problem, Journal of Computational and Applied Mathematics, 443 (2024), pp. 115753/1--115753/24, DOI 10.1016/j.cam.2024.115753 .
Abstract
For the Stokes--Darcy coupled problem, which models a fluid that flows from a free medium into a porous medium, we introduce and analyze an adaptive stabilized finite element method using Lagrange equal order element to approximate the velocity and pressure of the fluid. The interface conditions between the free medium and the porous medium are given by mass conservation, the balance of normal forces, and the Beavers--Joseph--Saffman conditions. We prove the well-posedness of the discrete problem and present a convergence analysis with optimal error estimates in natural norms. Next, we introduce and analyze a residual-based a posteriori error estimator for the stabilized scheme. Finally, we present numerical examples to demonstrate the performance and effectiveness of our scheme.

G.R. Barrenechea, V. John, P. Knobloch, Finite element methods respecting the discrete maximum principle for convection-diffusion equations, SIAM Review, 66 (2024), pp. 1--86, DOI 10.1137/22M1488934 .
Abstract
Convection-diffusion-reaction equations model the conservation of scalar quantities. From the analytic point of view, solutions of these equations satisfy, under certain conditions, maximum principles, which represent physical bounds of the solution. That the same bounds are respected by numerical approximations of the solution is often of utmost importance in practice. The mathematical formulation of this property, which contributes to the physical consistency of a method, is called the discrete maximum principle (DMP). In many applications, convection dominates diffusion by several orders of magnitude. It is well known that standard discretizations typically do not satisfy the DMP in this convection-dominated regime. In fact, in this case it turns out to be a challenging problem to construct discretizations that, on the one hand, respect the DMP and, on the other hand, compute accurate solutions. This paper presents a survey on finite element methods, with the main focus on the convection-dominated regime, that satisfy a local or a global DMP. The concepts of the underlying numerical analysis are discussed. The survey reveals that for the steady-state problem there are only a few discretizations, all of them nonlinear, that at the same time both satisfy the DMP and compute reasonably accurate solutions, e.g., algebraically stabilized schemes. Moreover, most of these discretizations have been developed in recent years, showing the enormous progress that has been achieved lately. Similarly, methods based on algebraic stabilization, both nonlinear and linear, are currently the only finite element methods that combine the satisfaction of the global DMP and accurate numerical results for the evolutionary equations in the convection-dominated scenario.

B. García-Archilla , V. John, S. Katz, J. Novo, POD-ROMs for incompressible flows including snapshots of the temporal derivative of the full order solution: Error bounds for the pressure, Journal of Numerical Mathematics, 32 (2024), pp. 301--329, DOI 10.1515/jnma-2023-0039 .
Abstract
Reduced order methods (ROMs) for the incompressible Navier?Stokes equations, based on proper orthogonal decomposition (POD), are studied that include snapshots which approach the temporal derivative of the velocity from a full order mixed finite element method (FOM). In addition, the set of snapshots contains the mean velocity of the FOM. Both the FOM and the POD-ROM are equipped with a grad-div stabilization. A velocity error analysis for this method can be found already in the literature. The present paper studies two different procedures to compute approximations to the pressure and proves error bounds for the pressure that are independent of inverse powers of the viscosity. Numerical studies support the analytic results and compare both methods.

T. Gutleb, S. Olver, I. Papadopoulos, R. Slevinsky, Building hierarchies of semiclassical Jacobi polynomials for spectral methods in annuli, SIAM Journal on Scientific Computing, 46 (2024), pp. A3448--A3476, DOI 10.1137/23M160846X .

W. Lei, S. Piani, P. Farrell, N. Rotundo, L. Heltai, A weighted hybridizable discontinuous Galerkin method for drift-diffusion problems, Journal of Scientific Computing, 99 (2024), pp. 33/1--33/26, DOI 10.1007/s10915-024-02481-w .
Abstract
In this work we propose a weighted hybridizable discontinuous Galerkin method (W-HDG) for drift-diffusion problems. By using specific exponential weights when computing the L2 product in each cell of the discretization, we are able to mimic the behavior of the Slotboom variables, and eliminate the drift term from the local matrix contributions, while still solving the problem for the primal variables. We show that the proposed numerical scheme is well-posed, and validate numerically that it has the same properties as classical HDG methods, including optimal convergence, and superconvergence of postprocessed solutions. For polynomial degree zero, dimension one, and vanishing HDG stabilization parameter, W-HDG coincides with the Scharfetter--Gummel finite volume scheme (i.e., it produces the same system matrix). The use of local exponential weights generalizes the Scharfetter-Gummel scheme (the state-of-the-art for finite volume discretization of transport dominated problems) to arbitrary high order approximations.

H. Liang, H. Rui, A parameter robust reconstruction nonconforming virtual element method for the incompressible poroelasticity model, Applied Numerical Mathematics. An IMACS Journal, 202 (2024), pp. 127--142, DOI 10.1016/j.apnum.2024.05.001 .
Abstract
A reconstruction nonconforming virtual element method for the incompressible poroelasticity model is developed and analyzed. We investigate to determine the divergence-free displacement in incompressible poroelasticity model on polygons. The presented method mainly involves the parameter robust for mu also can be considered as the real pressure robustness for the virtual element. Using the space on the polygons to build the reconstruction operator, the method can be applied on general polygonal meshes, satisfy pressure robustness and overcome Poisson locking in incompressible model. The results are corroborated by theoretical derivations as well as numerical results.

S. Piani, P. Farrell, W. Lei, N. Rotundo, L. Heltai, Data-driven solutions of ill-posed inverse problems arising from doping reconstruction in semiconductors, Applied Mathematics in Science and Engineering, 32 (2024), pp. 2323626/1--2323626/27, DOI 10.1080/27690911.2024.2323626 .
Abstract
The non-destructive estimation of doping concentrations in semiconductor devices is of paramount importance for many applications ranging from crystal growth, the recent redefinition of the 1kg to defect, and inhomogeneity detection. A number of technologies (such as LBIC, EBIC and LPS) have been developed which allow the detection of doping variations via photovoltaic effects. The idea is to illuminate the sample at several positions and detect the resulting voltage drop or current at the contacts. We model a general class of such photovoltaic technologies by ill-posed global and local inverse problems based on a drift-diffusion system that describes charge transport in a self-consistent electrical field. The doping profile is included as a parametric field. To numerically solve a physically relevant local inverse problem, we present three different data-driven approaches, based on least squares, multilayer perceptrons, and residual neural networks. Our data-driven methods reconstruct the doping profile for a given spatially varying voltage signal induced by a laser scan along the sample's surface. The methods are trained on synthetic data sets (pairs of discrete doping profiles and corresponding photovoltage signals at different illumination positions) which are generated by efficient physics-preserving finite volume solutions of the forward problem. While the linear least square method yields an average absolute l-infinity / displaystyle ell ^infty error around 10%, the nonlinear networks roughly halve this error to 5%, respectively. Finally, we optimize the relevant hyperparameters and test the robustness of our approach with respect to noise.

M. Demir, V. John, Pressure-robust approximation of the incompressible Navier--Stokes equations in a rotating frame of reference, BIT. Numerical Mathematics, 64 (2024), pp. 36/1--36/19, DOI 10.1007/s10543-024-01037-6 .
Abstract
A pressure-robust space discretization of the incompressible Navier--Stokes equations in a rotating frame of reference is considered. The discretization employs divergence-free, H1-conforming mixed finite element methods like Scott--Vogelius pairs. An error estimate for the velocity is derived that tracks the dependency of the error bound on the coefficients of the problem, in particular on the angular velocity. Numerical examples illustrate the theoretical results.

V. John, X. Li, Ch. Merdon, H. Rui, Inf-sup stabilized Scott--Vogelius pairs on general simplicial grids by Raviart--Thomas enrichment, Mathematical Models & Methods in Applied Sciences, 34 (2024), pp. 919--949, DOI 10.1142/S0218202524500180 .
Abstract
This paper considers the discretization of the Stokes equations with Scott--Vogelius pairs of finite element spaces on arbitrary shape-regular simplicial grids. A novel way of stabilizing these pairs with respect to the discrete inf-sup condition is proposed and analyzed. The key idea consists in enriching the continuous polynomials of order k of the Scott--Vogelius velocity space with appropriately chosen and explicitly given Raviart--Thomas bubbles. This approach is inspired by [Li/Rui, IMA J. Numer. Anal, 2021], where the case k=1 was studied. The proposed method is pressure-robust, with optimally converging H1-conforming velocity and a small H(div)-conforming correction rendering the full velocity divergence-free. For k>d, with d being the dimension, the method is parameter-free. Furthermore, it is shown that the additional degrees of freedom for the Raviart--Thomas enrichment and also all non-constant pressure degrees of freedom can be condensated, effectively leading to a pressure-robust, inf-sup stable, optimally convergent Pk×P0 scheme. Aspects of the implementation are discussed and numerical studies confirm the analytic results.

V. John, X. Li, Ch. Merdon, Pressure-robust L2 ($Omega$) error analysis for Raviart--Thomas enriched Scott--Vogelius pairs, Applied Mathematics Letters, 156 (2024), pp. 109138/1--109138/12, DOI 10.1016/j.aml.2024.109138 .
Abstract
Recent work shows that it is possible to enrich the Scott--Vogelius finite element pair by cer- tain Raviart--Thomas functions to obtain an inf-sup stable and divergence-free method on general shape-regular meshes. A skew-symmetric consistency term was suggested for avoiding an ad- ditional stabilization term for higher order elements, but no L² (Ω) error estimate was shown for the Stokes equations. This note closes this gap. In addition, the optimal choice of the stabilization parameter is studied numerically.

V. John, Ch. Merdon, M. Zainelabdeen, Augmenting the grad-div stabilization for Taylor--Hood finite elements with a vorticity stabilization, Journal of Numerical Mathematics, published online in Nov. 2024, DOI 10.1515/jnma-2023-0118 .
Abstract
The least squares vorticity stabilization (LSVS), proposed in Ahmed et al. for the Scott--Vogelius finite element discretization of the Oseen equations, is studied as an augmentation of the popular grad-div stabilized Taylor--Hood pair of spaces. An error analysis is presented which exploits the situation that the velocity spaces of Scott--Vogelius and Taylor--Hood are identical. Convection-robust error bounds are derived under the assumption that the Scott--Vogelius discretization is well posed on the considered grid. Numerical studies support the analytic results and they show that the LSVS-grad-div method might lead to notable error reductions compared with the standard grad-div method.

I. Papadopoulos, Numerical analysis of the SIMP model for the topology optimization problem of minimizing compliance in linear elasticity, Numerische Mathematik, (2024), DOI 10.1007/s00211-024-01438-3 .

D. Abdel, N.E. Courtier, P. Farrell, Volume exclusion effects in perovskite charge transport modeling, Optical and Quantum Electronics, 55 (2023), pp. 884/1--884/14, DOI 10.1007/s11082-023-05125-9 .
Abstract
Due to their flexible material properties, perovskite materials are a promising candidate for many semiconductor devices such as lasers, memristors, LEDs and solar cells. For example, perovskite-based solar cells have recently become one of the fastest growing photovoltaic technologies. Unfortunately, perovskite devices are far from commercialization due to challenges such as fast degradation. Mathematical models can be used as tools to explain the behavior of such devices, for example drift-diffusion equations portray the ionic and electric motion in perovskites. In this work, we take volume exclusion effects on ion migration within a perovskite crystal lattice into account. This results in the formulation of two different ionic current densities for such a drift-diffusion model -- treating either the mobility or the diffusivity as density-dependent while the other quantity remains constant. The influence of incorporating each current density description into a model for a typical perovskite solar cell configuration is investigated numerically, through simulations performed using two different open source tools.

D. Abdel, C. Chainais-Hillairet, P. Farrell, M. Herda, Numerical analysis of a finite volume scheme for charge transport in perovskite solar cells, IMA Journal of Numerical Analysis, published online on 10.6.2023, DOI 10.1093/imanum/drad034 .
Abstract
In this paper, we consider a drift-diffusion charge transport model for perovskite solar cells, where electrons and holes may diffuse linearly (Boltzmann approximation) or nonlinearly (e.g. due to Fermi-Dirac statistics). To incorporate volume exclusion effects, we rely on the Fermi-Dirac integral of order −1 when modeling moving anionic vacancies within the perovskite layer which is sandwiched between electron and hole transport layers. After non-dimensionalization, we first prove a continuous entropy-dissipation inequality for the model. Then, we formulate a corresponding two-point flux finite volume scheme on Voronoi meshes and show an analogous discrete entropy-dissipation inequality. This inequality helps us to show the existence of a discrete solution of the nonlinear discrete system with the help of a corollary of Brouwer's fixed point theorem and the minimization of a convex functional. Finally, we verify our theoretically proven properties numerically, simulate a realistic device setup and show exponential decay in time with respect to the L² error as well as a physically and analytically meaningful relative entropy.

S. Katz, A. Caiazzo, V. John, Impact of viscosity modeling on the simulation of aortic blood flow, Journal of Computational and Applied Mathematics, 425 (2023), pp. 115036/1--115036/18, DOI 10.1016/j.cam.2022.115036 .
Abstract
Modeling issues for the simulation of blood flow in an aortic coarctation are studied in this paper. From the physical point of view, several viscosity models for non-Newtonian fluids as well as a Newtonian fluid model will be considered. From the numerical point of view, two different turbulence models are utilized in the simulations. The impact of both, the physical and the numerical modeling, on clinically relevant biomarkers is investigated and compared.

S. Katz, A. Caiazzo, B. Moreau, U. Wilbrandt, J. Brüning, L. Goubergrits, V. John, Impact of turbulence modeling on the simulation of blood flow in aortic coarctation, International Journal of Numerical Methods in Biomedical Engineering, 39 (2023), pp. e3695/1--e3695/36, DOI 10.1002/cnm.3695 .
Abstract
Numerical simulations of pulsatile blood flow in an aortic coarctation require the use of turbulence modeling. This paper considers three models from the class of large eddy simulation (LES) models (Smagorinsky, Vreman, -model) and one model from the class of variational multiscale models (residual-based) within a finite element framework. The influence of these models on the estimation of clinically relevant biomarkers used to assess the degree of severity of the pathological condition (pressure difference, secondary flow degree, normalized flow displacement, wall shear stress) is investigated in detail. The simulations show that most methods are consistent in terms of severity indicators such as pressure difference and stenotic velocity. Moreover, using second-order velocity finite elements, different turbulence models might lead to considerably different results concerning other clinically relevant quantities such as wall shear stresses. These differences may be attributed to differences in numerical dissipation introduced by the turbulence models.

L. Schmeller, D. Peschka, Gradient flows for coupling order parameters and mechanics, SIAM Journal on Applied Mathematics, 83 (2023), pp. 225--253, DOI 10.1137/22M148478X .
Abstract
We construct a formal gradient flow structure for phase-field evolution coupled to mechanics in Lagrangian coordinates, present common ways to couple the evolution and provide an incremental minimization strategy. While the usual presentation of continuum mechanics is intentionally very brief, the focus of this paper is on an extensible functional analytical framework and a discretization approach that preserves an appropriate variational structure as much as possible. As examples, we first present phase separation and swelling of gels and then the approach of stationary states of multiphase systems with surface tension and show the robustness of the general approach.

A. Alphonse, D. Caetano, A. Djurdjevac, Ch.M. Elliot, Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs, Journal of Differential Equations, 353 (2023), pp. 268-338, DOI 10.1016/j.jde.2022.12.032 .
Abstract
We develop a functional framework suitable for the treatment of partial differential equations and variational problems on evolving families of Banach spaces. We propose a definition for the weak time derivative that does not rely on the availability of a Hilbertian structure and explore conditions under which spaces of weakly differentiable functions (with values in an evolving Banach space) relate to classical Sobolev?Bochner spaces. An Aubin?Lions compactness result is proved. We analyse concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we explicitly provide the definition of the time derivative and verify isomorphism properties with the aforementioned Sobolev?Bochner spaces. We conclude with the proof of well posedness for a class of nonlinear monotone problems on an abstract evolving space (generalising the evolutionary p-Laplace equation on a moving domain or surface) and identify some additional problems that can be formulated with the setting developed in this work.

D. Peschka, Partial and complete wetting of thin films with dynamic contact angle, Physics of Fluids, 35 (2023), pp. 041705/1--041705/6, DOI 10.1063/5.0146538 .
Abstract
The wetting of thin films depends critically on the sign of the spreading coefficient S. We discuss the cases S<0, S=0, and S>0 for transient models with contact line dissipation and find that the use of a dynamic contact angle solves problems for S>0 that models might otherwise have. For initial data with a non-zero slope and S>0, we show that there exists a finite time at which the contact angle of the thin film goes to zero. Then, a molecular precursor emerges from the thin film and moves outward at a constant velocity.

A.K. Barua, R. Chew, L. Shuwang, J. Lowengrub, A. Münch, B. Wagner, Sharp-interface problem of the Ohta--Kawasaki model for symmetric diblock copolymers, Journal of Computational Physics, 481 (2023), pp. 112032/1--112032/23, DOI 10.1016/j.jcp.2023.112032 .
Abstract
The Ohta-Kawasaki model for diblock-copolymers is well known to the scientific community of diffuse-interface methods. To accurately capture the long-time evolution of the moving interfaces, we present a derivation of the corresponding sharp-interface limit using matched asymptotic expansions, and show that the limiting process leads to a Hele-Shaw type moving interface problem. The numerical treatment of the sharp-interface limit is more complicated due to the stiffness of the equations. To address this problem, we present a boundary integral formulation corresponding to a sharp interface limit of the Ohta-Kawasaki model. Starting with the governing equations defined on separate phase domains, we develop boundary integral equations valid for multi-connected domains in a 2D plane. For numerical simplicity we assume our problem is driven by a uniform Dirichlet condition on a circular far-field boundary. The integral formulation of the problem involves both double- and single-layer potentials due to the modified boundary condition. In particular, our formulation allows one to compute the nonlinear dynamics of a non-equilibrium system and pattern formation of an equilibrating system. Numerical tests on an evolving slightly perturbed circular interface (separating the two phases) are in excellent agreement with the linear analysis, demonstrating that the method is stable, efficient and spectrally accurate in space.

R. Shiri, L. Schmeller, R. Seemann, D. Peschka, B. Wagner, Impact of noise on spinodal dewetting of liquid-liquid films, communications physics, 6 (2023), pp. 109/1--109/11, DOI 10.1038/s42005-023-01208-x .
Abstract
We investigate the spinodal dewetting of a thin liquid polystyrene (PS) film on a liquid polymethylmethacrylate (PMMA) subtrate. Following the evolution of the corrugations of the PS film via in situ measurements by atomic force microscopy (AFM) and those of the PS-PMMA interface via ex situ imaging, we provide a direct and detailed comparison of the experimentally determined spinodal wavelengths with the predictions from linear stability analysis of a thin-film continuum model for the bilayer system. The impact of rough interfaces and fluctuations is studied theoretically by investigating the impact of different choices of initial data on the unstable wavelength and on the rupture time. The key factor is the mode selection by initial data perturbed with correlated colored noise in the linearly unstable regime, which becomes relevant only for liquid bilayers to such an extent. By numerically solving the mathematical model, we further address the impact of nonlinear effects on rupture times and on the morphological evolution of the interfaces in comparison with experimental results.

G.L. Celora, M.G. Hennessy, A. Münch, B. Wagner, S.L. Waters, The dynamics of a collapsing polyelectrolyte gel, SIAM Journal on Applied Mathematics, 83 (2023), pp. 1146--1171, DOI 10.1137/21M1419726 .

C. Chainais-Hillairet, R. Eymard, J. Fuhrmann, A monotone numerical flux for quasilinear convection diffusion equations, Mathematics of Computation, 93 (2024), pp. 203--231 (published online in June 2023), DOI 10.1090/mcom/3870 .

R. Araya, C. Cárcamo, A.H. Poza, A stabilized finite element method for the Stokes--Temperature coupled problem, Applied Numerical Mathematics. An IMACS Journal, 187 (2023), pp. 24--49, DOI 10.1016/j.apnum.2023.02.002 .
Abstract
In this work, we introduce and analyze a new stabilized finite element scheme for the Stokes--Temperature coupled problem. This new scheme allows equal order of interpolation to approximate the quantities of interest, i.e. velocity, pressure, temperature, and stress. We analyze an equivalent variational formulation of the coupled problem inspired by the ideas proposed in [3]. The existence of the discrete solution is proved, decoupling the proposed stabilized scheme and using the help of continuous dependence results and Brouwer's theorem under the standard assumption of sufficiently small data. Optimal convergence is proved under classic regularity assumptions of the solution. Finally, we present some numerical examples to show the quality of our scheme, in particular, we compare our results with those coming from a standard reference in geosciences described in [38].

D. Budáč, V. Miloš, M. Carda, M. Paidar, J. Fuhrmann, K. Bouzek, Prediction of electrical conductivity of porous composites using a simplified Monte Carlo 3D equivalent electronic circuit network model: LSM--YSZ case study, Electrochimica Acta, 457 (2023), pp. 142512/1--142512/12, DOI 10.1016/j.electacta.2023.142512 .
Abstract
Multiphase electric charge conductors composed of materials with various properties are widely utilized in both research and industrial applications. The composite materials include porous electrodes and other components mainly applied in fuel cell and battery technologies. In this study, a simplified Monte Carlo equivalent electronic circuit (EEC) network model is presented. In comparison to similar models, the present EEC network model allows an accurate prediction of the electrical properties of such materials, thus saving time-consuming experimental determination. The distinct feature of this EEC network model is that it requires only experimentally easily obtainable data as the input parameters: phase composition, porosity and bulk electrical conductivity of the individual constituents. During its run, the model generates a large number of artificial cubically shaped specimens based on random distribution of individual phases according to the input composition. Each of the specimens generated was modelled by a corresponding EEC network. The EEC networks were solved using Kirchhoff's laws, resulting in impedance response simulation for the prediction of composite conductivity values. The EEC network model was validated using lanthanum strontium manganite mixed with yttria-stabilized zirconia. Excellent agreement was obtained between the experimentally determined and the calculated electrical conductivity for sample porosities of 0 to 60 %. Due to its variability, the EEC network model can be suitable for a wide range of practical applications. The presented approach has high potential to save an enormous amount of experimental effort, while maintaining sufficient accuracy, when designing corresponding multiphase electrode structures.

K. Fellner, J. Fischer, M. Kniely, B.Q. Tang, Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion, Journal of Nonlinear Science, 33 (2023), pp. 66/1--66/49, DOI 10.1007/s00332-023-09926-w .
Abstract
The global existence of renormalised solutions and convergence to equilibrium for reaction-diffusion systems with non-linear diffusion are investigated. The system is assumed to have quasi-positive non-linearities and to satisfy an entropy inequality. The difficulties in establishing global renormalised solutions caused by possibly degenerate diffusion are overcome by introducing a new class of weighted truncation functions. By means of the obtained global renormalised solutions, we study the large-time behaviour of complex balanced systems arising from chemical reaction network theory with non-linear diffusion. When the reaction network does not admit boundary equilibria, the complex balanced equilibrium is shown, by using the entropy method, to exponentially attract all renormalised solutions in the same compatibility class. This convergence extends even to a range of non-linear diffusion, where global existence is an open problem, yet we are able to show that solutions to approximate systems converge exponentially to equilibrium uniformly in the regularisation parameter.

R. Finn, M. O'Donovan, P. Farrell, J. Moatti, T. Streckenbach, Th. Koprucki, S. Schulz, Theoretical study of the impact of alloy disorder on carrier transport and recombination processes in deep UV (Al,Ga)N light emitters, Applied Physics Letters, 122 (2023), pp. 241104/1--241104/7, DOI 10.1063/5.0148168 .
Abstract
Aluminum gallium nitride [(Al,Ga)N] has gained significant attention in recent years due to its potential for highly efficient light emitters operating in the deep ultra-violet (UV) range (<280 nm). However, given that current devices exhibit extremely low efficiencies, understand- ing the fundamental properties of (Al,Ga)N-based systems is of key importance. Here, using a multi-scale simulation framework, we study the impact of alloy disorder on carrier transport, radiative and non-radiative recombination processes in a c-plane Al 0.7 Ga0.3 N/Al0.8 Ga0.2 N quantum well embedded in a p-n junction. Our calculations reveal that alloy fluctuations can open "percolative" pathways that promote transport for the electrons and holes into the quantum well region. Such an effect is neglected in conventional and widely used transport sim- ulations. Moreover, we find that the resulting increased carrier density and alloy induced carrier localization effects significantly increase non-radiative Auger--Meitner recombination in comparison to the radiative process. Thus, to suppress such non-radiative process and poten- tially related material degradation, a careful design (wider well, multi-quantum wells) of the active region is required to improve the effi- ciency of deep UV light emitters.

B. García-Archilla, V. John, J. Novo, Second order error bounds for POD-ROM methods based on first order divided differences, Applied Mathematics Letters, 146 (2023), pp. 108836/1--108836/7, DOI 10.1016/j.aml.2023.108836 .
Abstract
This note proves for the heat equation that using BDF2 as time stepping scheme in POD-ROM methods with snapshots based on difference quotients gives both the optimal second order error bound in time and pointwise estimates.

B. García-Archilla, V. John, J. Novo, POD-ROMs for incompressible flows including snapshots of the temporal derivative of the full order solution, SIAM Journal on Numerical Analysis, 61 (2023), pp. 1340--1368, DOI 10.1137/22M1503853 .
Abstract
In this paper we study the influence of including snapshots that approach the velocity time derivative in the numerical approximation of the incompressible Navier--Stokes equations by means of proper orthogonal decomposition (POD) methods. Our set of snapshots includes the velocity approximation at the initial time from a full order mixed finite element method (FOM) together with approximations to the time derivative at different times. The approximation at the initial velocity can be replaced by the mean value of the velocities at the different times so that when implementing the method to the fluctuations, as done mostly in practice, only approximations to the time derivatives are included in the set of snapshots. For the POD method we study the differences between projecting onto L2 and H1. In both cases pointwise in time error bounds can be proved. Including grad-div stabilization in both the FOM and the POD methods, error bounds with constants independent of inverse powers of the viscosity can be obtained.

L. Giacomelli, M. Gnann, D. Peschka, Droplet motion with contact-line friction: Long-time asymptotics in complete wetting, Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences, 479 (2023), pp. 20230090/1--20230090/23, DOI 10.1098/rspa.2023.0090 .
Abstract
We consider the thin-film equation for a class of free boundary conditions modelling friction at the contact line, as introduced by E and Ren. Our analysis focuses on formal long-time asymptotics of solutions in the perfect wetting regime. In particular, through the analysis of quasi-self-similar solutions, we characterize the profile and the spreading rate of solutions depending on the strength of friction at the contact line, as well as their (global or local) corrections, which are due to the dynamical nature of the free boundary conditions. These results are complemented with full transient numerical solutions of the free boundary problem.

G. Gilardi, A. Signori, J. Sprekels, Nutrient control for a viscous Cahn--Hilliard--Keller--Segel model with logistic source describing tumor growth, Discrete and Continuous Dynamical Systems -- Series S, 16 (2023), pp. 3552--3572, DOI 10.3934/dcdss.2023123 .
Abstract
In this paper, we address a distributed control problem for a system of partial differential equations describing the evolution of a tumor that takes the biological mechanism of chemotaxis into account. The system describing the evolution is obtained as a nontrivial combination of a Cahn--Hilliard type system accounting for the segregation between tumor cells and healthy cells, with a Keller--Segel type equation accounting for the evolution of a nutrient species and modeling the chemotaxis phenomenon. First, we develop a robust mathematical background that allows us to analyze an associated optimal control problem. This analysis forced us to select a source term of logistic type in the nutrient equation and to restrict the analysis to the case of two space dimensions. Then, the existence of an optimal control and first-order necessary conditions for optimality are established.

A. Jha, V. John, P. Knobloch, Adaptive grids in the context of algebraic stabilizations for convection-diffusion-reaction equations, SIAM Journal on Scientific Computing, 45 (2023), pp. B564--B589, DOI 10.1137/21M1466360 .
Abstract
Three algebraically stabilized finite element schemes for discretizing convection-diffusion-reaction equations are studied on adaptively refined grids. These schemes are the algebraicflux correction (AFC) scheme with the Kuzmin limiter, the AFC scheme with the Barrenechea-John-Knobloch limiter, and the recently proposed monotone upwind--type algebraically stabilizedmethod. Both conforming closure of the refined grids and grids with hanging vertices are considered.A nonstandard algorithmic step becomes necessary before these schemes can be applied on gridswith hanging vertices. The assessment of the schemes is performed with respect to the satisfactionof the global discrete maximum principle, the accuracy, e.g., smearing of layers, and the efficiency insolving the corresponding nonlinear problems.

A. Jha, O. Pártl, N. Ahmed, D. Kuzmin, An assessment of solvers for algebraically stabilized discretizations of convection-diffusion-reaction equations, Journal of Numerical Mathematics, 31 (2023), pp. 79--103, DOI 10.1515/jnma-2021-0123 .
Abstract
We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include flux-corrected transport schemes and monolithic limiters. We discretize in space using a continuous Galerkin method and P₁ or Q₁ finite elements. Time integration is performed using the Crank-Nicolson method or an explicit strong stability preserving Runge-Kutta method. Nonlinear systems are solved using a fixed-point iteration method, which requires solution of large linear systems at each iteration or time step. The great variety of options in the choice of discretization methods and solver components calls for a dedicated comparative study of existing approaches. To perform such a study, we define new 3D test problems for time dependent and stationary convection-diffusion-reaction equations. The results of our numerical experiments illustrate how the limiting technique, time discretization and solver impact on the overall performance.

B. Spetzler, D. Abdel, F. Schwierz, M. Ziegler, P. Farrell, The role of vacancy dynamics in two-dimensional memristive devices, Advanced Electronic Materials, published online on 08.11.2023, DOI 10.1002/aelm.202300635 .
Abstract
Two-dimensional (2D) layered transition metal dichalcogenides (TMDCs) are promising memristive materials for neuromorphic computing systems as they could solve the problem of the excessively high energy consumption of conventional von Neumann computer architectures. Despite extensive experimental work, the underlying switching mechanisms are still not understood, impeding progress in material and device functionality. This study reveals the dominant role of mobile defects in the switching dynamics of 2D TMDC materials. The switching process is governed by the formation and annihilation dynamics of a local vacancy depletion zone. Moreover, minor changes in the interface potential barriers cause fundamentally different device behavior previously thought to originate from multiple mechanisms. The key mechanisms are identified with a charge transport model for electrons, holes, and ionic point defects, including image-charge-induced Schottky barrier lowering (SBL). The model is validated by comparing simulations to measurements for various 2D MoS2-based devices, strongly corroborating the relevance of vacancies in TMDC devices and offering a new perspective on the switching mechanisms. The insights gained from this study can be used to extend the functional behavior of 2D TMDC memristive devices in future neuromorphic computing applications.

P. Farrell, J. Moatti, M. O'Donovan, S. Schulz, Th. Koprucki, Importance of satisfying thermodynamic consistency in optoelectronic device simulations for high carrier densities, Optical and Quantum Electronics, 55 (2023), pp. 978/1--978/12, DOI 10.1007/s11082-023-05234-5 .
Abstract
We show the importance of using a thermodynamically consistent flux discretization when describing drift-diffusion processes within light emitting diode simulations. Using the classical Scharfetter--Gummel scheme with Fermi--Dirac statistics is an example of such an inconsistent scheme. In this case, for an (In,Ga)N multi quantum well device, the Fermi levels show steep gradients on one side of the quantum wells which are not to be expected. This result originates from neglecting diffusion enhancement associated with Fermi--Dirac statistics in the numerical flux approximation. For a thermodynamically consistent scheme, such as the SEDAN scheme, the spikes in the Fermi levels disappear. We will show that thermodynamic inconsistency has far reaching implications on the current-voltage curves and recombination rates.

D. Frerichs-Mihov, L. Henning, V. John, Using deep neural networks for detecting spurious oscillations in discontinuous Galerkin solutions of convection-dominated convection-diffusion equations, Journal of Scientific Computing, 97 (2023), pp. 36/1--36/27, DOI 10.1007/s10915-023-02335-x .
Abstract
Standard discontinuous Galerkin (DG) finite element solutions to convection-dominated convection-diffusion equations usually possess sharp layers but also exhibit large spurious oscillations. Slope limiters are known as a post-processing technique to reduce these unphysical values. This paper studies the application of deep neural networks for detecting mesh cells on which slope limiters should be applied. The networks are trained with data obtained from simulations of a standard benchmark problem with linear finite elements. It is investigated how they perform when applied to discrete solutions obtained with higher order finite elements and to solutions for a different benchmark problem.

V. John, P. Knobloch, U. Wilbrandt, A posteriori optimization of parameters in stabilized methods for convection-diffusion problems -- Part II, Journal of Computational and Applied Mathematics, 428 (2023), pp. 115167/1--115167/17, DOI 10.1016/j.cam.2023.115167 .
Abstract
Extensions of algorithms for computing optimal stabilization parameters in finite element methods for convection-diffusion equations are presented. These extensions reduce the dimension of the control space, in comparison to available methods, and thus address the long computing times of these methods. One method is proposed that considers only relevant mesh cells, another method that uses groups of mesh cells, and the combination of both methods is also studied. The incorporation of these methods within a gradient-based optimization procedure, via solving an adjoint problem, is explained. Numerical studies provide impressions on the gain of efficiency as well as on the loss of accuracy if control spaces with reduced dimensions are utilized.

CH. Merdon, W. Wollner, Pressure-robustness in the context of optimal control, SIAM Journal on Control and Optimization, 61 (2023), pp. 342--360, DOI 10.1137/22M1482603 .
Abstract
This paper studies the benefits of pressure-robust discretizations in the scope of optimal control of incompressible flows. Gradient forces that may appear in the data can have a negative impact on the accuracy of state and control and can only be correctly balanced if their L²-orthogonality onto discretely divergence-free test functions is restored. Perfectly orthogonal divergence-free discretizations or divergence-free reconstructions of these test functions do the trick and lead to much better analytic a priori estimates that are also validated in numerical examples.

A.H. Erhardt, S. Solem, Bifurcation analysis of a modified cardiac cell model, SIAM Journal on Applied Dynamical Systems, 21 (2022), pp. 231--247, DOI 10.1137/21M1425359 .

F. Galarce Marín, D. Lombardi, O. Mula, State estimation with model reduction and shape variability: Application to biomedical problems, SIAM Journal on Scientific Computing, 44 (2022), pp. B805--B833, DOI 10.1137/21M1430480 .
Abstract
We develop a mathematical and numerical framework to solve state estimation problems for applications that present variations in the shape of the spatial domain. This situation arises typically in a biomedical context where inverse problems are posed on certain organs or portions of the body which inevitably involve morphological variations. If one wants to provide fast reconstruction methods, the algorithms must take into account the geometric variability. We develop and analyze a method which allows to take this variability into account without needing any a priori knowledge on a parametrization of the geometrical variations. For this, we rely on morphometric techniques involving Multidimensional Scaling, and couple them with reconstruction algorithms that make use of reduced model spaces pre-computed on a database of geometries. We prove the potential of the method on a synthetic test problem inspired from the reconstruction of blood flows and quantities of medical interest with Doppler ultrasound imaging.

D. Peschka, L. Heltai, Model hierarchies and higher-order discretisation of time-dependent thin-film free boundary problems with dynamic contact angle, Journal of Computational Physics, 464 (2022), pp. 111325/1--111325/22, DOI 10.1016/j.jcp.2022.111325 .
Abstract
We present a mathematical and numerical framework for the physical problem of thin-film fluid flows over planar surfaces including dynamic contact angles. In particular, we provide algorithmic details and an implementation of higher-order spatial and temporal discretisation of the underlying free boundary problem using the finite element method. The corresponding partial differential equation is based on a thermodynamic consistent energetic variational formulation of the problem using the free energy and viscous dissipation in the bulk, on the surface, and at the moving contact line. Model hierarchies for limits of strong and weak contact line dissipation are established, implemented and studied. We analyze the performance of the numerical algorithm and investigate the impact of the dynamic contact angle on the evolution of two benchmark problems: gravity-driven sliding droplets and the instability of a ridge.

P. Vágner, M. Pavelka, J. Fuhrmann, V. Klika, A multiscale thermodynamic generalization of Maxwell--Stefan diffusion equations and of the dusty gas model, International Journal of Heat and Mass Transfer, 199 (2022), pp. 123405/1--123405/14, DOI 10.1016/j.ijheatmasstransfer.2022.123405 .
Abstract
Despite the fact that the theory of mixtures has been part of non-equilibrium thermodynamics and engineering for a long time, it is far from complete. While it is well formulated and tested in the case of mechanical equilibrium (where only diffusion-like processes take place), the question how to properly describe homogeneous mixtures that flow with multiple independent velocities that still possess some inertia (before mechanical equilibrium is reached) is still open. Moreover, the mixtures can have several temperatures before they relax to a common value. In this paper, we derive a theory of mixtures from Hamiltonian mechanics in interaction with electromagnetic fields. The resulting evolution equations are then reduced to the case with only one momentum (classical irreversible thermodynamics), providing a generalization of the Maxwell-Stefan diffusion equations. In a next step, we reduce that description to the mechanical equilibrium (no momentum) and derive a non-isothermal variant of the dusty gas model. These reduced equations are solved numerically, and we illustrate the results on effciency analysis, showing where in a concentration cell effciency is lost. Finally, the theory of mixtures identifies the temperature difference between constituents as a possible new source of the Soret coeffcient. For the sake of clarity, we restrict the presentation to the case of binary mixtures; the generalization is straightforward.

A. Zafferi, D. Peschka, M. Thomas, GENERIC framework for reactive fluid flows, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 103 (2023), pp. e202100254/1--e202100254/70 (published online on 09.05.2022), DOI 10.1002/zamm.202100254 .
Abstract
We describe reactive fluid flows in terms of the formalism General Equation for Non-Equilibrium Reversible-Irreversible Coupling also known as GENERIC. Together with the formalism, we present the thermodynamical and mechanical foundations for the treatment of fluid flows using continuous fields and present a clear relation and transformation between a Lagrangian and an Eulerian formulation of the corresponding systems of partial differential equations. We bring the abstract framework to life by providing many physically relevant examples for reactive compressive fluid flows.

G.L. Celora, M.G. Hennessy, A. Münch, B. Wagner, S.L. Waters, A kinetic model of a polyelectrolyte gel undergoing phase separation, Journal of the Mechanics and Physics of Solids, 160 (2022), pp. 104771/1--104771/27, DOI 10.1016/j.jmps.2021.104771 .
Abstract
In this study we use non-equilibrium thermodynamics to systematically derive a phase-field model of a polyelectrolyte gel coupled to a thermodynamically consistent model for the salt solution surrounding the gel. The governing equations for the gel account for the free energy of the internal interfaces which form upon phase separation, as well as finite elasticity and multi-component transport. The fully time-dependent model describes the evolution of small changes in the mobile ion concentrations and follows their impact on the large-scale solvent flux and the emergence of long-time pattern formation in the gel. We observe a strong acceleration of the evolution of the free surface when the volume phase transition sets in, as well as the triggering of spinodal decomposition that leads to strong inhomogeneities in the lateral stresses, potentially leading to experimentally visible patterns.

D. Bothe, W. Dreyer, P.-É. Druet, Multicomponent incompressible fluids -- An asymptotic study, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, published online on 14.01.2022, DOI 10.1002/zamm.202100174 .
Abstract
This paper investigates the asymptotic behavior of the Helmholtz free energy of mixtures at small compressibility. We start from a general representation for the local free energy that is valid in stable subregions of the phase diagram. On the basis of this representation we classify the admissible data to construct a thermodynamically consistent constitutive model. We then analyze the incompressible limit, where the molar volume becomes independent of pressure. Here we are confronted with two problems:

(i) Our study shows that the physical system at hand cannot remain incompressible for arbitrary large deviations from a reference pressure unless its volume is linear in the composition. (ii) As a consequence of the 2nd law of thermodynamics, the incompressible limit implies that the molar volume becomes independent of temperature as well. Most applications, however, reveal the non-appropriateness of this property. According to our mathematical treatment, the free energy as a function of temperature and partial masses tends to a limit in the sense of epi-- or Gamma--convergence. In the context of the first problem, we study the mixing of two fluids to compare the linearity with experimental observations. The second problem will be treated by considering the asymptotic behavior of both a general inequality relating thermal expansion and compressibility and a PDE-system relying on the equations of balance for partial masses, momentum and the internal energy.

J. Fischer, K. Hopf, M. Kniely, A. Mielke, Global existence analysis of energy-reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 54 (2022), pp. 220--267, DOI 10.1137/20M1387237 .
Abstract
We establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear.

B. Gaudeul, J. Fuhrmann, Entropy and convergence analysis for two finite volume schemes for a Nernst--Planck--Poisson system with ion volume constraints, Numerische Mathematik, 151 (2022), pp. 99--149, DOI 10.1007/s00211-022-01279-y .
Abstract
In this paper, we consider a drift-diffusion system with cross-coupling through the chemical potentials comprising a model for the motion of finite size ions in liquid electrolytes. The drift term is due to the self-consistent electric field maintained by the ions and described by a Poisson equation. We design two finite volume schemes based on different formulations of the fluxes. We also provide a stability analysis of these schemes and an existence result for the corresponding discrete solutions. A convergence proof is proposed for non-degenerate solutions. Numerical experiments show the behavior of these schemes.

V. Miloš, P. Vágner, D. Budáč, M. Carda, M. Paidar, J. Fuhrmann, K. Bouzek, Generalized Poisson--Nernst--Planck-based physical model of the O$_2$ I LSM I YSZ electrode, Journal of The Electrochemical Society, 169 (2022), pp. 044505/1--044505/17, DOI 10.1149/1945-7111/ac4a51 .
Abstract
The paper presents a generalized Poisson--Nernst--Planck model of an yttria-stabilized zirconia electrolyte developed from first principles of nonequilibrium thermodynamics which allows for spatial resolution of the space charge layer. It takes into account limitations in oxide ion concentrations due to the limited availability of oxygen vacancies. The electrolyte model is coupled with a reaction kinetic model describing the triple phase boundary with electron conducting lanthanum strontium manganite and gaseous phase oxygen. By comparing the outcome of numerical simulations based on different formulations of the kinetic equations with results of EIS and CV measurements we attempt to discern the existence of separate surface lattice sites for oxygen adatoms and O^2- from the assumption of shared ones. Furthermore, we discern mass-action kinetics models from exponential kinetics models.

M. O'Donovan, P. Farrell, T. Streckenbach, Th. Koprucki, S. Schulz, Multiscale simulations of uni-polar hole transport in (In,Ga)N quantum well systems, Optical and Quantum Electronics, 54 (2022), pp. 405/1--405/23, DOI 10.1007/s11082-022-03752-2 .
Abstract
Understanding the impact of the alloy micro-structure on carrier transport becomes important when designing III-nitride-based LED structures. In this work, we study the impact of alloy fluctuations on the hole carrier transport in (In,Ga)N single and multi-quantum well systems. To disentangle hole transport from electron transport and carrier recombination processes, we focus our attention on uni-polar (p-i-p) systems. The calculations employ our recently established multi-scale simulation framework that connects atomistic tight-binding theory with a macroscale drift-diffusion model. In addition to alloy fluctuations, we pay special attention to the impact of quantum corrections on hole transport. Our calculations indicate that results from a virtual crystal approximation present an upper limit for the hole transport in a p-i-p structure in terms of the current-voltage characteristics. Thus we find that alloy fluctuations can have a detrimental effect on hole transport in (In,Ga)N quantum well systems, in contrast to uni-polar electron transport. However, our studies also reveal that the magnitude by which the random alloy results deviate from virtual crystal approximation data depends on several factors, e.g. how quantum corrections are treated in the transport calculations.

B. Takács, Y. Hadjimichael, High order discretization methods for spatial-dependent epidemic models, Mathematics and Computers in Simulation, 198 (2022), pp. 211--236, DOI 10.1016/j.matcom.2022.02.021 .
Abstract
In this paper, an SIR model with spatial dependence is studied and results regarding its stability and numerical approximation are presented. We consider a generalization of the original Kermack and McKendrick model in which the size of the populations differs in space. The use of local spatial dependence yields a system of integro-differential equations. The uniqueness and qualitative properties of the continuous model are analyzed. Furthermore, different choices of spatial and temporal discretizations are employed, and step-size restrictions for population conservation, positivity, and monotonicity preservation of the discrete model are investigated. We provide sufficient conditions under which high order numerical schemes preserve the discrete properties of the model. Computational experiments verify the convergence and accuracy of the numerical methods.

A.F.M. TER Elst, A. Linke, J. Rehberg, On the numerical range of sectorial forms, Pure and Applied Functional Analysis, 7 (2022), pp. 1931--1940.
Abstract
We provide a sharp and optimal generic bound for the angle of the sectorial form associated to a non-symmetric second-order elliptic differential operator with various boundary conditions. Consequently this gives an, in general, sharper H^∞-angle for the H^∞-calculus on L_p for all p ∈ (1, ∞) if the coefficients are real valued.

A. Glitzky, M. Liero, G. Nika, A coarse-grained electrothermal model for organic semiconductor devices, Mathematical Methods in the Applied Sciences, 45 (2022), pp. 4809--4833, DOI 10.1002/mma.8072 .
Abstract
We derive a coarse-grained model for the electrothermal interaction of organic semiconductors. The model combines stationary drift-diffusion based electrothermal models with thermistor type models on subregions of the device and suitable transmission conditions. Moreover, we prove existence of a solution using a regularization argument and Schauder's fixed point theorem. In doing so, we extend recent work by taking into account the statistical relation given by the Gauss--Fermi integral and mobility functions depending on the temperature, charge-carrier density, and field strength, which is required for a proper description of organic devices.

K. Hopf, Weak-strong uniqueness for energy-reaction-diffusion systems, Mathematical Models & Methods in Applied Sciences, 21 (2022), pp. 1015--1069, DOI 10.1142/S0218202522500233 .
Abstract
We establish weak-strong uniqueness and stability properties of renormalised solutions to a class of energy-reaction-diffusion systems, which genuinely feature cross-diffusion effects. The systems considered are motivated by thermodynamically consistent models, and their formal entropy structure allows us to use as a key tool a suitably adjusted relative entropy method. Weak-strong uniqueness is obtained for general entropy-dissipating reactions without growth restrictions, and certain models with a non-integrable diffusive flux. The results also apply to a class of (isoenergetic) reaction-cross-diffusion systems.

A. Alphonse, C.N. Rautenberg, J.F. Rodrigues, Analysis of a quasi-variational contact problem arising in thermoelasticity, Nonlinear Analysis. An International Mathematical Journal, 217 (2022), pp. 112728/1--112728/40, DOI 10.1016/j.na.2021.112728 .
Abstract
We formulate and study two mathematical models of a thermoforming process involving a membrane and a mould as implicit obstacle problems. In particular, the membrane-mould coupling is determined by the thermal displacement of the mould that depends in turn on the membrane through the contact region. The two models considered are a stationary (or elliptic) model and an evolutionary (or quasistatic) one. For the first model, we prove the existence of weak solutions by solving an elliptic quasi-variational inequality coupled to elliptic equations. By exploring the fine properties of the variation of the contact set under non-degenerate data, we give sufficient conditions for the existence of regular solutions, and under certain contraction conditions, also a uniqueness result. We apply these results to a series of semi-discretised problems that arise as approximations of regular solutions for the evolutionary or quasistatic problem. Here, under certain conditions, we are able to prove existence for the evolutionary problem and for a special case, also the uniqueness of time-dependent solutions.

P.-É. Druet, Maximal mixed parabolic-hyperbolic regularity for the full equations of multicomponent fluid dynamics, Nonlinearity, 35 (2022), pp. 3812--3882, DOI 10.1088/1361-6544/ac5679 .
Abstract
We consider a Navier--Stokes--Fick--Onsager--Fourier system of PDEs describing mass, energy and momentum balance in a Newtonian fluid with composite molecular structure. For the resulting parabolic-hyperbolic system, we introduce the notion of optimal regularity of mixed type, and we prove the short-time existence of strong solutions for a typical initial boundary-value-problem. By means of a partial maximum principle, we moreover show that such a solution cannot degenerate in finite time due to blow-up or vanishing of the temperature or the partial mass densities. This second result is however only valid under certain growth conditions on the phenomenological coefficients. In order to obtain some illustration of the theory, we set up a special constitutive model for volume-additive mixtures.

D. Frerichs-Mihov, V. John, On a technique for reducing spurious oscillations in DG solutions of convection-diffusion equations, Applied Mathematics Letters, 129 (2022), pp. 107969/1--107969/7, DOI 10.1016/j.aml.2022.107969 .
Abstract
This note studies a generalization of a post-processing technique and a novel method inspired by the same technique which significantly reduce spurious oscillations in discontinuous Galerkin solutions of convection-diffusion equations in the convection-dominated regime.

V. John, P. Knobloch, O. Pártl, A numerical assessment of finite element discretizations for convection-diffusion-reaction equations satisfying discrete maximum principles, Computational Methods in Applied Mathematics, 23 (2023), pp. 969--988 (published online on 30.09.2022), DOI 10.1515/cmam-2022-0125 .
Abstract
Numerical studies are presented that investigate finite element methods satisfying discrete maximum principles for convection-diffusion-reaction equations. Two linear methods and several nonlinear schemes, some of them proposed only recently, are included in these studies, which consider a number of two-dimensional examples. The evaluation of the results examines the accuracy of the numerical solutions with respect to quantities of interest, like layer widths, and the efficiency of the simulations.

V. John, P. Knobloch, On algebraically stabilized schemes for convection-diffusion-reaction problems, Numerische Mathematik, 152 (2022), pp. 553--585, DOI 10.1007/s00211-022-01325-9 .

V. John, B. Moreau, J. Novo, Error analysis of a SUPG-stabilized POD-ROM method for convection-diffusion-reaction equations, Computers & Mathematics with Applications. An International Journal, 122 (2022), pp. 48--60, DOI 10.1016/j.camwa.2022.07.017 .
Abstract
A reduced order model (ROM) method based on proper orthogonal decomposition (POD) is analyzed for convection-diffusion-reaction equations. The streamline-upwind Petrov--Galerkin (SUPG) stabilization is used in the practically interesting case of dominant convection, both for the full order method (FOM) and the ROM simulations. The asymptotic choice of the stabilization parameter for the SUPG-ROM is done as proposed in the literature. This paper presents a finite element convergence analysis of the SUPG-ROM method for errors in different norms. The constants in the error bounds are uniform with respect to small diffusion coefficients. Numerical studies illustrate the performance of the SUPG-ROM method.

M. Landstorfer, R. Müller, Thermodynamic models for a concentration and electric field dependent susceptibility in liquid electrolytes, Electrochimica Acta, 428 (2022), pp. 140368/1--140368/19, DOI 10.1016/j.electacta.2022.140368 .
Abstract
The dielectric susceptibility $chi$ is an elementary quantity of the electrochemical double layer and the associated Poisson equation. While most often $chi$ is treated as a material constant, its dependency on the salt concentration in liquid electrolytes is demonstrated by various bulk electrolyte experiments. This is usually referred to as dielectric decrement. Further, it is theoretically well accepted that the susceptibility declines for large electric fields. This effect is frequently termed dielectric saturation. We analyze the impact of a variable susceptibility in terms of species concentrations and electric fields based on non-equilibrium thermodynamics. This reveals some non-obvious generalizations compared to the case of a constant susceptibility. In particular the consistent coupling of the Poisson equation, the momentum balance and the chemical potentials functions are of ultimate importance. In a numerical study, we systematically analyze the effects of a concentration and field dependent susceptibility on the double layer of a planar electrode electrolyte interface. We compute the differential capacitance and the spatial structure of the electric potential, solvent concentration and ionic distribution for various non-constant models of $chi$.

M. Landstorfer, M. Ohlberger, S. Rave, M. Tacke, A modelling framework for efficient reduced order simulations of parametrised lithium-ion battery cells, European Journal of Applied Mathematics, 34 (2023), pp. 554--591 (published online on 29.11.2022), DOI 10.1017/S0956792522000353 .
Abstract
In this contribution we present a new modeling and simulation framework for parametrized Lithium-ion battery cells. We first derive a new continuum model for a rather general intercalation battery cell on the basis of non-equilibrium thermodynamics. In order to efficiently evaluate the resulting parameterized non-linear system of partial differential equations the reduced basis method is employed. The reduced basis method is a model order reduction technique on the basis of an incremental hierarchical approximate proper orthogonal decomposition approach and empirical operator interpolation. The modeling framework is particularly well suited to investigate and quantify degradation effects of battery cells. Several numerical experiments are given to demonstrate the scope and efficiency of the modeling framework.

A. Mielke, J. Naumann, On the existence of global-in-time weak solutions and scaling laws for Kolmogorov's two-equation model of turbulence, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 102 (2022), pp. e202000019/1--e202000019/31, DOI 10.1002/zamm.202000019 .
Abstract
This paper is concerned with Kolmogorov's two-equation model for free turbulence in space dimension 3, involving the mean velocity u, the pressure p, an average frequency omega, and a mean turbulent kinetic energy k. We first discuss scaling laws for a slightly more general two-equation models to highlight the special role of the model devised by Kolmogorov in 1942. The main part of the paper consists in proving the existence of weak solutions of Kolmogorov's two-equation model under space-periodic boundary conditions in cubes with positive side length l. To this end, we provide new a priori estimates and invoke existence result for pseudo-monotone operators.

D. Abdel, P. Vágner, J. Fuhrmann, P. Farrell, Modelling charge transport in perovskite solar cells: Potential-based and limiting ion depletion, Electrochimica Acta, 390 (2021), pp. 138696/1--138696/12, DOI 10.1016/j.electacta.2021.138696 .
Abstract
From Maxwell--Stefan diffusion and general electrostatics, we derive a drift-diffusion model for charge transport in perovskite solar cells (PSCs) where any ion in the perovskite layer may flexibly be chosen to be mobile or immobile. Unlike other models in the literature, our model is based on quasi Fermi potentials instead of densities. This allows to easily include nonlinear diffusion (based on Fermi--Dirac, Gauss--Fermi or Blakemore statistics for example) as well as limit the ion depletion (via the Fermi--Dirac integral of order-1). The latter will be motivated by a grand-canonical formalism of ideal lattice gas. Furthermore, our model allows to use different statistics for different species. We discuss the thermodynamic equilibrium, electroneutrality as well as generation/recombination. Finally, we present numerical finite volume simulations to underline the importance of limiting ion depletion.

D. Abdel, P. Farrell, J. Fuhrmann, Assessing the quality of the excess chemical potential flux scheme for degenerate semiconductor device simulation, Optical and Quantum Electronics, 53 (2021), pp. 163/1--163/10, DOI 10.1007/s11082-021-02803-4 .
Abstract
The van Roosbroeck system models current flows in (non-)degenerate semiconductor devices. Focusing on the stationary model, we compare the excess chemical potential discretization scheme, a flux approximation which is based on a modification of the drift term in the current densities, with another state-of-the-art Scharfetter-Gummel scheme, namely the diffusion-enhanced scheme. Physically, the diffusion-enhanced scheme can be interpreted as a flux approximation which modifies the thermal voltage. As a reference solution we consider an implicitly defined integral flux, using Blakemore statistics. The integral flux refers to the exact solution of a local two point boundary value problem for the continuous current density and can be interpreted as a generalized Scharfetter-Gummel scheme. All numerical discretization schemes can be used within a Voronoi finite volume method to simulate charge transport in (non-)degenerate semiconductor devices. The investigation includes the analysis of Taylor expansions, a derivation of error estimates and a visualization of errors in local flux approximations to extend previous discussions. Additionally, drift-diffusion simulations of a p-i-n device are performed.

P.-É. Druet, Global-in-time existence for liquid mixtures subject to a generalised incompressibility constraint, Journal of Mathematical Analysis and Applications, 499 (2021), pp. 125059/1--125059/56, DOI 10.1016/j.jmaa.2021.125059 .
Abstract
We consider a system of partial differential equations describing diffusive and convective mass transport in a fluid mixture of N > 1 chemical species. A weighted sum of the partial mass densities of the chemical species is assumed to be constant, which expresses the incompressibility of the fluid, while accounting for different reference sizes of the involved molecules. This condition is different from the usual assumption of a constant total mass density, and it leads in particular to a non-solenoidal velocity field in the Navier-Stokes equations. In turn, the pressure gradient occurs in the diffusion fluxes, so that the PDE-system of mass transport equations and momentum balance is fully coupled. Another striking feature of such incompressible mixtures is the algebraic formula connecting the pressure and the densities, which can be exploited to prove a pressure bound in L¹. In this paper, we consider incompressible initial states with bounded energy and show the global existence of weak solutions with defect measure.

M. Heida, M. Kantner, A. Stephan, Consistency and convergence for a family of finite volume discretizations of the Fokker--Planck operator, ESAIM: Mathematical Modelling and Numerical Analysis, 55 (2021), pp. 3017--3042, DOI 10.1051/m2an/2021078 .
Abstract
We introduce a family of various finite volume discretization schemes for the Fokker--Planck operator, which are characterized by different weight functions on the edges. This family particularly includes the well-established Scharfetter--Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the quality of the weights. We show by numerical experiments that for small gradients the choice of the optimal representative of the discretization family is highly non-trivial while for large gradients the Scharfetter--Gummel scheme stands out compared to the others.

S. Kayser, P. Farrell, N. Rotundo, Detecting striations via the lateral photovoltage scanning method without screening effect, Optical and Quantum Electronics, 53 (2021), pp. 288/1--288/10, DOI 10.1007/s11082-021-02911-1 .
Abstract
The lateral photovoltage scanning method (LPS) detects doping inhomogeneities in semiconductors such as Si, Ge and Si(x)Ge(1-x) in a cheap, fast and nondestructive manner. LPS relies on the bulk photovoltaic effect and thus can detect any physical quantity affecting the band profiles of the sample. LPS finite volume simulation using commercial software suffer from long simulation times and convergence instabilities. We present here an open-source finite volume simulation for a 2D Si sample using the ddfermi simulator. For low injection conditions we show that the LPS voltage is proportional to the doping gradient as previous theory suggested under certain conditions. For higher injection conditions we directly show how the LPS voltage and the doping gradient differ and link the physical effect of lower local resolution to the screening effect. Previously, the loss of local resolution was assumed to be only connected to the enlargement of the excess charge carrier distribution.

O. Marquardt, Simulating the electronic properties of semiconductor nanostructures using multiband $kcdot p$ models, Computational Materials Science, 194 (2021), pp. 110318/1--110318/11, DOI 10.1016/j.commatsci.2021.110318 .
Abstract
The eight-band $kcdot p$ formalism been successfully applied to compute the electronic properties of a wide range of semiconductor nanostructures in the past and can be considered the backbone of modern semiconductor heterostructure modelling. However, emerging novel material systems and heterostructure fabrication techniques raise questions that cannot be answered using this well-established formalism, due to its intrinsic limitations. The present article reviews recent studies on the calculation of electronic properties of semiconductor nanostructures using a generalized multiband $kcdot p$ approach that allows both the application of the eight-band model as well as more sophisticated approaches for novel material systems and heterostructures.

G. Nika, Derivation of effective models from heterogenous Cosserat media via periodic unfolding, Ricerche di Matematica. A Journal of Pure and Applied Mathematics, published online on 01.07.2021, DOI 10.1007/s11587-021-00610-3 .
Abstract
We derive two different effective models from a heterogeneous Cosserat continuum taking into account the Cosserat intrinsic length of the constituents. We pass to the limit using homogenization via periodic unfolding and in doing so we provide rigorous proof to the results introduced by Forest, Pradel, and Sab (Int. J. Solids Structures 38 (26-27): 4585-4608 '01). Depending on how different characteristic lengths of the domain scale with respect to the Cosserat intrinsic length, we obtain either an effective classical Cauchy continuum or an effective Cosserat continuum. Moreover, we provide some corrector type results for each case.

A. Stephan, EDP-convergence for a linear reaction-diffusion system with fast reversible reaction, Calculus of Variations and Partial Differential Equations, 60 (2021), pp. 226/1--226/35, DOI 10.1007/s00526-021-02089-0 .
Abstract
We perform a fast-reaction limit for a linear reaction-diffusion system consisting of two diffusion equations coupled by a linear reaction. We understand the linear reaction-diffusion system as a gradient flow of the free energy in the space of probability measures equipped with a geometric structure, which contains the Wasserstein metric for the diffusion part and cosh-type functions for the reaction part. The fast-reaction limit is done on the level of the gradient structure by proving EDP-convergence with tilting. The limit gradient system induces a diffusion system with Lagrange multipliers on the linear slow-manifold. Moreover, the limit gradient system can be equivalently described by a coarse-grained gradient system, which induces a diffusion equation with a mixed diffusion constant for the coarse-grained slow variable.

L. Lilaj, H. Harthum, T. Meyer, M. Shahrayari, G. Bertalan, A. Caiazzo, J. Braun, Th. Fischer, S. Hirsch, I. Sack, Inversion-recovery MR elastography of the human brain for improved stiffness quantification near fluid-solid boundaries, Magnetic Resonance in Medicine, (2021), published online on 28.06.2021, DOI 10.1002/mrm.28898 .

N. Ahmed, G.R. Barrenechea, E. Burman, J. Guzmán, A. Linke, Ch. Merdon, A pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation, SIAM Journal on Numerical Analysis, 59 (2021), pp. 2746--2774, DOI 10.1137/20M1351230 .
Abstract
Discretization of Navier--Stokes' equations using pressure-robust finite element methods is considered for the high Reynolds number regime. To counter oscillations due to dominating convection we add a stabilization based on a bulk term in the form of a residual-based least squares stabilization of the vorticity equation supplemented by a penalty term on (certain components of) the gradient jump over the elements faces. Since the stabilization is based on the vorticity equation, it is independent of the pressure gradients, which makes it pressure-robust. Thus, we prove pressureindependent error estimates in the linearized case, known as Oseen's problem. In fact, we prove an O(h^k+¹/2) error estimate in the L²-norm that is known to be the best that can be expected for this type of problem. Numerical examples are provided that, in addition to confirming the theoretical results, show that the present method compares favorably to the classical residual-based SUPG stabilization.

TH. Apel, V. Kempf, A. Linke, Ch. Merdon, A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes, IMA Journal of Numerical Analysis, 42 (2022), pp. 392--416 (published online on 14.01.2021), DOI 10.1093/imanum/draa097 .
Abstract
Most classical finite element schemes for the (Navier--)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors, whenever the Stokes momentum balance is dominated by a strong and complicated pressure gradient. It is a consequence of a method, which does not exactly satisfy the divergence constraint. However, inf-sup stable schemes can often be made pressure-robust just by a recent, modified discretization of the exterior forcing term, using H(div)-conforming velocity reconstruction operators. This approach has so far only been analyzed on shape-regular triangulations. The novelty of the present contribution is that the reconstruction approach for the Crouzeix--Raviart method, which has a stable Fortin operator on arbitrary meshes, is combined with results on the interpolation error on anisotropic elements for reconstruction operators of Raviart--Thomas and Brezzi--Douglas--Marini type, generalizing the method to a large class of anisotropic triangulations. Numerical examples confirm the theoretical results in a 2D and a 3D test case.

L. Baňas, R. Lasarzik, A. Prohl, Numerical analysis for nematic electrolytes, IMA Journal of Numerical Analysis, 41 (2021), pp. 2186--2254, DOI 10.1093/imanum/draa082 .
Abstract
We consider a system of nonlinear PDEs modeling nematic electrolytes, and construct a dissipative solution with the help of its implementable, structure-inheriting space-time discretization. Computational studies are performed to study the mutual effects of electric, elastic, and viscous effects onto the molecules in a nematic electrolyte.

D. Bothe, P.-É. Druet, Mass transport in multicomponent compressible fluids: Local and global well-posedness in classes of strong solutions for general class-one models, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 210 (2021), pp. 112389/1--112389/53, DOI 10.1016/j.na.2021.112389 .
Abstract
We consider a system of partial differential equations describing mass transport in a multicomponent isothermal compressible fluid. The diffusion fluxes obey the Fick-Onsager or Maxwell- Stefan closure approach. Mechanical forces result into one single convective mixture velocity, the barycentric one, which obeys the Navier-Stokes equations. The thermodynamic pressure is defined by the Gibbs-Duhem equation. Chemical potentials and pressure are derived from a thermodynamic potential, the Helmholtz free energy, with a bulk density allowed to be a general convex function of the mass densities of the constituents. The resulting PDEs are of mixed parabolic-hyperbolic type. We prove two theoretical results concerning the well-posedness of the model in classes of strong solutions: 1. The solution always exists and is unique for short-times and 2. If the initial data are sufficiently near to an equilibrium solution, the well-posedness is valid on arbitrary large, but finite time intervals. Both results rely on a contraction principle valid for systems of mixed type that behave like the compressible Navier- Stokes equations. The linearised parabolic part of the operator possesses the self map property with respect to some closed ball in the state space, while being contractive in a lower order norm only. In this paper, we implement these ideas by means of precise a priori estimates in spaces of exact regularity.

D. Bothe, P.-É. Druet, Well-posedness analysis of multicomponent incompressible flow models, Journal of Evolution Equations, 21 (2021), pp. 4039--4093, DOI 10.1007/s00028-021-00712-3 .
Abstract
In this paper, we extend our study of mass transport in multicomponent isothermal fluids to the incompressible case. For a mixture, incompressibility is defined as the independence of average volume on pressure, and a weighted sum of the partial mass densities of the species stays constant. In this type of models, non solenoidal effects affect the velocity field in the Navier--Stokes equations and, due to different specific volumes of the species, the pressure remains connected to the densities by algebraic formula. By means of a change of variables in the transport problem, we equivalently reformulate the PDE system as to eliminate positivity and incompressibility constraints affecting the density, and prove two type of results: the local-in-time well-posedness in classes of strong solutions, and the global-in-time existence of solutions for initial data sufficiently close to a smooth equilibrium solution.

D. Chaudhuri, M. O'Donovan, T. Streckenbach, O. Marquardt, P. Farrell, S.K. Patra, Th. Koprucki, S. Schulz, Multiscale simulations of the electronic structure of III-nitride quantum wells with varied indium content: Connecting atomistic and continuum-based models, Journal of Applied Physics, 129 (2021), pp. 073104/1--073104/16, DOI 10.1063/5.0031514 .

O.C. Ernst, D. Uebel, S. Kayser, F. Lange, Th. Teubner, T. Boeck, Revealing all states of dewetting of a thin gold layer on a silicon surface by nanosecond laser conditioning, Applied Surface Science Advances, 3 (2021), pp. 100040/1--100040/10, DOI 10.1016/j.apsadv.2020.100040 .
Abstract
Dewetting is a ubiquitous phenomenon which can be applied to the laser synthesis of nanoparticles. A classical spinodal dewetting process takes place in four successive states, which differ from each other in their morphology. In this study all states are revealed by interaction of pulsed nanosecond UV laser light with thin gold layers with thicknesses between 1 nm and 10 nm on (100) silicon wafers. The specific morphologies of the dewetting states are discussed with particular emphasis on the state boundaries. The main parameter determining which state is formed is not the duration for which the gold remains liquid, but rather the input energy provided by the laser. This shows that each state transition has a separate measurable activation energy. The temperature during the nanosecond pulses and the duration during which the gold remains liquid was determined by simulation using the COMSOL Multiphysics software package. Using these calculations, an accurate local temperature profile and its development over time was simulated. An analytical study of the morphologies and formed structures was performed using Minkowski measures. With aid of this tool, the laser induced structures were compared with thermally annealed samples, with perfectly ordered structures and with perfectly random structures. The results show that both, structures of the laser induced and the annealed samples, strongly resemble the perfectly ordered structures. This reveals a close relationship between these structures and suggests that the phenomenon under investigation is indeed a spinodal dewetting generated by an internal material wave function.
The purposeful generation of these structures and the elucidation of the underlying mechanism of dewetting by ultrashort pulse lasers may assist the realisation of various technical elements such as nanowires in science and industry.

G. Fu, Ch. Lehrenfeld, A. Linke, T. Streckenbach, Locking free and gradient robust H(div)-conforming HDG methods for linear elasticity, Journal of Scientific Computing, 86 (2021), DOI 10.1007/s10915-020-01396-6 .
Abstract
Robust discretization methods for (nearly-incompressible) linear elasticity are free of volume-locking and gradient-robust. While volume-locking is a well-known problem that can be dealt with in many different discretization approaches, the concept of gradient-robustness for linear elasticity is new. We discuss both aspects and propose novel Hybrid Discontinuous Galerkin (HDG) methods for linear elasticity. The starting point for these methods is a divergence-conforming discretization. As a consequence of its well-behaved Stokes limit the method is gradient-robust and free of volume-locking. To improve computational efficiency, we additionally consider discretizations with relaxed divergence-conformity and a modification which re-enables gradient-robustness, yielding a robust and quasi-optimal discretization also in the sense of HDG superconvergence.

B. García-Archilla, V. John, J. Novo, On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows, Computer Methods in Applied Mechanics and Engineering, 385 (2021), pp. 114032/1--114032/54, DOI 10.1016/j.cma.2021.114032 .

L. Heltai, A. Caiazzo, L.O. Müller, Multiscale coupling of one-dimensional vascular models and elastic tissues, Annals of Biomedical Engineering (ABME), published online on 20.07.2021, DOI 10.1007/s10439-021-02804-0 .
Abstract
We present a computational multiscale model for the efficient simulation of vascularized tissues, composed of an elastic three-dimensional matrix and a vascular network. The effect of blood vessel pressure on the elastic tissue is surrogated via hyper-singular forcing terms in the elasticity equations, which depend on the fluid pressure. In turn, the blood flow in vessels is treated as a one-dimensional network. The pressure and velocity of the blood in the vessels are simulated using a high-order finite volume scheme, while the elasticity equations for the tissue are solved using a finite element method. This work addresses the feasibility and the potential of the proposed coupled multiscale model. In particular, we assess whether the multiscale model is able to reproduce the tissue response at the effective scale (of the order of millimeters) while modeling the vasculature at the microscale. We validate the multiscale method against a full scale (three-dimensional) model, where the fluid/tissue interface is fully discretized and treated as a Neumann boundary for the elasticity equation. Next, we present simulation results obtained with the proposed approach in a realistic scenario, demonstrating that the method can robustly and efficiently handle the one-way coupling between complex fluid microstructures and the elastic matrix.

P.L. Lederer, Ch. Merdon, Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations, Journal of Numerical Mathematics, published online on 6.11.2021, DOI https://doi.org/10.1515/jnma-2021-0078 .
Abstract
This paper improves guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the errors of divergence-free primal and H(div)-conforming dual mixed methods (for the velocity gradient) with an equilibration constraint that needs special care when discretised. To relax the constraints on the primal and dual method, a more general result is derived that enables the use of a recently developed mass conserving mixed stress discretisation to design equilibrated fluxes that yield pressure-independent guaranteed upper bounds for any pressure-robust (but not necessarily divergence-free) primal discretisation. Moreover, a provably efficient local design of the equilibrated fluxes is presented that reduces the numerical costs of the error estimator. All theoretical findings are verified by numerical examples which also show that the efficiency indices of our novel guaranteed upper bounds for the velocity error are close to 1.

M. O'Donovan, D. Chaudhuri, T. Streckenbach, P. Farrell, S. Schulz, Th. Koprucki, From atomistic tight-binding theory to macroscale drift-diffusion: Multiscale modeling and numerical simulation of uni-polar charge transport in (In,Ga)N devices with random fluctuations, Journal of Applied Physics, 130 (2021), pp. 065702/1--065702/13, DOI 10.1063/5.0059014 .

M. Thomas, S. Tornquist, Discrete approximation of dynamic phase-field fracture in visco-elastic materials, Discrete and Continuous Dynamical Systems -- Series S, 14 (2021), pp. 3865--3924, DOI 10.3934/dcdss.2021067 .
Abstract
This contribution deals with the analysis of models for phase-field fracture in visco-elastic materials with dynamic effects. The evolution of damage is handled in two different ways: As a viscous evolution with a quadratic dissipation potential and as a rate-independent law with a positively 1-homogeneous dissipation potential. Both evolution laws encode a non-smooth constraint that ensures the unidirectionality of damage, so that the material cannot heal. Suitable notions of solutions are introduced in both settings. Existence of solutions is obtained using a discrete approximation scheme both in space and time. Based on the convexity properties of the energy functional and on the regularity of the displacements thanks to their viscous evolution, also improved regularity results with respect to time are obtained for the internal variable: It is shown that the damage variable is continuous in time with values in the state space that guarantees finite values of the energy functional.

A. Alphonse, C.N. Rautenberg, J.F. Rodrigues, Analysis of a quasi-variational contact problem arising in thermoelasticity, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 217 (2022), pp. 112728/1--112728/40 (published online on 13.12.2021), DOI 10.1016/j.na.2021.112728 .
Abstract
We formulate and study two mathematical models of a thermoforming process involving a membrane and a mould as implicit obstacle problems. In particular, the membrane-mould coupling is determined by the thermal displacement of the mould that depends in turn on the membrane through the contact region. The two models considered are a stationary (or elliptic) model and an evolutionary (or quasistatic) one. For the first model, we prove the existence of weak solutions by solving an elliptic quasi-variational inequality coupled to elliptic equations. By exploring the fine properties of the variation of the contact set under non-degenerate data, we give sufficient conditions for the existence of regular solutions, and under certain contraction conditions, also a uniqueness result. We apply these results to a series of semi-discretised problems that arise as approximations of regular solutions for the evolutionary or quasistatic problem. Here, under certain conditions, we are able to prove existence for the evolutionary problem and for a special case, also the uniqueness of time-dependent solutions.

P. Farrell, S. Kayser, N. Rotundo, Modeling and simulation of the lateral photovoltage scanning method, Computers & Mathematics with Applications. An International Journal, 102 (2021), pp. 248--260, DOI 10.1016/j.camwa.2021.10.017 .
Abstract
The fast, cheap and nondestructive lateral photovoltage scanning (LPS) method detects inhomogeneities in semiconductors crystals. The goal of this paper is to model and simulate this technique for a given doping profile. Our model is based on the semiconductor device equations combined with a nonlinear boundary condition, modelling a volt meter. To validate our 2D and 3D finite volume simulations, we use theory developed by Tauc [21] to derive three analytical predictions which our simulation results corroborate, even for anisotropic 2D and 3D meshes. Our code runs about two orders of magnitudes faster than earlier implementations based on commercial software [15]. It also performs well for small doping concentrations which previously could not be simulated at all due to numerical instabilities. Our simulations provide experimentalists with reference laser powers for which meaningful voltages can still be measured. For higher laser power the screening effect does not allow this anymore.

D. Frerichs, V. John, On reducing spurious oscillations in discontinuous Galerkin (DG) methods for steady-state convection-diffusion equations, Journal of Computational and Applied Mathematics, 393 (2021), pp. 113487/1--113487/20, DOI 10.1016/j.cam.2021.113487 .

TH. Koprucki, A. Maltsi, A. Mielke, On the Darwin--Howie--Whelan equations for the scattering of fast electrons described by the Schrödinger equation, SIAM Journal on Applied Mathematics, 81 (2021), pp. 1552--1578, DOI 10.1137/21M139164X .
Abstract
The Darwin-Howie-Whelan equations are commonly used to describe and simulate the scattering of fast electrons in transmission electron microscopy. They are a system of infinitely many envelope functions, derived from the Schrödinger equation. However, for the simulation of images only a finite set of envelope functions is used, leading to a system of ordinary differential equations in thickness direction of the specimen. We study the mathematical structure of this system and provide error estimates to evaluate the accuracy of special approximations, like the two-beam and the systematic-row approximation.

A. Maltsi, T. Niermann, T. Streckenbach, K. Tabelow, Th. Koprucki, Numerical simulation of TEM images for In(Ga)As/GaAs quantum dots with various shapes, Optical and Quantum Electronics, 52 (2020), pp. 257/1--257/11, DOI 10.1007/s11082-020-02356-y .
Abstract
We present a mathematical model and a tool chain for the numerical simulation of TEM images of semiconductor quantum dots (QDs). This includes elasticity theory to obtain the strain profile coupled with the Darwin-Howie-Whelan equations, describing the propagation of the electron wave through the sample. We perform a simulation study on indium gallium arsenide QDs with different shapes and compare the resulting TEM images to experimental ones. This tool chain can be applied to generate a database of simulated TEM images, which is a key element of a novel concept for model-based geometry reconstruction of semiconductor QDs, involving machine learning techniques.

W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 71 (2020), pp. 119/1--119/68, DOI 10.1007/s00033-020-01341-5 .
Abstract
We consider an improved Nernst--Planck--Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convection--diffusion--reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross--diffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper we establish the existence of a global--in--time weak solution for the full model, allowing for a general structure of the mobility tensor and for chemical reactions with highly non linear rates in the bulk and on the active boundary. We characterise the singular states of the system, showing that the chemical species can vanish only globally in space, and that this phenomenon must be concentrated in a compact set of measure zero in time. With respect to our former study [DDGG16], we also essentially improve the a priori estimates, in particular concerning the relative chemical potentials.

P.-É. Druet, A. Jüngel, Analysis of cross-diffusion systems for fluid mixtures driven by a pressure gradient, SIAM Journal on Mathematical Analysis, 52 (2020), pp. 2179--2197, DOI 10.1137/19M1301473 .
Abstract
The convective transport in a multicomponent isothermal compressible fluid subject to the mass continuity equations is considered. The velocity is proportional to the negative pressure gradient, according to Darcy?s law, and the pressure is defined by a state equation imposed by the volume extension of the mixture. These model assumptions lead to a parabolic-hyperbolic system for the mass densities. The global-in-time existence of classical and weak solutions is proved in a bounded domain with no-penetration boundary conditions. The idea is to decompose the system into a porous-medium-type equation for the volume extension and transport equations for the modified number fractions. The existence proof is based on parabolic regularity theory, the theory of renormalized solutions, and an approximation of the velocity field.

M. Landstorfer, B. Prifling, V. Schmidt, Mesh generation for periodic 3D microstructure models and computation of effective properties, Journal of Computational Physics, 431 (2021), pp. 110071/1--110071/20 (published online on 23.12.2020), DOI https://doi.org/10.1016/j.jcp.2020.110071 .
Abstract
Understanding and optimizing effective properties of porous functional materials, such as permeability or conductivity, is one of the main goals of materials science research with numerous applications. For this purpose, understanding the underlying 3D microstructure is crucial since it is well known that the materials? morphology has an significant impact on their effective properties. Because tomographic imaging is expensive in time and costs, stochastic microstructure modeling is a valuable tool for virtual materials testing, where a large number of realistic 3D microstructures can be generated and used as geometry input for spatially-resolved numerical simulations. Since the vast majority of numerical simulations is based on solving differential equations, it is essential to have fast and robust methods for generating high-quality volume meshes for the geometrically complex microstructure domains. The present paper introduces a novel method for generating volume-meshes with periodic boundary conditions based on an analytical representation of the 3D microstructure using spherical harmonics. Due to its generality, the present method is applicable to many scientific areas. In particular, we present some numerical examples with applications to battery research by making use of an already existing stochastic 3D microstructure model that has been calibrated to eight differently compacted cathodes.

P. Vágner, M. Pavelka, O. Esen, Multiscale thermodynamics of charged mixtures, Continuum Mechanics and Thermodynamics, published online on 25.07.2020, DOI 10.1007/s00161-020-00900-5 .
Abstract
A multiscale theory of interacting continuum mechanics and thermodynamics of mixtures of fluids, electrodynamics, polarization and magnetization is proposed. The mechanical (reversible) part of the theory is constructed in a purely geometric way by means of semidirect products. This leads to a complex Hamiltonian system with a new Poisson bracket, which can be used in principle with any energy functional. The thermodynamic (irreversible) part is added as gradient dynamics, generated by derivatives of a dissipation potential, which makes the theory part of the GENERIC framework. Subsequently, Dynamic MaxEnt reductions are carried out, which lead to reduced GENERIC models for smaller sets of state variables. Eventually, standard engineering models are recovered as the low-level limits of the detailed theory. The theory is then compared to recent literature.

M. Akbas, Th. Gallouët, A. Gassmann, A. Linke, Ch. Merdon, A gradient-robust well-balanced scheme for the compressible isothermal Stokes problem, Computer Methods in Applied Mechanics and Engineering, 367 (2020), pp. 113069/1--113069/25, DOI 10.1016/j.cma.2020.113069 .
Abstract
A novel notion for constructing a well-balanced scheme --- a gradient-robust scheme --- is introduced and a showcase application for a steady compressible, isothermal Stokes equations is presented. Gradient-robustness means that arbitrary gradient fields in the momentum balance are well-balanced by the discrete pressure gradient --- if there is enough mass in the system to compensate the force. The scheme is asymptotic-preserving in the sense that it degenerates for low Mach numbers to a recent inf-sup stable and pressure-robust discretization for the incompressible Stokes equations. The convergence of the coupled FEM-FVM scheme for the nonlinear, isothermal Stokes equations is proved by compactness arguments. Numerical examples illustrate the numerical analysis, and show that the novel approach can lead to a dramatically increased accuracy in nearly-hydrostatic low Mach number flows. Numerical examples also suggest that a straight-forward extension to barotropic situations with nonlinear equations of state is feasible.

I. Selmer, P. Farrell, I. Smirnova, P. Gurikov, Comparison of finite difference method and finite volume method simulations for a mass transport model describing the supercritical drying kinetic of gel particles in a packed bed, Gels, 6 (2020), pp. 45/1--45/26, DOI 10.3390/gels6040045 .

N. Ahmed, V. John, An assessment of two classes of variational multiscale methods for the simulation of incompressible turbulent flows, Computer Methods in Applied Mechanics and Engineering, 365 (2020), pp. 112997/1--112997/20, DOI 10.1016/j.cma.2020.112997 .
Abstract
A numerical assessment of two classes of variational multiscale (VMS) methods for the simulation of incompressible flows is presented. Two types of residual-based VMS methods and two types of projection-based VMS methods are included in this assessment. The numerical simulations are performed at turbulent channel flow problems with various friction Reynolds numbers. It turns out the the residual-based VMS methods, in particular when used with a pair of inf-sup stable finite elements, give usually the most accurate results for second order statistics. For this pair of finite element spaces, a flexible GMRES method with a Least Squares Commutator (LSC) preconditioner proved to be an efficient solver.

C. Cancès, C. Chainais-Hillairet, J. Fuhrmann, B. Gaudeul, A numerical analysis focused comparison of several finite volume schemes for an unipolar degenerated drift-diffusion model, IMA Journal of Numerical Analysis, 41 (2021), pp. 271--314 (published online on 17.07.2020), DOI 10.1093/imanum/draa002 .
Abstract
In this paper, we consider an unipolar degenerated drift-diffusion system where the relation between the concentration of the charged species c and the chemical potential h is h(c) = log ^c/_1-c. We design four different finite volume schemes based on four different formulations of the fluxes. We provide a stability analysis and existence results for the four schemes. The convergence proof with respect to the discretization parameters is established for two of them. Numerical experiments illustrate the behaviour of the different schemes.

J.A. Carrillo, K. Hopf, M.-Th. Wolfram, Numerical study of Bose--Einstein condensation in the Kaniadakis--Quarati model for bosons, Kinetic and Related Models, 13 (2020), pp. 507--529, DOI 10.3934/krm.2020017 .
Abstract
Kaniadakis and Quarati (1994) proposed a Fokker--Planck equation with quadratic drift as a PDE model for the dynamics of bosons in the spatially homogeneous setting. It is an open question whether this equation has solutions exhibiting condensates in finite time. The main analytical challenge lies in the continuation of exploding solutions beyond their first blow-up time while having a linear diffusion term. We present a thoroughly validated time-implicit numerical scheme capable of simulating solutions for arbitrarily large time, and thus enabling a numerical study of the condensation process in the Kaniadakis--Quarati model. We show strong numerical evidence that above the critical mass rotationally symmetric solutions of the Kaniadakis--Quarati model in 3D form a condensate in finite time and converge in entropy to the unique minimiser of the natural entropy functional at an exponential rate. Our simulations further indicate that the spatial blow-up profile near the origin follows a universal power law and that transient condensates can occur for sufficiently concentrated initial data.

R. Chill, H. Meinlschmidt, J. Rehberg, On the numerical range of second order elliptic operators with mixed boundary conditions in L$^p$, Journal of Evolution Equations, 21 (2021), pp. 3267--3288 (published online on 20.10.2020), DOI 10.1007/s00028-020-00642-6 .
Abstract
We consider second order elliptic operators with real, nonsymmetric coefficient functions which are subject to mixed boundary conditions. The aim of this paper is to provide uniform resolvent estimates for the realizations of these operators on L^p in a most direct way and under minimal regularity assumptions on the domain. This is analogous to the main result in [7]. Ultracontractivity of the associated semigroups is also considered. All results are for two different form domains realizing mixed boundary conditions. We further consider the case of Robin- instead of classical Neumann boundary conditions and also allow for operators inducing dynamic boundary conditions. The results are complemented by an intrinsic characterization of elements of the form domains inducing mixed boundary conditions.

D.H. Doan, A. Fischer, J. Fuhrmann, A. Glitzky, M. Liero, Drift-diffusion simulation of S-shaped current-voltage relations for organic semiconductor devices, Journal of Computational Electronics, 19 (2020), pp. 1164--1174, DOI 10.1007/s10825-020-01505-6 .
Abstract
We present an electrothermal drift-diffusion model for organic semiconductor devices with Gauss-Fermi statistics and positive temperature feedback for the charge carrier mobilities. We apply temperature dependent Ohmic contact boundary conditions for the electrostatic potential and discretize the system by a finite volume based generalized Scharfetter-Gummel scheme. Using path-following techniques we demonstrate that the model exhibits S-shaped current-voltage curves with regions of negative differential resistance, which were only recently observed experimentally.

B. García-Archilla, V. John, J. Novo, Symmetric pressure stabilization for equal-order finite element approximations to the time-dependent Navier--Stokes equations, IMA Journal of Numerical Analysis, 41 (2021), pp. 1093--1129 (published online on 23.06.2020), DOI 10.1093/imanum/draa037 .

M.G. Hennessy, A. Münch, B. Wagner, Phase separation in swelling and deswelling hydrogels with a free boundary, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 101 (2020), pp. 032501/1--032501/14, DOI 10.1103/PhysRevE.101.032501 .
Abstract
We present a full kinetic model of a hydrogel that undergoes phase separation during swelling and deswelling. The model accounts for the interfacial energy of coexisting phases, finite strain of the polymer network, andsolvent transport across free boundaries. For the geometry of an initially dry layer bonded to a rigid substrate,the model predicts that forcing solvent into the gel at a fixed rate can induce a volume phase transition, whichgives rise to coexisting phases with different degrees of swelling, in systems where this cannot occur in the free-swelling case. While a nonzero shear modulus assists in the propagation of the transition front separating thesephases in the driven-swelling case, increasing it beyond a critical threshold suppresses its formation. Quenchinga swollen hydrogel induces spinodal decomposition, which produces several highly localized, highly swollenphases which coarsen and are then ejected from free boundary. The wealth of dynamic scenarios of this systemis discussed using phase-plane analysis and numerical solutions in a one-dimensional setting.

D. Janke, A. Caiazzo, N. Ahmed, N. Alia, O. Knoth, B. Moreau, U. Wilbrandt, D. Willink, Th. Amon, V. John, On the feasibility of using open source solvers for the simulation of a turbulent air flow in a dairy barn, Computers and Electronics in Agriculture, 175 (2020), pp. 105546/1--105546/16, DOI 10.1016/j.compag.2020.105546 .
Abstract
Two transient open source solvers, OpenFOAM and ParMooN, are assessed with respect to the simulation of the turbulent air flow inside and around a dairy barn. For this purpose, data were obtained in an experimental campaign at a 1:100 scaled wind tunnel model. Both solvers used different meshes, discretization schemes, and turbulence models. The experimental data and numerical results agree well for time-averaged stream-wise and vertical-wise velocities. In particular, the air exchange was predicted with high accuracy by both solvers with relative errors less than 5 % compared to the experimental results. With respect to the turbulent quantities, good agreements at the second (downwind) half of the barn inside and especially outside the barn could be achieved, where both codes accurately predicted the flow separation and the root-mean-square velocities. Deviations between simulations and experimental results regarding turbulent quantities could be observed in the first part of the barn, due to different inlet conditions between the experimental setup and the numerical simulations. Both solvers proved to be promising tools for the accurate prediction of time-dependent phenomena in an agricultural context, e.g., like the transport of particulate matter or pathogen-laden aerosols in and around agricultural buildings.

C.K. Macnamara, A. Caiazzo, I. Ramis-Conde, M.A.J. Chaplain, Computational modelling and simulation of cancer growth and migration within a 3D heterogeneous tissue: The effects of fibre and vascular structure, Journal of Computational Science, 40 (2020), pp. 101067/1--101067/24, DOI 10.1016/j.jocs.2019.101067 .
Abstract
The term cancer covers a multitude of bodily diseases, broadly categorised by having cells which do not behave normally. Since cancer cells can arise from any type of cell in the body, cancers can grow in or around any tissue or organ making the disease highly complex. Our research is focused on understanding the specific mechanisms that occur in the tumour microenvironment via mathematical and computational modeling. We present a 3D individual-based model which allows one to simulate the behaviour of, and spatio-temporal interactions between, cells, extracellular matrix fibres and blood vessels. Each agent (a single cell, for example) is fully realised within the model and interactions are primarily governed by mechanical forces between elements. However, as well as the mechanical interactions we also consider chemical interactions, for example, by coupling the code to a finite element solver to model the diffusion of oxygen from blood vessels to cells. The current state of the art of the model allows us to simulate tumour growth around an arbitrary blood-vessel network or along the striations of fibrous tissue.

L.G. Ramos, R. Kehl, R. Nabben, Projections, deflation and multigrid for non-symmetric matrices, SIAM Journal on Matrix Analysis and Applications, 41 (2020), pp. 83--105, DOI 10.1137/18M1180268 .

O. Souček, M. Heida, J. Málek, On a thermodynamic framework for developing boundary conditions for Korteweg-type fluids, International Journal of Engineering Science, 154 (2020), pp. 103316/1--103316/28, DOI 10.1016/j.ijengsci.2020.103316 .
Abstract
We provide a derivation of several classes of boundary conditions for fluids of Korteweg-type using a simple and transparent thermodynamic approach that automatically guarentees that the derived boundary conditions are compatible with the second law of thermodynamics. The starting assumption of our approach is to describe the boundary of the domain as the membrane separating two different continua, one inside the domain, and the other outside the domain. With this viewpoint one may employ the framework of continuum thermodynamics involving singular surfaces. This approach allows us to identify, for various classes of surface Helmholtz free energies, the corresponding surface entropy production mechanisms. By establishing the constitutive relations that guarantee that the surface entropy production is non-negative, we identify a new class of boundary conditions, which on one hand generalizes in a nontrivial manner the Navier's slip boundary conditions, and on the other hand describes dynamic and static contact angle conditions. We explore the general model in detail for a particular case of Korteweg fluid where the Helmholtz free energy in the bulk is that of a van der Waals fluid. We perform a series of numerical experiments to document the basic qualitative features of the novel boundary conditions and their practical applicability to model phenomena such as the contact angle hysteresis.

M. Tlidi, E. Berríos-Caro , D. Pinto-Ramo, A.G. Vladimirov, M.G. Clerc, Interaction between vegetation patches and gaps: A self-organized response to water scarcity, Physica D. Nonlinear Phenomena, 414 (2020), pp. 132708/1--132708/12, DOI 10.1016/j.physd.2020.132708 .
Abstract
The dynamics of ecological systems are often described by integrodifferential equations that incorporate nonlocal interactions associated with facilitative, competitive interactions between plants, and seed dispersion. In the weak-gradient limit, these models can be reduced to a simple partial-differential equation in the form of a nonvariational Swift?Hohenberg equation. In this contribution, we perform this reduction for any type of kernels provided that their Taylor series converge. Some parameters such as linear and nonlinear diffusion coefficients are affected by the spatial form of the kernel. In particular, Gaussian and exponential kernels are used to evaluate all coefficients of the reduced model. This weak gradient approximation is greatly useful for the investigation of periodic and localized vegetation patches, and gaps. Based on this simple model, we investigate the interaction between two-well separated patches and gaps. In the case of patches, the interaction is always repulsive. As a consequence, bounded states of patches are excluded. However, when two gaps are close to one another, they start to interact through their oscillatory tails. The interaction alternates between attractive and repulsive depending on the distance separating them. This allows for the stabilization of bounded gaps and clusters of them. The analytical formula of the interaction potential is derived for both patches and gaps interactions and checked by numerical investigation of the model equation.

D. Uebel, S. Kayser, T. Markurt, O.C. Ernst, Th. Teubner, T. Boeck, Fast Raman mapping and in situ TEM observation of metal induced crystallization of amorphous silicon, CrystEngComm, 22 (2020), pp. 7983--7991, DOI 10.1039/D0CE00960A .
Abstract
Crystalline silicon is grown onto an amorphous silicon (a-Si) seed layer from liquid tin solution (steady state liquid phase epitaxy, SSLPE). To investigate the crystallization of embedded a-Si during our process, we adapted Raman measurements for fast mapping, with dwell times of just one second per single measurement. A purposely developed imaging algorithm which performs point-by-point gauss fitting provides adequate visualization of the data. We produced scans of a-Si layers showing crystalline structures formed in the a-Si matrix during processing. Compared to scanning electron microscopy images which reveal merely the topography of the grown layer, new insights are gained into the role of the seed layer by Raman mapping. As part of a series of SSLPE experiments, which were interrupted at various stages of growth, we show that plate-like crystallites grow laterally over the a-Si layer while smaller, randomly orientated crystals arise from the a-Si layer. Results are confirmed by an in situ TEM experiment of the metal-induced crystallization. Contrary to presumptions, initially formed surface crystallites do not originate from the seed layer and are irrelevant to the final growth morphology, since they dissolve within minutes due to Ostwald ripening. The a-Si layer crystallizes within minutes as well, and crystallites of the final morphology originate from seeds of this layer.

J.-P. Köster, A. Putz, H. Wenzel, H.-J. Wünsche, M. Radziunas, H. Stephan, M. Wilkens, A. Zeghuzi, A. Knigge, Mode competition in broad-ridge-waveguide lasers, Semiconductor Science and Technology, 36 (2020), pp. 015014/1--015014/12, DOI 10.1088/1361-6641/abc6e7 .
Abstract
The lateral brightness achievable with high-power GaAs-based laser diodes having long and broad waveguides is commonly regarded to be limited by the onset of higher-order lateral modes. For the study of the lateral-mode competition two complementary simulation tools are applied, representing different classes of approximations. The first tool bases on a completely incoherent superposition of mode intensities and disregards longitudinal effects like spatial hole burning, whereas the second tool relies on a simplified carrier transport and current flow. Both tools yield agreeing power-current characteristics that fit the data measured for 5 to 23 µm wide ridges. Also, a similarly good qualitative conformance of the near and far fields is found. However, the threshold of individual modes, the partition of power between them at a given current, and details of the near and far fields show differences. These differences are the consequence of a high sensitivity of the mode competition to details of the models and of the device structure. Nevertheless, it can be concluded concordantly that the brightness rises with increasing ridge width irrespective of the onset of more and more lateral modes. The lateral brightness 2W · mm¯¹ 1mrad¯¹ at 10MW · cm¯²2 power density on the front facet of the investigated laser with widest ridge (23 µm) is comparable with best values known from much wider broad-area lasers. In addition, we show that one of the simulation tools is able to predict beam steering and coherent beam

M. Thomas, C. Bilgen, K. Weinberg, Analysis and simulations for a phase-field fracture model at finite strains based on modified invariants, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 100 (2020), pp. e201900288/1--e201900288/51, DOI 10.1002/zamm.201900288 .
Abstract
Phase-field models have already been proven to predict complex fracture patterns in two and three dimensions for brittle fracture at small strains. In this paper we discuss a model for phase-field fracture at finite deformations in more detail. Among the identification of crack location and projection of crack growth the numerical stability is one of the main challenges in solid mechanics. We here present a phase-field model at finite strains, which takes into account the anisotropy of damage by applying an anisotropic split and the modified invariants of the right Cauchy-Green strain tensor. We introduce a suitable weak notion of solution that also allows for a spatial and temporal discretization of the model. In this framework we study the existence of solutions and we show that the time-discrete solutions converge in a weak sense to a solution of the time-continuous formulation of the model. Numerical examples in two and three space dimensions are carried out in the range of validity of the analytical results.

A. Alphonse, M. Hintermüller, C.N. Rautenberg, Existence, iteration procedures and directional differentiability for parabolic QVIs, Calculus of Variations and Partial Differential Equations, 59 (2020), pp. 95/1--95/53, DOI 10.1007/s00526-020-01732-6 .
Abstract
We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities (VIs). Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.

A. Caiazzo, R. Maier, D. Peterseim, Reconstruction of quasi-local numerical effective models from low-resolution measurements, Journal of Scientific Computing, 85 (2020), pp. 10/1--10/23, DOI 10.1007/s10915-020-01304-y .
Abstract
We consider the inverse problem of reconstructing an effective model for a prototypical diffusion process in strongly heterogeneous media based on low-resolution measurements. We rely on recent quasi-local numerical effective models that, in contrast to conventional homogenized models, are provably reliable beyond periodicity assumptions and scale separation. The goal of this work is to show that the identification of the matrix representation of these effective models is possible. Algorithmic aspects of the inversion procedure and its performance are illustrated in a series of numerical experiments.

P.-É. Druet, A theory of generalised solutions for ideal gas mixtures with Maxwell--Stefan diffusion, Discrete and Continuous Dynamical Systems -- Series S, 14 (2021), pp. 4035--4067 (published online in Nov 2020), DOI 10.3934/dcdss.2020458 .
Abstract
After the pioneering work by Giovangigli on mathematics of multicomponent flows, several attempts were made to introduce global weak solutions for the PDEs describing the dynamics of fluid mixtures. While the incompressible case with constant density was enlighted well enough due to results by Chen and Jüngel (isothermal case), or Marion and Temam, some open questions remain for the weak solution theory of gas mixtures with their corresponding equations of mixed parabolic-hyperbolic type. For instance, Mucha, Pokorny and Zatorska showed the possibility to stabilise the hyperbolic component by means of the Bresch-Desjardins technique and a regularisation of pressure preventing vacuum. The result by Dreyer, Druet, Gajewski and Guhlke avoids emphex machina stabilisations, but the mathematical assumption that the Onsager matrix is uniformly positive on certain subspaces leads, in the dilute limit, to infinite diffusion velocities which are not compatible with the Maxwell-Stefan form of diffusion fluxes. In this paper, we prove the existence of global weak solutions for isothermal and ideal compressible mixtures with natural diffusion. The main new tool is an asymptotic condition imposed at low pressure on the binary Maxwell-Stefan diffusivities, which compensates possibly extreme behaviour of weak solutions in the rarefied regime.

D. Frerichs, Ch. Merdon, Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem, IMA Journal of Numerical Analysis, 42 (2022), pp. 597--619 (published online on 09.11.2020), DOI 10.1093/imanum/draa073 .
Abstract
Non divergence-free discretisations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterised by large discretisations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods (VEM) on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretised in a careful manner. To be able to evaluate the right-hand side for the testfunctions, some explicit interpolation of the virtual testfunctions is needed that can be evaluated pointwise everywhere. The standard discretisation via an L² -bestapproximation does not preserve the divergence and so destroys the orthogonality between divergence-free testfunctions and possibly eminent gradient forces in the right-hand side. To repair this orthogonality and restore pressure-robustness another divergence-preserving reconstruction is suggested based on Raviart--Thomas approximations on local subtriangulations of the polygons. All findings are proven theoretically and are demonstrated numerically in two dimensions. The construction is also interesting for hybrid high-order methods on polygonal or polyhedral meshes.

J. Fuhrmann, M. Landstorfer, R. Müller, Modeling polycrystalline electrode-electrolyte interfaces: The differential capacitance, Journal of The Electrochemical Society, 167 (2020), pp. 106512/1--106512/15, DOI 10.1149/1945-7111/ab9cca .
Abstract
We present and analyze a model for polycrystalline electrode surfaces based on an improved continuum model that takes finite ion size and solvation into account. The numerical simulation of finite size facet patterns allows to study two limiting cases: While for facet size diameter $d^facet to 0$ we get the typical capacitance of a spatially homogeneous but possible amorphous or liquid surface, in the limit $L^Debye << d^facet$ , an ensemble of non-interacting single crystal surfaces is approached. Already for moderate size of the facet diameters, the capacitance is remarkably well approximated by the classical approach of adding the single crystal capacities of the contributing facets weighted by their respective surface fraction. As a consequence, the potential of zero charge is not necessarily attained at a local minimum of capacitance, but might be located at a local capacitance maximum instead. Moreover, the results show that surface roughness can be accurately taken into account by multiplication of the ideally flat polycrystalline surface capacitance with a single factor. In particular, we find that the influence of the actual geometry of the facet pattern in negligible and our theory opens the way to a stochastic description of complex real polycrystal surfaces.

V. John, P. Knobloch, P. Korsmeier, On the solvability of the nonlinear problems in an algebraically stabilized finite element method for evolutionary transport-dominated equations, Mathematics of Computation, 90 (2021), pp. 595--611 (published online on 16.11.2020), DOI 10.1090/mcom/3576 .

V. John, P. Knobloch, Existence of solutions of a finite element flux-corrected-transport scheme, Applied Mathematics Letters, 115 (2021), pp. 106932/1--106932/6 (published online on 01.12.2020), DOI 10.1016/j.aml.2020.106932 .
Abstract
The existence of a solution is proved for a nonlinear finite element flux-corrected-transport (FEM-FCT) scheme with arbitrary time steps for evolutionary convection-diffusion-reaction equations and transport equations.

A. Linke, Ch. Merdon, M. Neilan, Pressure-robustness in quasi-optimal a priori estimates for the Stokes problem, Electronic Transactions on Numerical Analysis, 52 (2020), pp. 281--294, DOI 10.1553/etna_vol52s281 .

A. Mielke, T. Roubíček, Thermoviscoelasticity in Kelvin--Voigt rheology at large strains, Archive for Rational Mechanics and Analysis, 238 (2020), pp. 1--45, DOI 10.1007/s00205-020-01537-z .
Abstract
The frame-indifferent thermodynamically-consistent model of thermoviscoelasticity at large strain is formulated in the reference configuration with using the concept of the second-grade nonsimple materials. We focus on physically correct viscous stresses that are frame indifferent under time-dependent rotations. Also elastic stresses are frame indifferent under rotations and respect positivity of the determinant of the deformation gradient. The heat transfer is governed by the Fourier law in the actual deformed configuration, which leads to a nontrivial description when pulled back into the reference configuration. Existence of weak solutions in the quasistatic setting, i.e. inertial forces are ignored, is shown by time discretization.

D.H. Doan, A. Glitzky, M. Liero, Analysis of a drift-diffusion model for organic semiconductor devices, Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 70 (2019), pp. 55/1--55/18, DOI 10.1007/s00033-019-1089-z .
Abstract
We discuss drift-diffusion models for charge-carrier transport in organic semiconductor devices. The crucial feature in organic materials is the energetic disorder due to random alignment of molecules and the hopping transport of carriers between adjacent energetic sites. The former leads to so-called Gauss-Fermi statistics, which describe the occupation of energy levels by electrons and holes. The latter gives rise to complicated mobility models with a strongly nonlinear dependence on temperature, density of carriers, and electric field strength. We present the state-of-the-art modeling of the transport processes and provide a first existence result for the stationary drift-diffusion model taking all of the peculiarities of organic materials into account. The existence proof is based on Schauder's fixed-point theorem.

M. Kantner, Generalized Scharfetter--Gummel schemes for electro-thermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient, Journal of Computational Physics, published online on 07.11.2019, urlhttps://doi.org/10.1016/j.jcp.2019.109091, DOI 10.1016/j.jcp.2019.109091 .
Abstract
Many challenges faced in today's semiconductor devices are related to self-heating phenomena. The optimization of device designs can be assisted by numerical simulations using the non-isothermal drift-diffusion system, where the magnitude of the thermoelectric cross effects is controlled by the Seebeck coefficient. We show that the model equations take a remarkably simple form when assuming the so-called Kelvin formula for the Seebeck coefficient. The corresponding heat generation rate involves exactly the three classically known self-heating effects, namely Joule, recombination and Thomson--Peltier heating, without any further (transient) contributions. Moreover, the thermal driving force in the electrical current density expressions can be entirely absorbed in the (nonlinear) diffusion coefficient via a generalized Einstein relation. The efficient numerical simulation relies on an accurate and robust discretization technique for the fluxes (finite volume Scharfetter--Gummel method), which allows to cope with the typically stiff solutions of the semiconductor device equations. We derive two non-isothermal generalizations of the Scharfetter--Gummel scheme for degenerate semiconductors (Fermi--Dirac statistics) obeying the Kelvin formula. The approaches differ in the treatment of degeneration effects: The first is based on an approximation of the discrete generalized Einstein relation implying a specifically modified thermal voltage, whereas the second scheme follows the conventionally used approach employing a modified electric field. We present a detailed analysis and comparison of both schemes, indicating a superior performance of the modified thermal voltage scheme.

R. Kehl, R. Nabben, D.B. Szyld, Adaptive multilevel Krylov methods, Electronic Transactions on Numerical Analysis, 51 (2019), pp. 512--528, DOI 10.1553/etna_vol51s512 .

G. Nika, B. Vernescu, Multiscale modeling of magnetorheological suspensions, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 71 (2020), pp. 14/1--14/19 (published online on 23.12.2019), DOI 10.1007/s00033-019-1238-4 .
Abstract
We develop a multiscale approach to describe the behavior of a suspension of solid magnetizable particles in a viscous non-conducting fluid in the presence of an externally applied magnetic field. By upscaling the quasi-static Maxwell equations coupled with the Stokes' equations we are able to capture the magnetorheological effect. The model we obtain generalizes the one introduced by Neuringer & Rosensweig for quasistatic phenomena. We derive the macroscopic constitutive properties explicitly in terms of the solutions of local problems. The effective coefficients have a nonlinear dependence on the volume fraction when chain structures are present. The velocity profiles computed for some simple flows, exhibit an apparent yield stress and the flowprofile resembles a Bingham fluid flow.

P. Vágner, C. Guhlke, V. Miloš, R. Müller, J. Fuhrmann, A continuum model for yttria-stabilised zirconia incorporating triple phase boundary, lattice structure and immobile oxide ions, Journal of Solid State Electrochemistry, 23 (2019), pp. 2907--2926, DOI 10.1007/s10008-019-04356-9 .
Abstract
A continuum model for yttria-stabilised zirconia (YSZ) in the framework of non-equilibrium thermodynamics is developed. Particular attention is given to i) modeling of the YSZ-metal-gas triple phase boundary, ii) incorporation of the lattice structure and immobile oxide ions within the free energy model and iii) surface reactions. A finite volume discretization method based on modified Scharfetter-Gummel fluxes is derived in order to perform numerical simulations.
The model is used to study the impact of yttria and immobile oxide ions on the structure of the charged boundary layer and the double layer capacitance. Cyclic voltammograms of an air-half cell are simulated to study the effect of parameter variations on surface reactions, adsorption and anion diffusion.

I. Selmer, A.-S. Behnecke, P. Farrell, A. Bueno, P. Gurikov, I. Smirnova, Model development for sc-drying kinetics of aerogels: Part 2. Packed bed of spherical particles, The Journal of Supercritical Fluids, 147 (2019), pp. 147/149--147/161, DOI https://doi.org/10.1016/j.supflu.2018.07.006 .
Abstract
We apply here the previously developed mass transfer model (Part 1) for drying of aerogel monoliths to the supercritical drying of particles in a packed bed. We analyzed the influence of the operating conditions, flow rate, particle and autoclave size on the drying time and consumption of carbon dioxide. Although the model requires certain improvement to quantitatively predict the experimental drying kinetics, it captures all features of the supercritical drying: (i) transition of the limiting mass transfer step between diffusion in the gel and convection in the bulk fluid; (ii) role of CO2 density as natural variable for analysis of the CO2 consumption; (iii) influence of the autoclave dimensions on the drying process. We suggest a unified approach towards the analysis of drying process based on a dimensionless number. Practical recommendations on the rational selection of process parameters to achieve appropriate combinations of drying time and CO2 consumption are formulated.

F. Agnelli, A. Constantinescu, G. Nika, Design and testing of 3D-printed micro-architectured polymer materials exhibiting a negative Poisson's ratio, Continuum Mechanics and Thermodynamics, 32 (2020), pp. 433--449 (published online on 20.11.2019), DOI 10.1007/s00161-019-00851-6 .
Abstract
This work proposes the complete design cycle for several auxetic materials where the cycle consists of three steps (i) the design of the micro-architecture, (ii) the manufacturing of the material and (iii) the testing of the material. We use topology optimization via a level-set method and asymptotic homogenization to obtain periodic micro-architectured materials with a prescribed effective elasticity tensor and Poisson's ratio. The space of admissible micro-architectural shapes that carries orthotropic material symmetry allows to attain shapes with an effective Poisson's ratio below -1. Moreover, the specimens were manufactured using a commercial stereolithography Ember printer and are mechanically tested. The observed displacement and strain fields during tensile testing obtained by digital image correlation match the predictions from the finite element simulations and demonstrate the efficiency of the design cycle.

N.R. Gauger, A. Linke, P. Schroeder, On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond, SMAI Journal of Computational Mathematics, 5 (2019), pp. 89--129.
Abstract
Recently, high-order space discretisations were proposed for the numerical simulation of the incompressible Navier--Stokes equations at high Reynolds numbers, even for complicated domains from simulation practice. Although the overall spatial approximation order of the algorithms depends on the approximation quality of the boundary (often not better than third order), competitively accurate and efficient results were reported. In this contribution, first, a possible explanation for this somewhat surprising result is proposed: the velocity error of high-order space discretisations is more robust against quantitatively large and complicated pressure fields than low-order methods. Second, it is demonstrated that novel pressure-robust methods are significantly more accurate than comparable classical, non-pressure-robust space discretisations, whenever the quadratic, nonlinear convection term is a nontrivial gradient field like in certain generalised Beltrami flows at high Reynolds number. Then, pressure-robust methods even allow to halve the (formal) approximation order without compromising the accuracy. Third, classical high-order space discretisations are outperformed by pressure-robust methods whenever the boundary is not approximated with high-order accuracy. This improved accuracy of (low-order) pressure-robust mixed methods is explained in terms of a Helmholtz--Hodge projector, which cancels out the nonlinear convection term in any generalised Beltrami flow, since it is a gradient field. The numerical results are illustrated by a novel numerical analysis for pressure-robust and classical space discretisations. Further, the relevance of these results is discussed for flows that are not of Beltrami type.

L. Heltai, A. Caiazzo, Multiscale modeling of vascularized tissues via non-matching immersed methods, International Journal of Numerical Methods in Biomedical Engineering, 35 (2019), pp. 3264/1--3264/32, DOI 10.1002/cnm.3264 .
Abstract
We consider a multiscale approach based on immersed methods for the efficient computational modeling of tissues composed of an elastic matrix (in two or three-dimensions) and a thin vascular structure (treated as a co-dimension two manifold) at a given pressure. We derive different variational formulations of the coupled problem, in which the effect of the vasculature can be surrogated in the elasticity equations via singular or hyper-singular forcing terms. These terms only depends on information defined on co-dimension two manifolds (such as vessel center line, cross sectional area, and mean pressure over cross section), thus drastically reducing the complexity of the computational model. We perform several numerical tests, ranging from simple cases with known exact solutions to the modeling of materials with random distributions of vessels. In the latter case, we use our immersed method to perform an in silico characterization of the mechanical properties of the effective biphasic material tissue via statistical simulations.

L. Heltai, N. Rotundo, Error estimates in weighted Sobolev norms for finite element immersed interface methods, Computers & Mathematics with Applications. An International Journal, 78 (2019), pp. 3586--3604, DOI 10.1016/j.camwa.2019.05.029 .
Abstract
When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using an independent background discretisation, not aligned with the interface itself. Optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface, at the cost of a higher complexity in the implementation. A much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions, as done in immersed boundary methods. These methods are known to have inferior convergence properties, depending on the global regularity of the solution across the interface, when compared to enriched methods. In this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon, restricted to a small neighbourhood of the interface. In particular we show that optimal approximations can be constructed in a natural and inexpensive way, simply by reformulating the problem in a distributionally consistent way, and by resorting to weighted norms when computing the global error of the approximation.

V. Klika , M. Pavelka , P. Vágner, M. Grmela, Dynamic maximum entropy reduction, Entropy. An International and Interdisciplinary Journal of Entropy and Information Studies, 21 (2019), pp. 1--27.

P.L. Lederer, Ch. Merdon, J. Schöberl, Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods, Journal of Numerical Mathematics, 142 (2019), pp. 713--748.
Abstract
Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure-independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness.

L.O. Müller, A. Caiazzo, P.J. Blanco, Reduced-order unscented Kalman filter with observations in the frequency domain: Application to computational hemodynamics, IEEE Transactions on Biomedical Engineering, 66 (2019), pp. 1269--1276, DOI 10.1109/TBME.2018.2872323 .
Abstract
Objective: The aim of this work is to assess the potential of the reduced order unscented Kalman filter (ROUKF) in the context of computational hemodynamics, in order to estimate cardiovascular model parameters when employing real patient-specific data. Methods: The approach combines an efficient blood flow solver for one-dimensional networks (for the forward problem) with the parameter estimation problem cast in the frequency space. Namely, the ROUKF is used to correct model parameter after each cardiac cycle, depending on the discrepancies of model outputs with respect to available observations properly mapped into the frequency space. Results: First we validate the filter in frequency domain applying it in the context of a set of experimental measurements for an in vitro model. Second, we perform different numerical experiments aiming at parameter estimation using patient-specific data. Conclusion: Our results demonstrate that the filter in frequency domain allows a faster and more robust parameter estimation, when compared to its time domain counterpart. Moreover, the proposed approach allows to estimate parameters that are not directly related to the network but are crucial for targeting inter-individual parameter variability (e.g., parameters that characterize the cardiac output). Significance: The ROUKF in frequency domain provides a robust and flexible tool for estimating parameters related to cardiovascular mathematical models using in vivo data.

A. Zeghuzi, H.-J. Wünsche, H. Wenzel, M. Radziunas, J. Fuhrmann, A. Klehr, U. Bandelow, A. Knigge, Time-dependent simulation of thermal lensing in high-power broad-area semiconductor lasers, IEEE J. Select. Topics Quantum Electron., 25 (2019), pp. 1502310/1--1502310/10, DOI 10.1109/JSTQE.2019.2925926 .
Abstract
We propose a physically realistic and yet numerically applicable thermal model to account for short and long term self-heating within broad-area lasers. Although the temperature increase is small under pulsed operation, a waveguide that is formed within a few-ns-long pulse can result in a transition from a gain-guided to an index-guided structure, leading to near and far field narrowing. Under continuous wave operation the longitudinally varying temperature profile is obtained self-consistently. The resulting unfavorable narrowing of the near field can be successfully counteracted by etching trenches.

A. Glitzky, M. Liero, Instationary drift-diffusion problems with Gauss--Fermi statistics and field-dependent mobility for organic semiconductor devices, Communications in Mathematical Sciences, 17 (2019), pp. 33--59, DOI 10.4310/cms.2019.v17.n1.a2 .
Abstract
This paper deals with the analysis of an instationary drift-diffusion model for organic semiconductor devices including Gauss--Fermi statistics and application-specific mobility functions. The charge transport in organic materials is realized by hopping of carriers between adjacent energetic sites and is described by complicated mobility laws with a strong nonlinear dependence on temperature, carrier densities and the electric field strength. To prove the existence of global weak solutions, we consider a problem with (for small densities) regularized state equations on any arbitrarily chosen finite time interval. We ensure its solvability by time discretization and passage to the time-continuous limit. Positive lower a priori estimates for the densities of its solutions that are independent of the regularization level ensure the existence of solutions to the original problem. Furthermore, we derive for these solutions global positive lower and upper bounds strictly below the density of transport states for the densities. The estimates rely on Moser iteration techniques.

W. Dreyer, C. Guhlke, R. Müller, The impact of solvation and dissociation on the transport parameters of liquid electrolytes: Continuum modeling and numerical study, European Physical Journal Special Topics, 227 (2019), pp. 2515--2538, DOI 10.1140/epjst/e2019-800133-2 .
Abstract
Electro-thermodynamics provides a consistent framework to derive continuum models for electrochemical systems. For the application to a specific experimental system, the general model must be equipped with two additional ingredients: a free energy model to calculate the chemical potentials and a kinetic model for the kinetic coefficients. Suitable free energy models for liquid electrolytes incorporating ion-solvent interaction, finite ion sizes and solvation already exist and have been validated against experimental measurements. In this work, we focus on the modeling of the mobility coefficients based on Maxwell--Stefan setting and incorporate them into the general electro-thermodynamic framework. Moreover, we discuss the impact of model parameter on conductivity, transference numbers and salt diffusion coefficient. In particular, the focus is set on the solvation of ions and incomplete dissociation of a non-dilute electrolyte.

P. Farrell, D. Peschka, Nonlinear diffusion, boundary layers and nonsmoothness: Analysis of challenges in drift-diffusion semiconductor simulations, Computers & Mathematics with Applications. An International Journal, 78 (2019), pp. 3731--3747, DOI 10.1016/j.camwa.2019.06.007 .
Abstract
We analyze and benchmark the error and the convergence order of finite difference, finite-element as well as Voronoi finite-volume discretization schemes for the drift-diffusion equations describing charge transport in bulk semiconductor devices. Three common challenges, that can corrupt the precision of numerical solutions, will be discussed: boundary layers at Ohmic contacts, discontinuties in the doping profile, and corner singularities in L-shaped domains. The influence on the order of convergence is assessed for each computational challenge and the different discretization schemes. Additionally, we provide an analysis of the inner boundary layer asymptotics near Ohmic contacts to support our observations.

J. Fuhrmann, C. Guhlke, Ch. Merdon, A. Linke, R. Müller, Induced charge electroosmotic flow with finite ion size and solvation effects, Electrochimica Acta, 317 (2019), pp. 778--785, DOI 10.1016/j.electacta.2019.05.051 .

A. Linke, L.G. Rebholz, Pressure-induced locking in mixed methods for time-dependent (Navier--)Stokes equations, Journal of Computational Physics, 388 (2019), pp. 350--356, DOI 10.1016/j.jcp.2019.03.010 .
Abstract
We consider inf-sup stable mixed methods for the time-dependent incompressible Stokes and Navier--Stokes equations, extending earlier work on the steady (Navier-)Stokes Problem. A locking phenomenon is identified for classical inf-sup stable methods like the Taylor-Hood or the Crouzeix-Raviart elements by a novel, elegant and simple numerical analysis and corresponding numerical experiments, whenever the momentum balance is dominated by forces of a gradient type. More precisely, a reduction of the L² convergence order for high order methods, and even a complete stall of the L² convergence order for lowest-order methods on preasymptotic meshes is predicted by the analysis and practically observed. On the other hand, it is also shown that (structure-preserving) pressure-robust mixed methods do not suffer from this locking phenomenon, even if they are of lowest-order. A connection to well-balanced schemes for (vectorial) hyperbolic conservation laws like the shallow water or the compressible Euler equations is made.

A. Alphonse, Ch.M. Elliott, J. Terra, A coupled ligand-receptor bulk-surface system on a moving domain: Well posedness, regularity and convergence to equilibrium, SIAM Journal on Mathematical Analysis, 50 (2018), pp. 1544--1592, DOI 10.1137/16M110808X .
Abstract
We prove existence, uniqueness, and regularity for a reaction-diffusion system of coupled bulk-surface equations on a moving domain modelling receptor-ligand dynamics in cells. The nonlinear coupling between the three unknowns is through the Robin boundary condition for the bulk quantity and the right hand sides of the two surface equations. Our results are new even in the non-moving setting, and in this case we also show exponential convergence to a steady state. The primary complications in the analysis are indeed the nonlinear coupling and the Robin boundary condition. For the well posedness and essential boundedness of solutions we use several De Giorgi-type arguments, and we also develop some useful estimates to allow us to apply a Steklov averaging technique for time-dependent operators to prove that solutions are strong. Some of these auxiliary results presented in this paper are of independent interest by themselves.

M. Heida, On convergences of the squareroot approximation scheme to the Fokker--Planck operator, Mathematical Models & Methods in Applied Sciences, 28 (2018), pp. 2599--2635, DOI 10.1142/S0218202518500562 .
Abstract
We study the qualitative convergence properties of a finite volume scheme that recently was proposed by Lie, Fackeldey and Weber [SIAM Journal on Matrix Analysis and Applications 2013 (34/2)] in the context of conformation dynamics. The scheme was derived from physical principles and is called the squareroot approximation (SQRA) scheme. We show that solutions to the SQRA equation converge to solutions of the Fokker-Planck equation using a discrete notion of G-convergence. Hence the squareroot approximation turns out to be a usefull approximation scheme to the Fokker-Planck equation in high dimensional spaces. As an example, in the special case of stationary Voronoi tessellations we use stochastic two-scale convergence to prove that this setting satisfies the G-convergence property. In particular, the class of tessellations for which the G-convergence result holds is not trivial.

L. Adam, M. Hintermüller, Th.M. Surowiec, A semismooth Newton method with analytical path-following for the $H^1$-projection onto the Gibbs simplex, IMA Journal of Numerical Analysis, 39 (2019), pp. 1276--1295 (published online on 07.06.2018), DOI 10.1093/imanum/dry034 .
Abstract
An efficient, function-space-based second-order method for the $H^1$-projection onto the Gibbs-simplex is presented. The method makes use of the theory of semismooth Newton methods in function spaces as well as Moreau-Yosida regularization and techniques from parametric optimization. A path-following technique is considered for the regularization parameter updates. A rigorous first and second-order sensitivity analysis of the value function for the regularized problem is provided to justify the update scheme. The viability of the algorithm is then demonstrated for two applications found in the literature: binary image inpainting and labeled data classification. In both cases, the algorithm exhibits mesh-independent behavior.

T. Ahnert, A. Münch, B. Niethammer, B. Wagner, Stability of concentrated suspensions under Couette and Poiseuille flow, Journal of Engineering Mathematics, 111 (2018), pp. 51--77, DOI 10.1007/s10665-018-9954-x .
Abstract
The stability of two-dimensional Poiseuille flow and plane Couette flow for concentrated suspensions is investigated. Linear stability analysis of the two-phase flow model for both flow geometries shows the existence of a convectively driven instability with increasing growth rates of the unstable modes as the particle volume fraction of the suspension increases. In addition it is shown that there exists a bound for the particle phase viscosity below which the two-phase flow model may become ill-posed as the particle phase approaches its maximum packing fraction. The case of two-dimensional Poiseuille flow gives rise to base state solutions that exhibit a jammed and unyielded region, due to shear-induced migration, as the maximum packing fraction is approached. The stability characteristics of the resulting Bingham-type flow is investigated and connections to the stability problem for the related classical Bingham-flow problem are discussed.

C. Bertoglio, A. Caiazzo, Y. Bazilevs, M. Braack, M. Esmaily-Moghadam, V. Gravemeier, A.L. Marsden, O. Pironneau, I.E. Vignon-Clementel, W.A. Wall, Benchmark problems for numerical treatment of backflow at open boundaries, International Journal of Numerical Methods in Biomedical Engineering, 34 (2018), pp. e2918/1--e2918/34, DOI 10.1002/cnm.2918 .
Abstract
In computational fluid dynamics, incoming velocity at open boundaries, or backflow, often yields to unphysical instabilities already for moderate Reynolds numbers. Several treatments to overcome these backflow instabilities have been proposed in the literature. However, these approaches have not yet been compared in detail in terms of accuracy in different physiological regimes, in particular due to the difficulty to generate stable reference solutions apart from analytical forms. In this work, we present a set of benchmark problems in order to compare different methods in different backflow regimes (with a full reversal flow and with propagating vortices after a stenosis). The examples are implemented in FreeFem++ and the source code is openly available, making them a solid basis for future method developments.

A. Bradji, J. Fuhrmann, On the convergence and convergence order of finite volume gradient schemes for oblique derivative boundary value problems, Computational & Applied Mathematics, 37 (2018), pp. 2533--2565, DOI 10.1007/s40314-017-0463-8 .

A. Fischer, M. Pfalz, K. Vandewal, M. Liero, A. Glitzky, S. Lenk, S. Reineke, Full electrothermal OLED model including nonlinear self-heating effects, Physical Review Applied, 10 (2018), pp. 014023/1--014023/12, DOI 10.1103/PhysRevApplied.10.014023 .
Abstract
Organic light-emitting diodes (OLEDs) are widely studied semiconductor devices for which a simple description by a diode equation typically fails. In particular, a full description of the current-voltage relation, including temperature effects, has to take the low electrical conductivity of organic semiconductors into account. Here, we present a temperature-dependent resistive network, incorporating recombination as well as electron and hole conduction to describe the current-voltage characteristics of an OLED over the entire operation range. The approach also reproduces the measured nonlinear electrothermal feedback upon Joule self-heating in a self-consistent way. Our model further enables us to learn more about internal voltage losses caused by the charge transport from the contacts to the emission layer which is characterized by a strong temperature-activated electrical conductivity, finally determining the strength of the electrothermal feedback. In general, our results provide a comprehensive picture to understand the electrothermal operation of an OLED which will be essential to ensure and predict especially long-term stability and reliability in superbright OLED applications.

J. Haskovec, S. Hittmeir, P. Markowich, A. Mielke, Decay to equilibrium for energy-reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 50 (2018), pp. 1037--1075, DOI 10.1137/16M1062065 .
Abstract
We derive thermodynamically consistent models of reaction-diffusion equations coupled to a heat equation. While the total energy is conserved, the total entropy serves as a driving functional such that the full coupled system is a gradient flow. The novelty of the approach is the Onsager structure, which is the dual form of a gradient system, and the formulation in terms of the densities and the internal energy. In these variables it is possible to assume that the entropy density is strictly concave such that there is a unique maximizer (thermodynamical equilibrium) given linear constraints on the total energy and suitable density constraints. We consider two particular systems of this type, namely, a diffusion-reaction bipolar energy transport system, and a drift-diffusion-reaction energy transport system with confining potential. We prove corresponding entropy-entropy production inequalities with explicitely calculable constants and establish the convergence to thermodynamical equilibrium, at first in entropy and further in L¹ using Cziszar-Kullback-Pinsker type inequalities.

G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Rate-independent damage in thermo-viscoelastic materials with inertia, Journal of Dynamics and Differential Equations, 30 (2018), pp. 1311--1364, DOI 10.1007/s10884-018-9666-y .
Abstract
We present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio-Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is independent of temperature.

P.W. Schroeder, Ch. Lehrenfeld, A. Linke, G. Lube, Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier--Stokes equations, SeMA Journal. Boletin de la Sociedad Espannola de Matematica Aplicada, 75 (2018), pp. 629--653, DOI 10.1007/s40324-018-0157-1 .
Abstract
Inf-sup stable FEM applied to time-dependent incompressible Navier--Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure-robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, Re-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on an essential regularity assumption for the gradient of the velocity, which is discussed in detail. In the sense of best practice, we review and establish pressure- and Re-semi-robust estimates for pointwise divergence-free H1-conforming FEM (like Scott--Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free H(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.

L. Blank, A. Caiazzo, F. Chouly, A. Lozinski, J. Mura, Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes, and Darcy problems, ESAIM: Mathematical Modelling and Numerical Analysis, 52 (2018), pp. 2149--2185, DOI 10.1051/m2an/2018063 .

M. Thomas, C. Bilgen, K. Weinberg, Phase-field fracture at finite strains based on modified invariants: A note on its analysis and simulations, GAMM-Mitteilungen, 40 (2018), pp. 207--237, DOI 10.1002/gamm.201730004 .
Abstract
Phase-field models have already been proven to predict complex fracture patterns in two and three dimensions for brittle fracture at small strains. In this paper we discuss a model for phase-field fracture at finite deformations in more detail. Among the identification of crack location and projection of crack growth the numerical stability is one of the main challenges in solid mechanics. We here present a phase-field model at finite strains, which takes into account the anisotropy of damage by applying an anisotropic split and the modified invariants of the right Cauchy-Green strain tensor. We introduce a suitable weak notion of solution that also allows for a spatial and temporal discretization of the model. In this framework we study the existence of solutions %Second the mathematical background of the approach is examined and and we show that the time-discrete solutions converge in a weak sense to a solution of the time-continuous formulation of the model. Numerical examples in two and three space dimensions are carried out in the range of validity of the analytical results.

N. Ahmed, C. Bartsch, V. John, U. Wilbrandt, An assessment of solvers for some saddle point problems emerging from the incompressible Navier--Stokes equations, Computer Methods in Applied Mechanics and Engineering, 331 (2018), pp. 492--513, DOI 10.1016/j.cma.2017.12.004 .
Abstract
Efficient incompressible flow simulations, using inf-sup stable pairs of finite element spaces, require the application of efficient solvers for the arising linear saddle point problems. This paper presents an assessment of different solvers: the sparse direct solver UMFPACK, the flexible GMRES (FGMRES) method with different coupled multigrid preconditioners, and FGMRES with Least Squares Commutator (LSC) preconditioners. The assessment is performed for steady-state and time-dependent flows around cylinders in 2d and 3d. Several pairs of inf-sup stable finite element spaces with second order velocity and first order pressure are used. It turns out that for the steady-state problems often FGMRES with an appropriate multigrid preconditioner was the most efficient method on finer grids. For the time-dependent problems, FGMRES with LSC preconditioners that use an inexact iterative solution of the velocity subproblem worked best for smaller time steps.

N. Ahmed, A. Linke, Ch. Merdon, On really locking-free mixed finite element methods for the transient incompressible Stokes equations, SIAM Journal on Numerical Analysis, 56 (2018), pp. 185--209.
Abstract
Inf-sup stable mixed methods for the steady incompressible Stokes equations that relax the divergence constraint are often claimed to deliver locking-free discretizations. However, this relaxation leads to a pressure-dependent contribution in the velocity error, which is proportional to the inverse of the viscosity, thus giving rise to a (different) locking phenomenon. However, a recently proposed modification of the right hand side alone leads to a discretization that is really locking-free, i.e., its velocity error converges with optimal order and is independent of the pressure and the smallness of the viscosity. In this contribution, we extend this approach to the transient incompressible Stokes equations, where besides the right hand side also the velocity time derivative requires an improved space discretization. Semi-discrete and fully-discrete a-priori velocity and pressure error estimates are derived, which show beautiful robustness properties. Two numerical examples illustrate the superior accuracy of pressure-robust space discretizations in the case of small viscosities.

P. Farrell, M. Patriarca, J. Fuhrmann, Th. Koprucki, Comparison of thermodynamically consistent charge carrier flux discretizations for Fermi--Dirac and Gauss--Fermi statistics, Optical and Quantum Electronics, 50 (2018), pp. 101/1--101/10, DOI 10.1007/s11082-018-1349-8 .
Abstract
We compare three thermodynamically consistent Scharfetter--Gummel schemes for different distribution functions for the carrier densities, including the Fermi--Dirac integral of order 1/2 and the Gauss--Fermi integral. The most accurate (but unfortunately also most costly) generalized Scharfetter--Gummel scheme requires the solution of an integral equation. We propose a new method to solve this integral equation numerically based on Gauss quadrature and Newton's method. We discuss the quality of this approximation and plot the resulting currents for Fermi--Dirac and Gauss--Fermi statistics. Finally, by comparing two modified (diffusion-enhanced and inverse activity based) Scharfetter--Gummel schemes with the more accurate generalized scheme, we show that the diffusion-enhanced ansatz leads to considerably lower flux errors, confirming previous results (J. Comp. Phys. 346:497-513, 2017).

M. Hintermüller, M. Hinze, Ch. Kahle, T. Keil, A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn--Hilliard--Navier--Stokes system, Optimization and Engineering. International Multidisciplinary Journal to Promote Optimization Theory & Applications in Engineering Sciences, 19 (2018), pp. 629--662, DOI 10.1007/s11081-018-9393-6 .
Abstract
This paper is concerned with the development and implementation of an adaptive solution algorithm for the optimal control of a time-discrete Cahn--Hilliard--Navier--Stokes system with variable densities. The free energy density associated to the Cahn--Hilliard system incorporates the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier--Stokes equation. A dual-weighed residual approach for goal-oriented adaptive finite elements is presented which is based on the concept of C-stationarity. The overall error representation depends on primal residual weighted by approximate dual quantities and vice versa as well as various complementary mismatch errors. Details on the numerical realization of the adaptive concept and a report on numerical tests are given.

V. John, P. Knobloch, J. Novo, Finite elements for scalar convection-dominated equations and incompressible flow problems -- A never ending story?, Computing and Visualization in Science, 19 (2018), pp. 47--63.
Abstract
The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important problems in these fields are discussed.

A. Linke, Ch. Merdon, M. Neilan, F. Neumann, Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes problem, Mathematics of Computation, 87 (2018), pp. 1543--1566, DOI 10.1090/mcom/3344 .
Abstract
Nearly all classical inf-sup stable mixed finite element methods for the incompressible Stokes equations are not pressure-robust, i.e., the velocity error is dependent on the pressure. However, recent results show that pressure-robustness can be recovered by a non-standard discretization of the right hand side alone. This variational crime introduces a consistency error in the method which can be estimated in a straightforward manner provided that the exact velocity solution is sufficiently smooth. The purpose of this paper is to analyze the pressure-robust scheme with low regularity. The numerical analysis applies divergence-free H¹-conforming Stokes finite element methods as a theoretical tool. As an example, pressure-robust velocity and pressure a-priori error estimates will be presented for the (first order) nonconforming Crouzeix--Raviart element. A key feature in the analysis is the dependence of the errors on the Helmholtz projector of the right hand side data, and not on the entire data term. Numerical examples illustrate the theoretical results.

P. Farrell, A. Linke, Uniform second order convergence of a complete flux scheme on unstructured 1D grids for a singularly perturbed advection-diffusion equation and some multidimensional extensions, Journal of Scientific Computing, 72 (2017), pp. 373--395, DOI 10.1007/s10915-017-0361-7 .
Abstract
The accurate and efficient discretization of singularly perturbed advection-diffusion equations on arbitrary 2D and 3D domains remains an open problem. An interesting approach to tackle this problem is the complete flux scheme (CFS) proposed by G. D. Thiart and further investigated by J. ten Thije Boonkkamp. For the CFS, uniform second order convergence has been proven on structured grids. We extend a version of the CFS to unstructured grids for a steady singularly perturbed advection-diffusion equation. By construction, the novel finite volume scheme is nodally exact in 1D for piecewise constant source terms. This property allows to use elegant continuous arguments in order to prove uniform second order convergence on unstructured one-dimensional grids. Numerical results verify the predicted bounds and suggest that by aligning the finite volume grid along the velocity field uniform second order convergence can be obtained in higher space dimensions as well.

M. Kantner, M. Mittnenzweig, Th. Koprucki, Hybrid quantum-classical modeling of quantum dot devices, Phys. Rev. B., 96 (2017), pp. 205301/1--205301/17, DOI 10.1103/PhysRevB.96.205301 .
Abstract
The design of electrically driven quantum dot devices for quantum optical applications asks for modeling approaches combining classical device physics with quantum mechanics. We connect the well-established fields of semi-classical semiconductor transport theory and the theory of open quantum systems to meet this requirement. By coupling the van Roosbroeck system with a quantum master equation in Lindblad form, we obtain a new hybrid quantum-classical modeling approach, which enables a comprehensive description of quantum dot devices on multiple scales: It allows the calculation of quantum optical figures of merit and the spatially resolved simulation of the current flow in realistic semiconductor device geometries in a unified way. We construct the interface between both theories in such a way, that the resulting hybrid system obeys the fundamental axioms of (non-)equilibrium thermodynamics. We show that our approach guarantees the conservation of charge, consistency with the thermodynamic equilibrium and the second law of thermodynamics. The feasibility of the approach is demonstrated by numerical simulations of an electrically driven single-photon source based on a single quantum dot in the stationary and transient operation regime.

A. Linke, M. Neilan, L.G. Rebholz, N. Wilson, A connection between coupled and penalty projection timestepping schemes with FE spatial discretization for the Navier--Stokes equations, Journal of Numerical Mathematics, 25 (2017), pp. 229--248, DOI 10.1515/jnma-2016-1024 .
Abstract
We prove that in finite element settings where the divergence-free subspace of the velocity space has optimal approximation properties, the solution of Chorin/Temam projection methods for Navier-Stokes equations equipped with grad-div stabilization with parameter gamma, converge to the associated coupled method solution with rate 1/gamma as gamma goes to infinity. We prove this first for backward Euler schemes, and then extend the results to BDF2 schemes, and finally to schemes with outflow boundary conditions. Several numerical experiments are given which verify the convergence rate, and show how using projection methods in this setting with large grad-div stabilization parameters can dramatically improve accuracy.

U. Wilbrandt, C. Bartsch, N. Ahmed, N. Alia, F. Anker, L. Blank, A. Caiazzo, S. Ganesan, S. Giere, G. Matthies, R. Meesala, A. Shamim, J. Venkatensan, V. John, ParMooN -- A modernized program package based on mapped finite elements, Computers & Mathematics with Applications. An International Journal, 74 (2017), pp. 74--88, DOI 10.1016/j.camwa.2016.12.020 .

R. Rossi, M. Thomas, Coupling rate-independent and rate-dependent processes: Existence results, SIAM Journal on Mathematical Analysis, 49 (2017), pp. 1419--1494.
Abstract
We address the analysis of an abstract system coupling a rate-independet process with a second order (in time) nonlinear evolution equation. We propose suitable weak solution concepts and obtain existence results by passing to the limit in carefully devised time-discretization schemes. Our arguments combine techniques from the theory of gradient systems with the toolbox for rate-independent evolution, thus reflecting the mixed character of the problem. Finally, we discuss applications to a class of rate-independent processes in visco-elastic solids with inertia, and to a recently proposed model for damage with plasticity.

G.R. Barrenechea, V. John, P. Knobloch, An algebraic flux correction scheme satisfying the discrete maximum principle and linearity preservation on general meshes, Mathematical Models & Methods in Applied Sciences, 27 (2017), pp. 525--548, DOI 10.1142/S0218202517500087 .

W. Huang, L. Kamenski, On the mesh nonsingularity of the moving mesh PDE method, Mathematics of Computation, 87 (2018), pp. 1887--1911 (published online on 02.10.2017), DOI 10.1090/mcom/3271 .
Abstract
The moving mesh PDE (MMPDE) method for variational mesh generation and adaptation is studied theoretically at the discrete level, in particular the nonsingularity of the obtained meshes. Meshing functionals are discretized geometrically and the MMPDE is formulated as a modified gradient system of the corresponding discrete functionals for the location of mesh vertices. It is shown that if the meshing functional satisfies a coercivity condition, then the mesh of the semi-discrete MMPDE is nonsingular for all time if it is nonsingular initially. Moreover, the altitudes and volumes of its elements are bounded below by positive numbers depending only on the number of elements, the metric tensor, and the initial mesh. Furthermore, the value of the discrete meshing functional is convergent as time increases, which can be used as a stopping criterion in computation. Finally, the mesh trajectory has limiting meshes which are critical points of the discrete functional. The convergence of the mesh trajectory can be guaranteed when a stronger condition is placed on the meshing functional. Two meshing functionals based on alignment and equidistribution are known to satisfy the coercivity condition. The results also hold for fully discrete systems of the MMPDE provided that the time step is sufficiently small and a numerical scheme preserving the property of monotonically decreasing energy is used for the temporal discretization of the semi-discrete MMPDE. Numerical examples are presented

P.L. Lederer, A. Linke, Ch. Merdon, J. Schöberl, Divergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements, SIAM Journal on Numerical Analysis, 55 (2017), pp. 1291--1314.
Abstract
Classical inf-sup stable mixed finite elements for the incompressible (Navier--)Stokes equations are not pressure-robust, i.e., their velocity errors depend on the continuous pressure. How-ever, a modification only in the right hand side of a Stokes discretization is able to reestablish pressure-robustness, as shown recently for several inf-sup stable Stokes elements with discontinuous discrete pressures. In this contribution, this idea is extended to low and high order Taylor--Hood and mini elements, which have continuous discrete pressures. For the modification of the right hand side a velocity reconstruction operator is constructed that maps discretely divergence-free test functions to exactly divergence-free ones. The reconstruction is based on local H(div)-conforming flux equilibration on vertex patches, and fulfills certain orthogonality properties to provide consistency and optimal a-priori error estimates. Numerical examples for the incompressible Stokes and Navier--Stokes equations confirm that the new pressure-robust Taylor--Hood and mini elements converge with optimal order and outperform signi--cantly the classical versions of those elements when the continuous pressure is comparably large.

N. Ahmed, S. Becher, G. Matthies, Higher-order discontinuous Galerkin time stepping and local projection stabilization techniques for the transient Stokes problem, Computer Methods in Applied Mechanics and Engineering, 313 (2017), pp. 28--52.
Abstract
We introduce and analyze discontinuous Galerkin time discretizations coupled with continuous finite element methods based on equal-order interpolation in space for velocity and pressure in transient Stokes problems. Spatial stability of the pressure is ensured by adding a stabilization term based on local projection. We present error estimates for the semi-discrete problem after discretization in space only and for the fully discrete problem. The fully discrete pressure shows an instability in the limit of small time step length. Numerical tests are presented which confirm our theoretical results including the pressure instability.

N. Ahmed, T.Ch. Rebollo, V. John, S. Rubino, A review of variational multiscale methods for the simulation of turbulent incompressible flows, Archives of Computational Methods in Engineering. State of the Art Reviews, 24 (2017), pp. 115--164.
Abstract
Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: They are based on the variational formulation of the incompressible Navier--Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used mathematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized.

N. Ahmed, T.Ch. Rebollo, V. John, S. Rubino, Analysis of a full space-time discretization of the Navier--Stokes equations by a local projection stabilization method, IMA Journal of Numerical Analysis, 37 (2017), pp. 1437--1467, DOI 10.1093/imanum/drw048 .
Abstract
A finite element error analysis of a local projection stabilization (LPS) method for the time-dependent Navier--Stokes equations is presented. The focus is on the high-order term-by-term stabilization method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard LPS methods is replaced by an interpolation-stabilized structure. The main contribution is on proving, theoretically and numerically, the optimal convergence order of the arising fully discrete scheme. In addition, the asymptotic energy balance is obtained for slightly smooth flows. Numerical studies support the analytical results and illustrate the potential of the method for the simulation of turbulent flows. Smooth unsteady flows are simulated with optimal order of accuracy.

N. Ahmed, A. Linke, Ch. Merdon, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, Computational Methods in Applied Mathematics, 18 (2018), pp. 353--372 (published online on 18.11.2017), DOI 10.1515/cmam-2017-0047 .
Abstract
In this contribution, classical mixed methods for the incompressible Navier-Stokes equations that relax the divergence constraint and are discretely inf-sup stable, are reviewed. Though the relaxation of the divergence constraint was claimed to be harmless since the beginning of the 1970ies, Poisson locking is just replaced by another more subtle kind of locking phenomenon, which is sometimes called poor mass conservation. Indeed, divergence-free mixed methods and classical mixed methods behave qualitatively in a different way: divergence-free mixed methods are pressure-robust, which means that, e.g., their velocity error is independent of the continuous pressure. The lack of pressure-robustness in classical mixed methods can be traced back to a consistency error of an appropriately defined discrete Helmholtz projector. Numerical analysis and numerical examples reveal that really locking-free mixed methods must be discretely inf-sup stable and pressure-robust, simultaneously. Further, a recent discovery shows that locking-free, pressure-robust mixed methods do not have to be divergence-free. Indeed, relaxing the divergence constraint in the velocity trial functions is harmless, if the relaxation of the divergence constraint in some velocity test functions is repaired, accordingly.

N. Ahmed, On the grad-div stabilization for the steady Oseen and Navier--Stokes equations, Calcolo. A Quarterly on Numerical Analysis and Theory of Computation, 54 (2017), pp. 471--501, DOI 10.1007/s10092-016-0194-z .
Abstract
This paper studies the parameter choice in the grad-div stabilization applied to the generalized problems of Oseen type. Stabilization parameters based on minimizing the H¹(Ω) error of the velocity are derived which do not depend on the viscosity parameter. For the proposed parameter choices, the H¹(Ω) error of the velocity is derived that shows a direct dependence on the viscosity parameter. Differences and common features to the situation for the Stokes equations are discussed. Numerical studies are presented which confirm the theoretical results. Moreover, for the Navier- Stokes equations, numerical simulations were performed on a two-dimensional ow past a circular cylinder. It turns out, for the MINI element, that the best results can be obtained without grad-div stabilization.

W. Dreyer, C. Guhlke, M. Landstorfer, R. Müller, New insights on the interfacial tension of electrochemical interfaces and the Lippmann equation, European Journal of Applied Mathematics, 29 (2018), pp. 708--753, DOI 10.1017/S0956792517000341 .
Abstract
The Lippmann equation is considered as universal relationship between interfacial tension, double layer charge, and cell potential. Based on the framework of continuum thermo-electrodynamics we provide some crucial new insights to this relation. In a previous work we have derived a general thermodynamic consistent model for electrochemical interfaces, which showed a remarkable agreement to single crystal experimental data. Here we apply the model to a curved liquid metal electrode. If the electrode radius is large compared to the Debye length, we apply asymptotic analysis methods and obtain the Lippmann equation. We give precise definitions of the involved quantities and show that the interfacial tension of the Lippmann equation is composed of the surface tension of our general model, and contributions arising from the adjacent space charge layers. This finding is confirmed by a comparison of our model to experimental data of several mercury-electrolyte interfaces. We obtain qualitative and quantitative agreement in the 2V potential range for various salt concentrations. We also discuss the validity of our asymptotic model when the electrode curvature radius is comparable to the Debye length.

P. Farrell, Th. Koprucki, J. Fuhrmann, Computational and analytical comparison of flux discretizations for the semiconductor device equations beyond Boltzmann statistics, Journal of Computational Physics, 346 (2017), pp. 497--513, DOI 10.1016/j.jcp.2017.06.023 .
Abstract
For a Voronoï finite volume discretization of the van Roosbroeck system with general charge carrier statistics we compare three thermodynamically consistent numerical fluxes known in the literature. We discuss an extension of the Scharfetter--Gummel scheme to non-Boltzmann (e.g. Fermi--Dirac) statistics. It is based on the analytical solution of a two-point boundary value problem obtained by projecting the continuous differential equation onto the interval between neighboring collocation points. Hence, it serves as a reference flux. The exact solution of the boundary value problem can be approximated by computationally cheaper fluxes which modify certain physical quantities. One alternative scheme averages the nonlinear diffusion (caused by the non-Boltzmann nature of the problem), another one modifies the effective density of states. To study the differences between these three schemes, we analyze the Taylor expansions, derive an error estimate, visualize the flux error and show how the schemes perform for a carefully designed p-i-n benchmark simulation. We present strong evidence that the flux discretization based on averaging the nonlinear diffusion has an edge over the scheme based on modifying the effective density of states.

V. John, A. Linke, Ch. Merdon, M. Neilan, L.G. Rebholz, On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Review, 59 (2017), pp. 492--544, DOI 10.1137/15M1047696 .
Abstract
The divergence constraint of the incompressible Navier--Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which influences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This paper reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, $bH(mathrmdiv)$-conforming finite elements, and mixed methods with an appropriate reconstruction of the test functions. Numerical examples illustrate both the potential effects of using non-robust discretizations and the improvements obtained by utilizing pressure-robust discretizations.

M. Liero, J. Fuhrmann, A. Glitzky, Th. Koprucki, A. Fischer, S. Reineke, 3D electrothermal simulations of organic LEDs showing negative differential resistance, Optical and Quantum Electronics, 49 (2017), pp. 330/1--330/8, DOI 10.1007/s11082-017-1167-4 .
Abstract
Organic semiconductor devices show a pronounced interplay between temperature-activated conductivity and self-heating which in particular causes inhomogeneities in the brightness of large-area OLEDs at high power. We consider a 3D thermistor model based on partial differential equations for the electrothermal behavior of organic devices and introduce an extension to multiple layers with nonlinear conductivity laws, which also take the diode-like behavior in recombination zones into account. We present a numerical simulation study for a red OLED using a finite-volume approximation of this model. The appearance of S-shaped current-voltage characteristics with regions of negative differential resistance in a measured device can be quantitatively reproduced. Furthermore, this simulation study reveals a propagation of spatial zones of negative differential resistance in the electron and hole transport layers toward the contact.

A. Linke, Ch. Merdon, W. Wollner, Optimal L2 velocity error estimate for a modified pressure-robust Crouzeix--Raviart Stokes element, IMA Journal of Numerical Analysis, 37 (2017), pp. 354--374, DOI 10.1093/imanum/drw019 .
Abstract
Recently, a novel approach for the robust discretization of the incompressible Stokes equations was proposed that slightly modifies the nonconforming Crouzeix--Raviart element such that its velocity error becomes pressure-independent. The modification results in an O(h) consistency error that allows straightforward proofs for the optimal convergence of the discrete energy norm of the velocity and of the L2 norm of the pressure. However, though the optimal convergence of the velocity in the L2 norm was observed numerically, it appeared to be nontrivial to prove. In this contribution, this gap is closed. Moreover, the dependence of the error estimates on the discrete inf-sup constant is traced in detail, which shows that classical error estimates are extremely pessimistic on domains with large aspect ratios. Numerical experiments in 2D and 3D illustrate the theoretical findings.

E. Meca Álvarez, A. Münch, B. Wagner, Sharp-interface formation during lithium intercalation into silicon, European Journal of Applied Mathematics, 29 (2018), pp. 118--145, DOI 10.1017/S0956792517000067 .
Abstract
In this study we present a phase-field model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as a-Si. The governing equations couple a viscous Cahn-Hilliard-Reaction model with elasticity in the framework of the Cahn-Larché system. We discuss the parameter settings and flux conditions at the free boundary that lead to the formation of phase boundaries having a sharp gradient in ion concentration between the initial state of the solid layer and the intercalated region. We carry out a matched asymptotic analysis to derive the corresponding sharp-interface model that also takes into account the dynamics of triple points where the sharp interface in the bulk of the layer intersects the free boundary. We numerically compare the interface motion predicted by the sharp-interface model with the long-time dynamics of the phase-field model.

A. Mielke, M. Mittnenzweig, Convergence to equilibrium in energy-reaction-diffusion systems using vector-valued functional inequalities, Journal of Nonlinear Science, 28 (2018), pp. 765--806 (published online on 11.11.2017), DOI 10.1007/s00332-017-9427-9 .
Abstract
We discuss how the recently developed energy-dissipation methods for reactiondi usion systems can be generalized to the non-isothermal case. For this we use concave entropies in terms of the densities of the species and the internal energy, where the importance is that the equilibrium densities may depend on the internal energy. Using the log-Sobolev estimate and variants for lower-order entropies as well as estimates for the entropy production of the nonlinear reactions we give two methods to estimate the relative entropy by the total entropy production, namely a somewhat restrictive convexity method, which provides explicit decay rates, and a very general, but weaker compactness method.

K. Disser, G.P. Galdi, G. Mazzone, P. Zunino, Inertial motions of a rigid body with a cavity filled with a viscous liquid, Archive for Rational Mechanics and Analysis, 221 (2016), pp. 487--526.
Abstract
We consider the system of equations modeling the free motion of a rigid body with a cavity filled by a viscous (Navier-Stokes) liquid. Zhukovskiy's Theorem states that in the limit of time going to infinity, the relative fluid velocity tends to 0 and the rigid velocity of the full structure tends to a steady rotation around one of the principle axes of inertia. We give a rigorous proof of this result.
In particular, we prove that every global weak solution in a suitable class is subject to Zhukovskiy's Theorem, and note that existence of these solutions has been established. Independently of the geometry and of parameters, this shows that the presence of fluid prevents precession of the body in the limit. In general, we cannot predict which axis will be attained, but we can show stability of the largest axis and provide criteria on the initial data which are decisive in special cases.

M. Kantner, Th. Koprucki, Numerical simulation of carrier transport in semiconductor devices at cryogenic temperatures, Optical and Quantum Electronics, 48 (2016), pp. 543/1--543/7, DOI 10.1007/s11082-016-0817-2 .
Abstract
At cryogenic temperatures the electron?hole plasma in semiconductors becomes strongly degenerate, leading to very sharp internal layers, extreme depletion in intrinsic domains and strong nonlinear diffusion. As a result, the numerical simulation of the drift?diffusion system suffers from serious convergence issues using standard methods. We consider a one-dimensional p-i-n diode to illustrate these problems and present a simple temperature-embedding scheme to enable the numerical simulation at cryogenic temperatures. The method is suitable for forward-biased devices as they appear e.g. in optoelectronic applications. Moreover, the method can be applied to wide band gap semiconductors where similar numerical issues occur already at room temperature.

M. Kantner, U. Bandelow, Th. Koprucki, J.-H. Schulze, A. Strittmatter, H.-J. Wünsche, Efficient current injection into single quantum dots through oxide-confined pn-diodes, IEEE Transactions on Electron Devices, 63 (2016), pp. 2036--2042.
Abstract
Current injection into single quantum dots embedded in vertical pn-diodes featuring oxide apertures is analyzed in the low-injection regime suitable for single-photon emitters. Experimental and theoretical evidence is found for a rapid lateral spreading of the carriers after passing the oxide aperture in the conventional pin-design. By an alternative design employing p-doping up to the oxide aperture the current spreading can be suppressed resulting in an enhanced current confinement and increased injection efficiencies, both, in the continuous wave and under pulsed excitation.

A. Linke, G. Matthies, L. Tobiska, Robust arbitrary order mixed finite element methods for the incompressible Stokes equations with pressure independent velocity errors, ESAIM: Mathematical Modelling and Numerical Analysis, 50 (2016), pp. 289--309.
Abstract
Standard mixed finite element methods for the incompressible Navier-Stokes equations that relax the divergence constraint are not robust against large irrotational forces in the momentum balance and the velocity error depends on the continuous pressure. This robustness issue can be completely cured by using divergence-free mixed finite elements which deliver pressure-independent velocity error estimates. However, the construction of H1-conforming, divergence-free mixed finite element methods is rather difficult. Instead, we present a novel approach for the construction of arbitrary order mixed finite element methods which deliver pressure-independent velocity errors. The approach does not change the trial functions but replaces discretely divergence-free test functions in some operators of the weak formulation by divergence-free ones. This modification is applied to inf-sup stable conforming and nonconforming mixed finite element methods of arbitrary order in two and three dimensions. Optimal estimates for the incompressible Stokes equations are proved for the H1 and L2 errors of the velocity and the L2 error of the pressure. Moreover, both velocity errors are pressure-independent, demonstrating the improved robustness. Several numerical examples illustrate the results.

D. Peschka, M. Thomas, A. Glitzky, R. Nürnberg, M. Virgilio, S. Guha, Th. Schröder, G. Cappellini, Th. Koprucki, Robustness analysis of a device concept for edge-emitting lasers based on strained germanium, Optical and Quantum Electronics, 48 (2016), pp. 156/1--156/7, DOI 10.1007/s11082-016-0394-4 .
Abstract
We consider a device concept for edge-emitting lasers based on strained germanium microstrips. The device features an inhomogeneous tensile strain distribution generated by a SiN stressor deposited on top of the Ge microstrip. This geometry requires a lateral contact scheme and hence a full two-dimensional description. The two-dimensional simulations of the carrier transport and of the optical field, carried out in a cross section of the device orthogonal to the optical cavity, use microscopic calculations of the strained Ge material gain as an input. In this paper we study laser performance and robustness against Shockley-Read-Hall lifetime variations and device sensitivity to different strain distributions.

D. Peschka, N. Rotundo, M. Thomas, Towards doping optimization of semiconductor lasers, Journal of Computational and Theoretical Transport, 45 (2016), pp. 410--423.
Abstract
We discuss analytical and numerical methods for the optimization of optoelectronic devices by performing optimal control of the PDE governing the carrier transport with respect to the doping profile. First, we provide a cost functional that is a sum of a regularization and a contribution, which is motivated by the modal net gain that appears in optoelectronic models of bulk or quantum-well lasers. Then, we state a numerical discretization, for which we study optimized solutions for different regularizations and for vanishing weights.

G.R. Barrenechea, V. John, P. Knobloch, Analysis of algebraic flux correction schemes, SIAM Journal on Numerical Analysis, 54 (2016), pp. 2427--2451.
Abstract
A family of algebraic flux correction schemes for linear boundary value problems in any space dimension is studied. These methods' main feature is that they limit the fluxes along each one of the edges of the triangulation, and we suppose that the limiters used are symmetric. For an abstract problem, the existence of a solution, existence and uniqueness of the solution of a linearized problem, and an a priori error estimate, are proved under rather general assumptions on the limiters. For a particular (but standard in practice) choice of the limiters, it is shown that a local discrete maximum principle holds. The theory developed for the abstract problem is applied to convection-diffusion-reaction equations, where in particular an error estimate is derived. Numerical studies show its sharpness.

C. Bertoglio, A. Caiazzo, A Stokes-residual backflow stabilization method applied to physiological flows, Journal of Computational Physics, 313 (2016), pp. 260--278.
Abstract
In computational fluid dynamics incoming flow at open boundaries, or emphbackflow, often yields to unphysical instabilities for high Reynolds numbers. It is widely accepted that this is due to the incoming energy arising from the convection term, which cannot be empha priori controlled when the velocity field is unknown at the boundary. In order to improve the robustness of the numerical simulations, we propose a stabilized formulation based on a penalization of the residual of a weak Stokes problem on the open boundary, whose viscous part controls the incoming convective energy, while the inertial term contributes to the kinetic energy. We also present different strategies for the approximation of the boundary pressure gradient, which is needed for defining the stabilization term. The method has the advantage that it does not require neither artificial modifications or extensions of the computational domain. Moreover, it is consistent with the Womersley solution. We illustrate our approach on numerical examples  - both academic and real-life -  relevant to blood and respiratory flows. The results also show that the stabilization parameter can be reduced with the mesh size.

P. Bringmann, C. Carstensen, Ch. Merdon, Guaranteed error control for the pseudostress approximation of the Stokes equations, Numerical Methods for Partial Differential Equations. An International Journal, 32 (2016), pp. 1411--1432.
Abstract
The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in $L^2$. Any standard mixed finite element function space can be utilized for this mixed formulation, e.g. the Raviart-Thomas discretization which is related to the Crouzeix-Raviart nonconforming finite element scheme in the lowest-order case. The effective and guaranteed a posteriori error control for this nonconforming velocity-oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf-sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy.

A. Ern, D. Di Pietro, A. Linke, F. Schieweck, A discontinuous skeletal method for the viscosity-dependent Stokes problem, Computer Methods in Applied Mechanics and Engineering, 306 (2016), pp. 175--195.
Abstract
We devise and analyze arbitrary-order nonconforming methods for the discretization of the viscosity-dependent Stokes equations on simplicial meshes. We keep track explicitly of the viscosity and aim at pressure-robust schemes that can deal with the practically relevant case of body forces with large curl-free part in a way that the discrete velocity error is not spoiled by large pressures. The method is inspired from the recent Hybrid High-Order (HHO) methods for linear elasticity. After elimination of the auxiliary variables by static condensation, the linear system to be solved involves only discrete face-based velocities, which are polynomials of degree k >=0, and cell-wise constant pressures. Our main result is a pressure-independent energy-error estimate on the velocity of order (k+1). The main ingredient to achieve pressure-independence is the use of a divergence-preserving velocity reconstruction operator in the discretization of the body forces. We also prove an L2-pressure estimate of order (k+1) and an L2-velocity estimate of order (k+2), the latter under elliptic regularity. The local mass and momentum conservation properties of the discretization are also established. Finally, two- and three-dimensional numerical results are presented to support the analysis.

A. Fiebach, A. Glitzky, A. Linke, Convergence of an implicit Voronoi finite volume method for reaction-diffusion problems, Numerical Methods for Partial Differential Equations. An International Journal, 32 (2016), pp. 141--174.
Abstract
We investigate the convergence of an implicit Voronoi finite volume method for reaction- diffusion problems including nonlinear diffusion in two space dimensions. The model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. The numerical scheme uses boundary conforming Delaunay meshes and preserves positivity and the dissipative property of the continuous system. Starting from a result on the global stability of the scheme (uniform, mesh-independent global upper and lower bounds), we prove strong convergence of the chemical activities and their gradients to a weak solution of the continuous problem. In order to illustrate the preservation of qualitative properties by the numerical scheme, we present a long-term simulation of the Michaelis-Menten-Henri system. Especially, we investigate the decay properties of the relative free energy and the evolution of the dissipation rate over several magnitudes of time, and obtain experimental orders of convergence for these quantities.

R. Haller-Dintelmann, A. Jonsson, D. Knees, J. Rehberg, Elliptic and parabolic regularity for second order divergence operators with mixed boundary conditions, Mathematical Methods in the Applied Sciences, 39 (2016), pp. 5007--5026, DOI 10.1002/mma.3484/abstract .
Abstract
We study second order equations and systems on non-Lipschitz domains including mixed boundary conditions. The key result is interpolation for suitable function spaces.

W. Huang, L. Kamenski, J. Lang, Stability of explicit one-step methods for P1-finite element approximation of linear diffusion equations on anisotropic meshes, SIAM Journal on Numerical Analysis, 54 (2016), pp. 1612--1634.
Abstract
We study the stability of explicit Runge-Kutta integration schemes for the linear finite element approximation of linear parabolic equations. The derived bound on the largest permissible time step is tight for any mesh and any diffusion matrix within a factor of 2 (d + 1), where d is the spatial dimension. Both full mass matrix and mass lumping are considered. The bound reveals that the stability condition is affected by two factors. The first one depends on the number of mesh elements and corresponds to the classic bound for the Laplace operator on a uniform mesh. The other factor reflects the effects of the interplay of the mesh geometry and the diffusion matrix. It is shown that it is not the mesh geometry itself but the mesh geometry in relation to the diffusion matrix that is crucial to the stability of explicit methods. When the mesh is uniform in the metric specified by the inverse of the diffusion matrix, the stability condition is comparable to the situation with the Laplace operator on a uniform mesh. Numerical results are presented to verify the theoretical findings.

M. Khodayari, P. Reinsberg, A.A. Abd-El-Latif, Ch. Merdon, J. Fuhrmann, H. Baltruschat, Determining solubility and diffusivity by using a flow cell coupled to a mass spectrometer, ChemPhysChem, 17 (2016), pp. 1647--1655.

N. Ahmed, G. Matthies, Numerical study of SUPG and LPS methods combined with higher order variational time discretization schemes applied to time-dependent convection-diffusion-reaction equations, Journal of Scientific Computing, 67 (2016), pp. 988--1018.
Abstract
This paper considers the numerical solution of time-dependent convection-diffusion-reaction equations. We shall employ combinations of streamline-upwind Petrov-Galerkin (SUPG) and local projection stabilization (LPS) methods in space with the higher order variational time discretization schemes. In particular, we consider time discretizations by discontinuous Galerkin (dG) methods and continuous Galerkin-Petrov (cGP) methods. Several numerical tests have been performed to assess the accuracy of combinations of spatial and temporal discretization schemes. Furthermore, the dependence of the results on the stabilization parameters of the spatial discretizations are discussed. Finally the long-time behavior of overshoots and undershoots is investigated.

A. Caiazzo, R. Guibert, I.E. Vignon-Clementel, A reduced-order modeling for efficient design study of artificial valve in enlarged ventricular outflow tracts, Computer Methods in Biomechanics and Biomedical Engineering, 19 (2016), pp. 1314--1318.
Abstract
A computational approach is proposed for efficient design study of a reducer stent to be percutaneously implanted in enlarged right ventricular outflow tracts (RVOT). The need for such a device is driven by the absence of bovine or artificial valves which could be implanted in these RVOT to replace the absent or incompetent native valve, as is often the case over time after Tetralogy of Fallot repair. Hemodynamics are simulated in the stented RVOT via a reduce order model based on proper orthogonal decomposition (POD), while the artificial valve is modeled as a thin resistive surface. The reduced order model is obtained from the numerical solution on a reference device configuration, then varying the geometrical parameters (diameter) for design purposes. To validate the approach, forces exerted on the valve and on the reducer are monitored, varying with geometrical parameters, and compared with the results of full CFD simulations. Such an approach could also be useful for uncertainty quantification.

W. Dreyer, C. Guhlke, M. Landstorfer, Theory and structure of the metal/electrolyte interface incorporating adsorption and solvation effects, Electrochimica Acta, 201 (2016), pp. 187--219.
Abstract
In this work we present a continuum theory for the metal/electrolyte interface which explicitly takes into account adsorption and partial solvation on the metal surface. It is based on a general theory of coupled thermo-electrodynamics for volumes and surfaces, utilized here in equilibrium and a 1D approximation. We provide explicit free energy models for the volumetric metal and electrolyte phases and derive a surface free energy for the species present on the metal surface. This surface mixture theory explicitly takes into account the very different amount of sites an adsorbate requires, originating from solvation effects on the surface. Additionally we account for electron transfer reactions on the surface and the associated stripping of the solvation shell. Based on our overall surface free energy we thus provide explicit expressions of the surface chemical potentials of all constituents. The equilibrium representations of the coverages and the overall charge are briefly summarized.

Our model is then used to describe two examples: (i) a silver single crystal electrode with (100) face in contact to a (0.01M NaF + 0.01M KPF6) aqueous solution, and (ii) a general metal surface in contact to some electrolytic solution AC for which an electron transfer reaction occurs in the potential range of interest. We reflect the actual modeling procedure for these examples and discuss the respective model parameters. Due to the representations of the coverages in terms of the applied potential we provide an adsorption map and introduce adsorption potentials. Finally we investigate the structure of the space charge layer at the metal/surface/electrolyte interface by means of numerical solutions of the coupled Poisson-momentum equation system for various applied potentials. It turns out that various layers self-consistently form within the overall space charge region, which are compared to historic and recent pictures of the double layer. Based on this we present new interpretations of what is known as inner and outer Helmholtz-planes and finally provide a thermodynamic consistent picture of the metal/electrolyte interface structure.

J. Fuhrmann, A numerical strategy for Nernst--Planck systems with solvation effect, Fuel Cells, 16 (2016), pp. 704--714.

J. Fuhrmann, A. Linke, Ch. Merdon, F. Neumann, T. Streckenbach, H. Baltruschat, M. Khodayari, Inverse modeling of thin layer flow cells for detection of solubility, transport and reaction coefficients from experimental data, Electrochimica Acta, 211 (2016), pp. 1--10.
Abstract
Thin layer flow cells are used in electrochemical research as experimental devices which allow to perform investigations of electrocatalytic surface reactions under controlled conditions using reasonably small electrolyte volumes. The paper introduces a general approach to simulate the complete cell using accurate numerical simulation of the coupled flow, transport and reaction processes in a flow cell. The approach is based on a mass conservative coupling of a divergence-free finite element method for fluid flow and a stable finite volume method for mass transport. It allows to perform stable and efficient forward simulations that comply with the physical bounds namely mass conservation and maximum principles for the involved species. In this context, several recent approaches to obtain divergence-free velocities from finite element simulations are discussed. In order to perform parameter identification, the forward simulation method is coupled to standard optimization tools. After an assessment of the inverse modeling approach using known realistic data, first results of the identification of solubility and transport data for O2 dissolved in organic electrolytes are presented. A plausibility study for a more complex situation with surface reactions concludes the paper and shows possible extensions of the scope of the presented numerical tools.

CH. Heinemann, K. Sturm, Shape optimisation for a class of semilinear variational inequalities with applications to damage models, SIAM Journal on Mathematical Analysis, 48 (2016), pp. 3579--3617, DOI 10.1137/16M1057759 .
Abstract
The present contribution investigates shape optimisation problems for a class of semilinear elliptic variational inequalities with Neumann boundary conditions. Sensitivity estimates and material derivatives are firstly derived in an abstract operator setting where the operators are defined on polyhedral subsets of reflexive Banach spaces. The results are then refined for variational inequalities arising from minimisation problems for certain convex energy functionals considered over upper obstacle sets in $H^1$. One particularity is that we allow for dynamic obstacle functions which may arise from another optimisation problems. We prove a strong convergence property for the material derivative and establish state-shape derivatives under regularity assumptions. Finally, as a concrete application from continuum mechanics, we show how the dynamic obstacle case can be used to treat shape optimisation problems for time-discretised brittle damage models for elastic solids. We derive a necessary optimality system for optimal shapes whose state variables approximate desired damage patterns and/or displacement fields.

A. Linke, Ch. Merdon, On velocity errors due to irrotational forces in the Navier--Stokes momentum balance, Journal of Computational Physics, 313 (2016), pp. 654--661.
Abstract
This contribution studies the influence of the pressure on the velocity error in finite element discretisations of the Navier--Stokes equations. Three simple benchmark problems that are all close to real-world applications convey that the pressure can be comparably large and is not to be underestimated. For widely used finite element methods like the Taylor--Hood finite element method, such relatively large pressures can lead to spurious oscillations and arbitrarily large errors in the velocity, even if the exact velocity is in the ansatz space. Only mixed finite element methods, whose velocity error is pressure-independent, like the Scott--Vogelius finite element method can avoid this influence.

A. Linke, Ch. Merdon, Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier--Stokes equations, Computer Methods in Applied Mechanics and Engineering, 311 (2016), pp. 304--326.
Abstract
Recently, it was understood how to repair a certain L2-orthogonality of discretely-divergence-free vector fields and gradient fields such that the velocity error of inf-sup stable discretizations for the incompressible Stokes equations becomes pressure-independent. These new 'pressure-robust' Stokes discretizations deliver a small velocity error, whenever the continuous velocity field can be well approximated on a given grid. On the contrary, classical inf-sup stable Stokes discretizations can guarantee a small velocity error only, when both the velocity and the pressure field can be approximated well, simultaneously.
In this contribution, 'pressure-robustness' is extended to the time-dependent Navier--Stokes equations. In particular, steady and time-dependent potential flows are shown to build an entire class of benchmarks, where pressure-robust discretizations can outperform classical approaches significantly. Speedups will be explained by a new theoretical concept, the 'discrete Helmholtz projector' of an inf-sup stable discretization. Moreover, different discrete nonlinear convection terms are discussed, and skew-symmetric pressure-robust discretizations are proposed.

P.-É. Druet, Some mathematical problems related to the second order optimal shape of a crystallization interface, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 2443--2463.
Abstract
We consider the problem to optimize the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimization principle of the free energy, while the temperature is solving the heat equation with a radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallization, the interface is expected to have a global graph representation. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second surface derivatives, and for the surface temperature gradient.

M. Liero, Th. Koprucki, A. Fischer, R. Scholz, A. Glitzky, p-Laplace thermistor modeling of electrothermal feedback in organic semiconductors, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 66 (2015), pp. 2957--2977.
Abstract
In large-area Organic Light-Emitting Diodes (OLEDs) spatially inhomogeneous luminance at high power due to inhomogeneous current flow and electrothermal feedback can be observed. To describe these self-heating effects in organic semiconductors we present a stationary thermistor model based on the heat equation for the temperature coupled to a p-Laplace-type equation for the electrostatic potential with mixed boundary conditions. The p-Laplacian describes the non-Ohmic electrical behavior of the organic material. Moreover, an Arrhenius-like temperature dependency of the electrical conductivity is considered. We introduce a finite-volume scheme for the system and discuss its relation to recent network models for OLEDs. In two spatial dimensions we derive a priori estimates for the temperature and the electrostatic potential and prove the existence of a weak solution by Schauder's fixed point theorem.

E. Meca Álvarez, I. Mercader, L. Ramirez-Piscina, Transitions between symmetric and nonsymmetric regimes in binary-mixture convection, Physica D. Nonlinear Phenomena, 303 (2015), pp. 39--49.

D. Peschka, M. Thomas, A. Glitzky, R. Nürnberg, K. Gärtner, M. Virgilio, S. Guha, G. Capellini, Th. Koprucki, Th. Schröder, Modeling of edge-emitting lasers based on tensile strained germanium microstrips, IEEE Photonics Journal, 7 (2015), pp. 1502115/1--1502115/15, DOI 10.1109/JPHOT.2015.2427093 .
Abstract
In this paper we present a thorough modeling of an edge-emitting laser based on strained germanium microstrips. The full band structure of the tensile strained germanium (Ge) layer enters the calculation of optical properties. Material gain for strained Ge is used in the two-dimensional simulation of the carrier transport and of the optical field within a cross section of the microstrips orthogonal to the optical cavity. We study optoelectronic properties of the device for two different designs. The simulation results are very promising as they show feasible ways towards Ge emitter devices with lower threshold currents and higher efficiency as published insofar.

M. Radszuweit, E. Alvarez-Lacalle, M. Bär, B. Echebarria, Cardiac contraction induces discordant alternans and localized block, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 91 (2015), pp. 022703/1--022703/12.
Abstract
In this paper we use a simplified model of cardiac excitation-contraction coupling to study the effect of tissue deformation on the dynamics of alternans, i.e. alternations in the duration of the cardiac action potential, that occur at fast pacing rates and are known to be pro-arrhythmic. We show that small stretch-activated currents can produce large effects and cause a transition from in-phase to off-phase alternations (i.e. from concordant to discordant alternans) and to conduction blocks. We demonstrate numerically and analytically that this effect is the result of a generic change in the slope of the conduction velocity restitution curve due to electromechanical coupling. Thus, excitation-contraction coupling can potentially play a relevant role in the transition to reentry and fibrillation.

CH. Brennecke, A. Linke, Ch. Merdon, J. Schöberl, Optimal and pressure-independent $L^2$ velocity error estimates for a modified Crouzeix--Raviart Stokes element with BDM reconstructions, Journal of Computational Mathematics, 33 (2015), pp. 191--208.
Abstract
Nearly all inf-sup stable mixed finite elements for the incompressible Stokes equations relax the divergence constraint. The price to pay is that a priori estimates for the velocity error become pressure-dependent, while divergence-free mixed finite elements deliver pressure-independent estimates. A recently introduced new variational crime using lowest-order Raviart-Thomas velocity reconstructions delivers a much more robust modified Crouzeix-Raviart element, obeying an optimal pressure-independent discrete H¹ velocity estimate. Refining this approach, a more sophisticated variational crime employing the lowest-order BDM element is proposed, which also allows proving an optimal pressure independent L² velocity error. Numerical examples confirm the analysis and demonstrate the improved robustness in the Navier-Stokes case.

W. Huang, L. Kamenski, R.D. Russell, A comparative numerical study of meshing functionals for variational mesh adaptation, Journal of Mathematical Study, 48 (2015), pp. 168--186.
Abstract
We present a comparative numerical study for three functionals used for variational mesh adaptation. One of them is a generalization of Winslow's variable diffusion functional while the others are based on equidistribution and alignment. These functionals are known to have nice theoretical properties and work well for most mesh adaptation problems either as a stand-alone variational method or combined within the moving mesh framework. Their performance is investigated numerically in terms of equidistribution and alignment mesh quality measures. Numerical results in 2D and 3D are presented.

W. Huang, L. Kamenski, A geometric discretization and a simple implementation for variational mesh generation and adaptation, Journal of Computational Physics, 301 (2015), pp. 322--337.
Abstract
We present a simple direct discretization for functionals used in the variational mesh generation and adaptation. Meshing functionals are discretized on simplicial meshes and the Jacobian matrix of the continuous coordinate transformation is approximated by the Jacobian matrices of affine mappings between elements. The advantage of this direct geometric discretization is that it preserves the basic geometric structure of the continuous functional, which is useful in preventing strong decoupling or loss of integral constraints satisfied by the functional. Moreover, the discretized functional is a function of the coordinates of mesh vertices and its derivatives have a simple analytical form, which allows a simple implementation of variational mesh generation and adaptation on computer. Since the variational mesh adaptation is the base for a number of adaptive moving mesh and mesh smoothing methods, the result in this work can be used to develop simple implementations of those methods. Numerical examples are given.

R. Huth, S. Jachalski, G. Kitavtsev, D. Peschka, Gradient flow perspective on thin-film bilayer flows, Journal of Engineering Mathematics, 94 (2015), pp. 43--61.
Abstract
We study gradient flow formulations of thin-film bilayer flows with triple-junctions between liquid/liquid/air. First we highlight the gradient structure in the Stokes free-boundary flow and identify its solutions with the well known PDE with boundary conditions. Next we propose a similar gradient formulation for the corresponding thin-film model and formally identify solutions with those of the corresponding free-boundary problem. A robust numerical algorithm for the thin-film gradient flow structure is then provided. Using this algorithm we compare the sharp triple-junction model with precursor models. For their stationary solutions a rigorous connection is established using Gamma-convergence. For time-dependent solutions the comparison of numerical solutions shows a good agreement for small and moderate times. Finally we study spreading in the zero-contact angle case, where we compare numerical solutions with asymptotically exact source-type solutions.

T. Roubíček, M. Thomas, Ch. Panagiotopoulos, Stress-driven local-solution approach to quasistatic brittle delamination, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 22 (2015), pp. 645--663.
Abstract
A unilateral contact problem between elastic bodies at small strains glued by a brittle adhesive is addressed in the quasistatic rate-independent setting. The delamination process is modelled as governed by stresses rather than by energies. This results in a specific scaling of an approximating elastic adhesive contact problem, discretised by a semi-implicit scheme and regularized by a BV-type gradient term. An analytical zero-dimensional example motivates the model and a specific local-solution concept. Two-dimensional numerical simulations performed on an engineering benchmark problem of debonding a fiber in an elastic matrix further illustrate the validity of the model, convergence, and algorithmical efficiency even for very rigid adhesives with high elastic moduli.

N. Ahmed, G. Matthies, Higher order continuous Galerkin--Petrov time stepping schemes for transient convection-diffusion-reaction equations, ESAIM: Mathematical Modelling and Numerical Analysis, 49 (2015), pp. 1429--1450.
Abstract
We present the analysis for the higher order continuous Galerkin--Petrov (cGP) time discretization schemes in combination with the one-level local projection stabilization in space applied to time-dependent convection-diffusion-reaction problems. Optimal a-priori error estimates will be proved. Numerical studies support the theoretical results. Furthermore, a numerical comparison between continuous Galerkin--Petrov and discontinuous Galerkin time discretization schemes will be given.

N. Ahmed, V. John, Adaptive time step control for higher order variational time discretizations applied to convection-diffusion equations, Computer Methods in Applied Mechanics and Engineering, 285 (2015), pp. 83--101.
Abstract
Higher order variational time stepping schemes allow an efficient post-processing for computing a higher order solution. This paper presents an adaptive algorithm whose time step control utilizes the post-processed solution. The algorithm is applied to convection-dominated convection-diffusion equations. It is shown that the length of the time step properly reflects the dynamics of the solution. The numerical costs of the adaptive algorithm are discussed.

A. Caiazzo, G. Montecinos, L.O. Müller, E.M. Haacke, E.F. Toro, Computational haemodynamics in stenotic internal jugular veins, Journal of Mathematical Biology, 70 (2015), pp. 745--772.
Abstract
Stenosis in internal jugular veins (IJVs) are frequently associated to pathological venous circulation and insufficient cerebral blood drainage. In this work, we set up a computational framework to assess the relevance of IJV stenoses through numerical simulation, combining medical imaging, patient-specific data and a mathematical model for venous occlusions. Coupling a three-dimensional (3D) description of blood flow in IJVs with a reduced one-dimesional model (1D) for major intracranial veins, we are able to model different anatomical configurations, an aspect of importance to understand the impact of IJV stenosis in intracranial venous haemodynamics. We investigate several stenotic configurations in a physiologic patient-specific regime, quantifying the effect of the stenosis in terms of venous pressure increase and wall shear stress patterns. Simulation results are in qualitative agreement with reported pressure anomalies in pathological cases. Moreover, they demonstrate the potential of the proposed multiscale framework for individual-based studies and computer-aided diagnosis.

J. Fuhrmann, Comparison and numerical treatment of generalized Nernst--Planck models, Computer Physics Communications. An International Journal and Program Library for Computational Physics and Physical Chemistry, 196 (2015), pp. 166--178.
Abstract
In its most widespread, classical formulation, the Nernst-Planck-Poisson system for ion transport in electrolytes fails to take into account finite ion sizes. As a consequence, it predicts unphysically high ion concentrations near electrode surfaces. Historical and recent approaches to an approriate modification of the model are able to fix this problem. Several appropriate formulations are compared in this paper. The resulting equations are reformulated using absolute activities as basic variables describing the species amounts. This reformulation allows to introduce a straightforward generalisation of the Scharfetter-Gummel finite volume discretization scheme for drift-diffusion equations. It is shown that it is thermodynamically consistent in the sense that the solution of the corresponding discretized generalized Poisson-Boltzmann system describing the thermodynamic equilibrium is a stationary state of the discretized time-dependent generalized Nernst-Planck system. Numerical examples demonstrate the improved physical correctness of the generalised models and the feasibility of the numerical approach.

CH. Heinemann, Ch. Kraus, Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 2565--2590.
Abstract
In this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolic-parabolic system. To this end, we first generalize the notion of weak solution introduced in WIAS 1520. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques.

CH. Heinemann, E. Rocca, Damage processes in thermoviscoelastic materials with damage-dependent thermal expansion coefficients, Mathematical Methods in the Applied Sciences, 38 (2015), pp. 4587--4612.
Abstract
In this paper we prove existence of global in time weak solutions for a highly nonlinear PDE system arising in the context of damage phenomena in thermoviscoelastic materials. The main novelty of the present contribution with respect to the ones already present in the literature consists in the possibility of taking into account a damage-dependent thermal expansion coefficient. This term implies the presence of nonlinear couplings in the PDE system, which make the analysis more challenging.

A. Linke, Ch. Merdon, Guaranteed energy error estimators for a modified robust Crouzeix--Raviart Stokes element, Journal of Scientific Computing, 64 (2015), pp. 541--558.
Abstract
This paper provides guaranteed upper energy error bounds for a modified lowest-order nonconforming Crouzeix--Raviart finite element method for the Stokes equations. The modification from [A. Linke 2014, On the role of the Helmholtz-decomposition in mixed methods for incompressible flows and a new variational crime] is based on the observation that only the divergence-free part of the right-hand side should balance the vector Laplacian. The new method has optimal energy error estimates and can lead to errors that are smaller by several magnitudes, since the estimates are pressure-independent. An efficient a posteriori velocity error estimator for the modified method also should involve only the divergence-free part of the right-hand side. Some designs to approximate the Helmholtz projector are compared and verified by numerical benchmark examples. They show that guaranteed error control for the modified method is possible and almost as sharp as for the unmodified method.

A. Mielke, J. Naumann, Global-in-time existence of weak solutions to Kolmogorov's two-equation model of turbulence, Comptes Rendus Mathematique. Academie des Sciences. Paris, 353 (2015), pp. 321--326.
Abstract
We consider Kolmogorov's model for the turbulent motion of an incompressible fluid in ℝ³. This model consists in a Navier-Stokes type system for the mean flow u and two further partial differential equations: an equation for the frequency ω and for the kinetic energy k each. We investigate this system of partial differential equations in a cylinder Ω x ]0,T[ (Ω ⊂ ℝ³ cube, 0 < T < +∞) under spatial periodic boundary conditions on ∂Ω x ]0,T[ and initial conditions in Ω x {0}. We present an existence result for a weak solution {u, ω, k} to the problem under consideration, with ω, k obeying the inequalities and .

R.M. Arkhipov, I. Babushkin, M.K. Lebedev, Y.A. Tolmachev, M.V. Arkhipov, Transient Cherenkov radiation from an inhomogeneous string excited by an ultrashort laser pulse at superluminal velocity, Physical Review A, 89 (2014), pp. 043811/1--043811/10.
Abstract
An optical response of one-dimensional string made of dipoles with a periodically varying density excited by a spot of light moving along the string at the superluminal (subluminal) velocity is studied. We consider in details the spectral and temporal dynamics of the Cherenkov radiation, which occurs in such system in the transient regime. We point out the resonance character of radiation and the appearance of a new Doppler-like frequency in the spectrum of the transient Cherenkov radiation. Possible applications of the effect as well as different string topologies are discussed

A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Computer Methods in Applied Mechanics and Engineering, 268 (2014), pp. 782--800.
Abstract
According to the Helmholtz decomposition, the irrotational parts of the momentum balance equations of the incompressible Navier-Stokes equations are balanced by the pressure gradient. Unfortunately, nearly all mixed methods for incompressible flows violate this fundamental property, resulting in the well-known numerical instability of poor mass conservation. The origin of this problem is the lack of L2-orthogonality between discretely divergence-free velocities and irrotational vector fields. In order to cure this, a new variational crime using divergence-free velocity reconstructions is proposed. Applying lowest order Raviart-Thomas velocity reconstructions to the nonconforming Crouzeix-Raviart element allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence-free flow solvers. In the Stokes case, optimal a-priori error estimates for the velocity gradients and the pressure are derived. Moreover, the discrete velocity is independent of the continuous pressure. Several detailed linear and nonlinear numerical examples illustrate the theoretical findings.

E. Jenkins, V. John, A. Linke, L.G. Rebholz, On the parameter choice in grad-div stabilization for the Stokes equations, Advances in Computational Mathematics, 40 (2014), pp. 491--516.
Abstract
Grad-div stabilization has been proved to be a very useful tool in discretizations of incompressible flow problems. Standard error analysis for inf-sup stable conforming pairs of finite element spaces predicts that the stabilization parameter should be optimally chosen to be O(1). This paper revisits this choice for the Stokes equations on the basis of minimizing the $H^1$ error of the velocity and the $L^2$ error of the pressure. It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle than known so far in the literature. It depends on the used norm, the solution, the family of finite element spaces, and the type of mesh. Depending on the situation, the optimal stabilization parameter might range from being very small to very large. The analytic results are supported by numerical examples.

R. Eymard, J. Fuhrmann, A. Linke, On MAC schemes on triangular Delaunay meshes, their convergence and application to coupled flow problems, Numerical Methods for Partial Differential Equations. An International Journal, 30 (2014), pp. 1397--1424.
Abstract
We study two classical generalized MAC schemes on unstructured triangular Delaunay meshes for the incompressible Stokes and Navier-Stokes equations and prove their convergence for the first time. These generalizations use the duality between Voronoi and triangles of Delaunay meshes, in order to construct two staggered discretization schemes. Both schemes are especially interesting, since compatible finite volume discretizations for coupled convection-diffusion equations can be constructed which preserve discrete maximum principles. In the first scheme, called tangential velocity scheme, the pressures are defined at the vertices of the mesh, and the discrete velocities are tangential to the edges of the triangles. In the second scheme, called normal velocity scheme, the pressures are defined in the triangles, and the discrete velocities are normal to the edges of the triangles. For both schemes, we prove the convergence in $L^2$ for the velocities and the discrete rotations of the velocities for the Stokes and the Navier-Stokes problem. Further, for the normal velocity scheme, we also prove the strong convergence of the pressure in $L^2$. Linear and nonlinear numerical examples illustrate the theoretical predictions.

A. Fiebach, A. Glitzky, A. Linke, Uniform global bounds for solutions of an implicit Voronoi finite volume method for reaction-diffusion problems, Numerische Mathematik, 128 (2014), pp. 31--72.
Abstract
We consider discretizations for reaction-diffusion systems with nonlinear diffusion in two space dimensions. The applied model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. We propose an implicit Voronoi finite volume discretization on regular Delaunay meshes that allows to prove uniform, mesh-independent global upper and lower $L^infty$ bounds for the chemical potentials. These bounds provide the main step for a convergence analysis for the full discretized nonlinear evolution problem. The fundamental ideas are energy estimates, a discrete Moser iteration and the use of discrete Gagliardo-Nirenberg inequalities. For the proof of the Gagliardo-Nirenberg inequalities we exploit that the discrete Voronoi finite volume gradient norm in $2d$ coincides with the gradient norm of continuous piecewise linear finite elements.

A. Gloria, S. Neukamm, F. Otto, An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), pp. 325--346.
Abstract
We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L²-norm in probability of the H¹-norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Greens function by Marahrens and the third author.

R. Guibert, K. Mcleod, A. Caiazzo, T. Mansi, Group-wise construction of reduced models for understanding and characterization of pulmonary blood flows from medical images, Medical Image Analysis, 18 (2014), pp. 63--82.

A. Caiazzo, V. John, U. Wilbrandt, On classical iterative subdomain methods for the Stokes--Darcy problem, Computer & Geosciences, 18 (2014), pp. 711--728.
Abstract
Iterative subdomain methods for the Stokes--Darcy problem that use Robin boundary conditions on the interface are reviewed. Their common underlying structure and their main differences are identified. In particular, it is clarified that there are different updating strategies for the interface conditions. For small values of fluid viscosity and hydraulic permeability, which are relevant in applications from geosciences, it is shown in numerical studies that only one of these updating strategies leads to an efficient numerical method, if this strategy is used in combination with appropriate parameters in the Robin boundary conditions. In particular, it is observed that the values of appropriate parameters are larger than those proposed so far. Not only the size but also the ratio of appropriate Robin parameters depends on the coefficients of the problem.

A. Caiazzo, J. Mura, Multiscale modeling of weakly compressible elastic materials in harmonic regime and application to microscale structure estimation, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 12 (2014), pp. 514--537.
Abstract
This article is devoted to the modeling of elastic materials composed by an incompressible elastic matrix and small compressible gaseous inclusions, under a time harmonic excitation. In a biomedical context, this model describes the dynamics of a biological tissue (e.g. lung or liver) when wave analysis methods (such as Magnetic Resonance Elastography) are used to estimate tissue properties. Due to the multiscale nature of the problem, direct numerical simulations are prohibitive. We extend the homogenized model introduced in [Baffico, Grandmont, Maday, Osses, SIAM J. Mult. Mod. Sim., 7(1), 2008] to a time harmonic regime to describe the solid-gas mixture from a macroscopic point of view in terms of an effective elasticity tensor. Furthermore, we derive and validate numerically analytical approximations for the effective elastic coefficients in terms of macroscopic parameters. This simplified description is used to to set up an efficient variational approach for the estimation of the tissue porosity, using the mechanical response to external harmonic excitations.

L. Kamenski, W. Huang, A study on the conditioning of finite element equations with arbitrary anisotropic meshes via a density function approach, Journal of Mathematical Study, 47 (2014), pp. 151--172.

L. Kamenski, W. Huang, How a nonconvergent recovered Hessian works in mesh adaptation, SIAM Journal on Numerical Analysis, 52 (2014), pp. 1692--1708.

L. Kamenski, W. Huang, H. Xu, Conditioning of finite element equations with arbitrary anisotropic meshes, Mathematics of Computation, 83 (2014), pp. 2187--2211.

A. Pérez-Serrano, J. Javaloyes, S. Balle, Directional reversals and multimode dynamics in semiconductor ring lasers, Physical Review A, 89 (2014), pp. 023818/1--023818/14.
Abstract
We investigate the dynamics of longitudinal modes in quantum-well semiconductor ring lasers by means of a spatio-temporal travelling wave model. We report the existence of a novel multimode instability in such a system that provokes a periodic deterministic directional reversal involving jumps between consecutive longitudinal modes. The switching sequence follows the modal frequencies from blue to red, and every modal jump is accompanied by a reversal of the direction of emission. We characterize and analyze such instability via the bifurcation analysis of the full travelling wave model as well as by performing the linear stability analysis of the monochromatic solutions.

K. Götze, Strong solutions for the interaction of a rigid body and a viscoelastic fluid, Journal of Mathematical Fluid Mechanics, 15 (2013), pp. 663--688.
Abstract
We study a coupled system of equations describing the movement of a rigid body which is immersed in a viscoelastic fluid. It is shown that under natural assumptions on the data and for general goemetries of the rigid body, excluding contact scenarios, a unique local-in-time strong solution exists.

M. Liero, A. Mielke, Gradient structures and geodesic convexity for reaction-diffusion systems, Philosophical Transactions of the Royal Society A : Mathematical, Physical & Engineering Sciences, 371 (2013), pp. 20120346/1--20120346/28.
Abstract
We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory. We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory.

A. Pérez-Serrano, J. Javaloyes, S. Balle, Multi-channel wavelength conversion using four-wave mixing in semiconductor ring lasers, IEEE Phot. Tech. Letter, 25 (2013), pp. 476--479.
Abstract
We theoretically study all-optical simultaneous wavelength conversion of multiple channels by four-wave mixing in semiconductor ring lasers. Locking the semiconductor ring laser to a holding beam allows to achieve large conversion efficiencies with good signal-tonoise ratio in several channels at multi-Gb/s bit rates. Cross-talk between signals, arising from the peculiar four-wave mixing cascade of modes in semiconductor ring lasers and their cross-gain saturation, is studied in detail. We show that it can be controlled by adjusting the intensity of the holding beam, the bias current of the laser and the number, intensity and wavelength of signals that one wants to convert.

A. Pérez-Serrano, J. Javaloyes, S. Balle, Spectral delay algebraic equation approach to broad area laser diodes, IEEE J. Select. Topics Quantum Electron., 19 (2013), pp. 1502808/1--1502808/8.

M.V. Arkhipov, R.M. Arkhipov, S.A. Pulkin, Effects of inversionless oscillation in two-level media from the point of view of specificities of the spatiotemporal propagation dynamics of radiation, Optics and Spectroscopy, 114 (2013), pp. 831--837.
Abstract
We report the results of computer simulation of the emission of radiation by an extended twolevel medium in a ring cavity. The cases of using strong external monochromatic, quasimonochromatic, and biharmonic radiation for pumping the twolevel medium are analyzed. It is shown that the emission of radiation with spectral content different from that of the pump radiation, which is interpreted as the inversionless oscillation, is the result of the spatiotemporal dynamics of light propagation in an extended twolevel medium imbedded in a cavity. The appearance of this radiation is not related to known resonances of amplification of a weak probe field in a thin layer of the twolevel system (the effect of inversionless oscillation) induced by strong resonance monochromatic or biharmonic field, as was thought before.

C. Bertoglio, A. Caiazzo, M.A. Fernàndez, Fractional-step schemes for the coupling of distributed and lumped models in hemodynamics, SIAM Journal on Scientific Computing, 35 (2013), pp. B551--B575.

A. Bradji, J. Fuhrmann, Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes, Applications of Mathematics, 58 (2013), pp. 1--38.
Abstract
A general class of nonconforming meshes has been recently studied for stationary anisotropic heterogeneous diffusion problems by R. Eymard and coworkers. Thanks to these basic ideas developed for stationary problems, we derive a new discretization scheme in order to approximate the nonstationary heat problem. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. Although the numerical scheme stems from the finite volume method, its formulation is based on the discrete version for the weak formulation defined for the heat problem. We derive error estimates for the solution in discrete norm, and an error estimate for an approximation of the gradient, in a general framework in which the discrete bilinear form is satisfying ellipticity. We prove in particular, that, when the discrete flux is calculated using a stabilized discrete gradient, the convergence order is h+k , where h (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularity assumption that the exact solution is twice continuously differentiable in time and space. These error estimates are useful because they allow us to get error estimates for the approximations of the exact solution and its first derivatives.

B. Cousins, S. Le Borne, A. Linke, Z. Wang, Efficient linear solvers for incompressible flow simulations using Scott--Vogelius finite elements, Numerical Methods for Partial Differential Equations. An International Journal, 29 (2013), pp. 1217--1237.
Abstract
Recent research has shown that in some practically relevant situations like multiphysics flows (Galvin et al., Comput Methods Appl Mech Eng, 2012) divergence-free mixed finite elements may have a significantly smaller discretization error than standard nondivergence-free mixed finite elements. To judge the overall performance of divergence-free mixed finite elements, we investigate linear solvers for the saddle point linear systems arising in Scott-Vogelius finite element implementations of the incompressible Navier-Stokes equations. We investigate both direct and iterative solver methods. Due to discontinuous pressure elements in the case of Scott-Vogelius (SV) elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements such as Taylor-Hood elements. For direct methods, we extend recent preliminary work using sparse banded solvers on the penalty method formulation to finer meshes and discuss extensions. For iterative methods, we test augmented Lagrangian and H -LU preconditioners with GMRES, on both full and statically condensed systems. Several numerical experiments are provided that show these classes of solvers are well suited for use with SV elements and could deliver an interesting overall performance in several applications.

D. Knees, A. Schröder, Computational aspects of quasi-static crack propagation, Discrete and Continuous Dynamical Systems -- Series S, 6 (2013), pp. 63--99.
Abstract
The focus of this note lies on the numerical analysis of models describing the propagation of a single crack in a linearly elastic material. The evolution of the crack is modeled as a rate-independent process based on the Griffith criterion. We follow two different approaches for setting up mathematically well defined models: the global energetic approach and an approach based on a viscous regularization. We prove the convergence of solutions of fully discretized models (i.e. with respect to time and space) and derive relations between the discretization parameters (mesh size, time step size, viscosity parameter, crack increment) which guarantee the convergence of the schemes. Further, convergence rates are provided for the approximation of energy release rates by certain discrete energy release rates. Thereby we discuss both, models with self-contact conditions on the crack faces as well as models with pure Neumann conditions on the crack faces. The convergence proofs rely on regularity estimates for the elastic fields close to the crack tip and local and global finite element error estimates. Finally the theoretical results are illustrated with some numerical calculations.

W. Dreyer, C. Guhlke, R. Müller, Overcoming the shortcomings of the Nernst--Planck model, Physical Chemistry Chemical Physics, 15 (2013), pp. 7075--7086, DOI 10.1039/C3CP44390F .
Abstract
This is a study on electrolytes that takes a thermodynamically consistent coupling between mechanics and diffusion into account. It removes some inherent deficiencies of the popular Nernst-Planck model. A boundary problem for equilibrium processes is used to illustrate the new features of our model.

A. Linke, L. Rebholz, On a reduced sparsity stabilization of grad-div type for incompressible flow problems, Computer Methods in Applied Mechanics and Engineering, 261--262 (2013), pp. 142--153.
Abstract
We introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems. We show this new operator has a similar positive effect on velocity error as the well-known and very successful grad-div stabilization operator, but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix. That is, while grad-div produces fully coupled block matrices (i.e. block-full), the matrices arising from the new operator are block-upper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operator's matrices are identical to those of grad-div. We provide error estimates and numerical examples for finite element simulations with the new operator, which reveals the significant improvement in accuracy it can provide. Solutions found using the new operator are also compared to those using usual grad-div stabilization, and in all cases, solutions are found to be very similar.

A. Mielke, R. Rossi, G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations, Calculus of Variations and Partial Differential Equations, 46 (2013), pp. 253--310.
Abstract
In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional, for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a time-discretization scheme with variational techniques. Finally, we discuss an application to a material model in finite-strain elasticity.

A. Mielke, E. Rohan, Homogenization of elastic waves in fluid-saturated porous media using the Biot model, Mathematical Models & Methods in Applied Sciences, 23 (2013), pp. 873--916.
Abstract
We consider periodically heterogeneous fluid-saturated poroelastic media described by the Biot model with inertia effects. The weak and semistrong formulations for displacement, seepage and pressure fields involve three equations expressing the momentum and mass balance and the Darcy law. Using the two-scale homogenization method we obtain the limit two-scale problem and prove the existence and uniqueness of its weak solutions. The Laplace transformation in time is used to decouple the macroscopic and microscopic scales. It is shown that the seepage velocity is eliminated form the macroscopic equations involving strain and pressure fields only. The plane harmonic wave propagation is studied using an example of layered medium. Illustrations show some influence of the orthotropy on the dispersion phenomena.

P.N. Racec, S. Schade, H.-Chr. Kaiser, Eigensolutions of the Wigner--Eisenbud problem for a cylindrical nanowire within finite volume method, Journal of Computational Physics, 252 (2013), pp. 52--64.
Abstract
We present a finite volume method for computing a representative range of eigenvalues and eigenvectors of the Schrödinger operator on a three dimensional cylindrically symmetric bounded domain with mixed boundary conditions. More specifically, we deal with a semiconductor nanowire which consists of a dominant host material and contains heterostructure features such as double-barriers or quantum dots. The three dimensional Schrödinger operator is reduced to a family of two dimensional Schrödinger operators distinguished by a centrifugal potential. Ultimately, we numerically treat them by means of a finite volume method. We consider a uniform, boundary conforming Delaunay mesh, which additionally conforms to the material interfaces. The 1/r singularity is eliminated by approximating r at the vertexes of the Voronoi boxes. We study how the anisotropy of the effective mass tensor acts on the uniform approximation of the first K eigenvalues and eigenvectors and their sequential arrangement. There exists an optimal uniform Delaunay discretization with matching anisotropy. This anisotropic discretization yields best accuracy also in the presence of a mildly varying scattering potential, shown exemplarily for a nanowire resonant tunneling diode. For potentials with 1/r singularity one retrieves the theoretically established first order convergence, while the second order convergence is recovered only on uniform grids with an anisotropy correction.

S. Amiranashvili, U. Bandelow, A. Mielke, Calculation of ultrashort pulse propagation based on rational approximations for medium dispersion, Optical and Quantum Electronics, 44 (2012), pp. 241--246.
Abstract
Ultrashort optical pulses contain only a fewoptical cycles and exhibit broad spectra. Their carrier frequency is therefore not well defined and their description in terms of the standard slowly varying envelope approximation becomes questionable. Existing modeling approaches can be divided in two classes, namely generalized envelope equations, that stem from the nonlinear Schrödinger equation, and non-envelope equations which treat the field directly. Based on fundamental physical rules we will present an approach that effectively interpolates between these classes and provides a suitable setting for accurate and highly efficient

K. Galvin, A. Linke, L. Rebholz, N. Wilson, Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection, Computer Methods in Applied Mechanics and Engineering, 237--240 (2012), pp. 166--176.
Abstract
We consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotation-free forcing in the momentum equation. We provide analysis that suggests for such problems, obtaining accurate solutions necessitates either the use of pointwise divergence-free finite elements (such as Scott-Vogelius), or heavy grad-div stabilization of weakly divergence-free elements. The theory is demonstrated in numerical experiments for a benchmark natural convection problem, where large irrotational forcing occurs with high Rayleigh numbers.

K. Hackl, S. Heinz, A. Mielke, A model for the evolution of laminates in finite-strain elastoplasticity, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 92 (2012), pp. 888--909.
Abstract
We study the time evolution in elastoplasticity within the rate-independent framework of generalized standard materials. Our particular interest is the formation and the evolution of microstructure. Providing models where existence proofs are possible is a challenging task since the presence of microstructure comes along with a lack of convexity and, hence, compactness arguments cannot be applied to prove the existence of solutions. In order to overcome this problem, we will incorporate information on the microstructure into the internal variable, which is still compatible with generalized standard materials. More precisely, we shall allow for such microstructure that is given by simple or sequential laminates. We will consider a model for the evolution of these laminates and we will prove a theorem on the existence of solutions to any finite sequence of time-incremental minimization problems. In order to illustrate the mechanical consequences of the theory developed some numerical results, especially dealing with the rotation of laminates, are presented.

A. Glitzky, An electronic model for solar cells including active interfaces and energy resolved defect densities, SIAM Journal on Mathematical Analysis, 44 (2012), pp. 3874--3900.
Abstract
We introduce an electronic model for solar cells taking into account heterostructures with active interfaces and energy resolved volume and interface trap densities. The model consists of continuity equations for electrons and holes with thermionic emission transfer conditions at the interface and of ODEs for the trap densities with energy level and spatial position as parameters, where the right hand sides contain generation-recombination as well as ionization reactions. This system is coupled with a Poisson equation for the electrostatic potential. We show the thermodynamic correctness of the model and prove a priori estimates for the solutions to the evolution system. Moreover, existence and uniqueness of weak solutions of the problem are proven. For this purpose we solve a regularized problem and verify bounds of the corresponding solution not depending on the regularization level.

D. Knees, A. Schröder, Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints, Mathematical Methods in the Applied Sciences, 35 (2012), pp. 1859--1884.
Abstract
A global higher differentiability result in Besov spaces is proved for the displacement fields of linear elastic models with self contact. Domains with cracks are studied, where nonpenetration conditions/Signorini conditions are imposed on the crack faces. It is shown that in a neighborhood of crack tips (in 2D) or crack fronts (3D) the displacement fields are B ^3/2 _2,∞ regular. The proof relies on a difference quotient argument for the directions tangential to the crack. In order to obtain the regularity estimates also in the normal direction, an argument due to Ebmeyer/Frehse/Kassmann is modified. The methods are then applied to further examples like contact problems with nonsmooth rigid foundations, to a model with Tresca friction and to minimization problems with nonsmooth energies and constraints as they occur for instance in the modeling of shape memory alloys. Based on Falk's approximation Theorem for variational inequalities, convergence rates for FE-discretizations of contact problems are derived relying on the proven regularity properties. Several numerical examples illustrate the theoretical results.

A. Linke, L.G. Rebholz, N.E. Wilson, On the convergence rate of grad-div stabilized Taylor--Hood to Scott--Vogelius solutions for incompressible flow problems, Journal of Mathematical Analysis and Applications, 381 (2011), pp. 612--626.
Abstract
It was recently proven that, under mild restrictions, grad-div stabilized Taylor-Hood solutions of Navier-Stokes problems converge to the Scott-Vogelius solution of that same problem. However, even though the analytical rate was only shown to be $gamma^-frac12$ (where $gamma$ is the stabilization parameter), the computational results suggest the rate may be improvable $gamma^-1$. We prove herein the analytical rate is indeed $gamma^-1$, and extend the result to other incompressible flow problems including Leray-$alpha$ and MHD. Numerical results are given that verify the theory.

A.G. Vladimirov, R. Lefever, M. Tlidi, Relative stability of multipeak localized patterns of cavity solitons, Physical Review A, 84 (2011), pp. 043848/1--043848/4.
Abstract
We study the relative stability of different one-dimensional (1D) and two-dimensional (2D) clusters of closely packed localized peaks of the Swift-Hohenberg equation. In the 1D case, we demonstrate numerically the existence of a spatial Maxwell transition point where all clusters involving up to 15 peaks are equally stable. Above (below) this point, clusters become more (less) stable when their number of peaks increases. In the 2D case, since clusters involving more than two peaks may exhibit distinct spatial arrangements, this point splits into a set of Maxwell transition pointsWe study the relative stability of different one-dimensional (1D) and two-dimensional (2D) clusters of closely packed localized peaks of the Swift-Hohenberg equation. In the 1D case, we demonstrate numerically the existence of a spatial Maxwell transition point where all clusters involving up to 15 peaks are equally stable. Above (below) this point, clusters become more (less) stable when their number of peaks increases. In the 2D case, since clusters involving more than two peaks may exhibit distinct spatial arrangements, this point splits into a set of Maxwell transition points

M. Augustin, A. Caiazzo, A. Fiebach, J. Fuhrmann, V. John, A. Linke, R. Umla, An assessment of discretizations for convection-dominated convection-diffusion equations, Computer Methods in Applied Mechanics and Engineering, 200 (2011), pp. 3395--3409.
Abstract
The performance of several numerical schemes for discretizing convection-dominated convection-diffusion equations will be investigated with respect to accuracy and efficiency. Accuracy is considered in measures which are of interest in applications. The study includes an exponentially fitted finite volume scheme, the Streamline-Upwind Petrov--Galerkin (SUPG) finite element method, a spurious oscillations at layers diminishing (SOLD) finite element method, a finite element method with continuous interior penalty (CIP) stabilization, a discontinuous Galerkin (DG) finite element method, and a total variation diminishing finite element method (FEMTVD). A detailed assessment of the schemes based on the Hemker example will be presented.

M. Case, V. Ervin, A. Linke, L. Rebholz, A connection between Scott--Vogelius and grad-div stabilized Taylor--Hood FE approximations of the Navier--Stokes equations, SIAM Journal on Numerical Analysis, 49 (2011), pp. 1461--1481.
Abstract
This article studies two methods for obtaining excellent mass conservation in finite element computations of the Navier-Stokes equations using continuous velocity fields. Under mild restrictions, the Scott-Vogelius element pair has recently been shown to be inf-sup stable and have optimal approximation properties, while also providing pointwise mass conservation. We present herein the first numerical tests of this element pair for the time dependent Navier-Stokes equations. We also prove that, again under these mild restrictions, the limit of the grad-div stabilized Taylor-Hood solutions to the Navier-Stokes problem converges to the Scott-Vogelius solution as the stabilization parameter tends to infinity. That is, in this setting, we provide theoretical justification that choosing the parameter large does not destroy the solution. A limiting result is also proven for the general case. Numerical tests are provided which verify the theory, and show how both Scott-Vogelius and grad-div stabilized Taylor-Hood (with large stabilization parameter) elements can provide accurate results with excellent mass conservation for Navier-Stokes approximations.

A. Glitzky, Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems, Mathematische Nachrichten, 284 (2011), pp. 2159--2174.
Abstract
Our focus are energy estimates for discretized reaction-diffusion systems for a finite number of species. We introduce a discretization scheme (Voronoi finite volume in space and fully implicit in time) which has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For a class of Voronoi finite volume meshes we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the discrete free energy to its equilibrium value with a unified rate of decay for this class of discretizations. The essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly by taking into account sequences of Voronoi finite volume meshes. Essential ingredient in that proof is a discrete Sobolev-Poincaré inequality.

A. Caiazzo, M. Fernandez, J.-F. Gerbeau, V. Martin, Projection schemes for fluid flows through a porous interface, SIAM Journal on Scientific Computing, 33 (2011), pp. 541--564.

A. Caiazzo, M.A. Fernandez, V. Martin, Analysis of a stabilized finite element method for fluid flows through a porous interface, Applied Mathematics Letters, 24 (2011), pp. 2124--2127.

S. Amiranashvili, A.G. Vladimirov, U. Bandelow, A model equation for ultrashort optical pulses around the zero dispersion frequency, The European Physical Journal D. Atomic, Molecular, Optical and Plasma Physics, 58 (2010), pp. 219--226.
Abstract
The nonlinear Schrödinger equation based on the Taylor approximation of the material dispersion can become invalid for ultrashort and few-cycle optical pulses. Instead, we use a rational fit to the dispersion function such that the resonances are naturally accounted for. This approach allows us to derive a simple non-envelope model for short pulses propagating in one spatial dimension. This model is further investigated numerically and analytically.

S. Amiranashvili, U. Bandelow, A. Mielke, Padé approximant for refractive index and nonlocal envelope equations, Optics Communications, 283 (2010), pp. 480--485.
Abstract
Padé approximant is superior to Taylor expansion when functions contain poles. This is especially important for response functions in complex frequency domain, where singularities are present and intimately related to resonances and absorption. Therefore we introduce a diagonal Padé approximant for the complex refractive index and apply it to the description of short optical pulses. This yields a new nonlocal envelope equation for pulse propagation. The model offers a global representation of arbitrary medium dispersion and absorption, e.g., the fulfillment of the Kramers-Kronig relation can be established. In practice, the model yields an adequate description of spectrally broad pulses for which the polynomial dispersion operator diverges and can induce huge errors.

J. Härdtlein, C. Pflaum, A. Linke, C.H. Wolters, Advanced expression templates programming, Computing and Visualization in Science, 13 (2010), pp. 59--68.

H.-G. Purwins, H. Bödeker, S. Amiranashvili, Dissipative solitons, Advances in Physics, 59 (2010), pp. 485--701.

J. Fuhrmann, A. Fiebach, A. Erdmann, P. Trefonas, Acid diffusion effects between resists in freezing processes used for contact hole patterning, Microelectronic Engineering, 87 (2010), pp. 951--954.
Abstract
Double patterning following an litho?litho-etch scheme is a possible option to create structure widths below the nominal resolution of optical light with current exposure technology. Interactions between the first and second resist layers may influence the final structure created. In this paper, we perform a model based investigation of the possible consequences of the diffusion of photo-generated acid from the second resist to the first one. As a consequence, less acid is available for the deprotection reaction, and we observe a tendency to an increase of the CD values of the primary structure. We attempt to explain observed footing effects in contact holes by this effect.

A. Glitzky, J.A. Griepentrog, Discrete Sobolev--Poincaré inequalities for Voronoi finite volume approximations, SIAM Journal on Numerical Analysis, 48 (2010), pp. 372--391.
Abstract
We prove a discrete Sobolev-Poincare inequality for functions with arbitrary boundary values on Voronoi finite volume meshes. We use Sobolev's integral representation and estimate weakly singular integrals in the context of finite volumes. We establish the result for star shaped polyhedral domains and generalize it to the finite union of overlapping star shaped domains. In the appendix we prove a discrete Poincare inequality for space dimensions greater or equal to two.

A. Glitzky, K. Gärtner, Existence of bounded steady state solutions to spin-polarized drift-diffusion systems, SIAM Journal on Mathematical Analysis, 41 (2010), pp. 2489--2513.
Abstract
We study a stationary spin-polarized drift-diffusion model for semiconductor spintronic devices. This coupled system of continuity equations and a Poisson equation with mixed boundary conditions in all equations has to be considered in heterostructures. In 3D we prove the existence and boundedness of steady states. If the Dirichlet conditions are compatible or nearly compatible with thermodynamic equilibrium the solution is unique. The same properties are obtained for a space discretized version of the problem: Using a Scharfetter-Gummel scheme on 3D boundary conforming Delaunay grids we show existence, boundedness and, for small applied voltages, the uniqueness of the discrete solution.

V. John, M. Roland, On the impact of the scheme for solving the higher-dimensional equation in coupled population balance systems, International Journal for Numerical Methods in Engineering, 82 (2010), pp. 1450--1474.

A. Mielke, L. Paoli, A. Petrov, U. Stefanelli, Error estimates for space-time discretizations of a rate-independent variational inequality, SIAM Journal on Numerical Analysis, 48 (2010), pp. 1625--1646.
Abstract
This paper deals with error estimates for space-time discretizations in the context of evolutionary variational inequalities of rate-independent type. After introducing a general abstract evolution problem, we address a fully-discrete approximation and provide a priori error estimates. The application of the abstract theory to a semilinear case is detailed. In particular, we provide explicit space-time convergence rates for the isothermal Souza-Auricchio model for shape-memory alloys.

H. Si, J. Fuhrmann, K. Gärtner, Boundary conforming Delaunay mesh generation, Computational Mathematics and Mathematical Physics, 50 (2010), pp. 38--53.

S. Amiranashvili, U. Bandelow, A. Mielke, Padé approximant for refractive index and nonlocal envelope equations, Optics Communications, 283 (2009), pp. 480--485.
Abstract
Padé approximant is superior to Taylor expansion when functions contain poles. This is especially important for response functions in complex frequency domain, where singularities are present and intimately related to resonances and absorption. Therefore we introduce a diagonal Padé approximant for the complex refractive index and apply it to the description of short optical pulses. This yields a new nonlocal envelope equation for pulse propagation. The model offers a global representation of arbitrary medium dispersion and absorption, e.g., the fulfillment of the Kramers-Kronig relation can be established. In practice, the model yields an adequate description of spectrally broad pulses for which the polynomial dispersion operator diverges and can induce huge errors.

A.G. Vladimirov, A. Pimenov, D. Rachinskii, Numerical study of dynamical regimes in a monolithic passively mode locked semiconductor laser, IEEE J. Quantum Electron., 45 (2009), pp. 462--468.

G. Kozyreff, M. Tlidi, A. Mussot, E. Louvergneaux, M. Taki, A.G. Vladimirov, Localized beating between dynamically generated frequencies, Physical Review Letters, 102 (2009), pp. 043905/1--043905/4.

M. Tlidi, A.G. Vladimirov, D. Pieroux, D. Turaev, Spontaneous motion of cavity solitons induced by a delayed feedback, Physical Review Letters, 103 (2009), pp. 103904/1--103904/4.

M. Ehrhardt, H. Han, Ch. Zheng, Numerical simulation of waves in periodic structures, Communications in Computational Physics, 5 (2009), pp. 849--872.
Abstract
In this work we present a new numerical technique for solving periodic structure problems. This new approach possesses several advantages. First, it allows for a fast evaluation of the Robin-to-Robin operator for periodic array problems. Secondly, this computational method can also be used for bi-periodic structure problems with local defects. Our strategy is an improvement of the recently developed recursive doubling process by Yuan and Lu.
In this paper we consider several problems, such as the exterior elliptic problems with strong coercivity, the time-dependent Schrödinger equation and finally the Helmholtz equation with damping.

J. Fuhrmann, A. Erdmann, Ch.R. Szmanda, A. Fiebach, M. Uhle, A model of self-limiting residual acid diffusion for pattern doubling, Microelectronic Engineering, 86 (2009), pp. 792--795.

J. Fuhrmann, A. Linke, H. Langmach, H. Baltruschat, Numerical calculation of the limiting current for a cylindrical thin layer flow cell, Electrochimica Acta, 55 (2009), pp. 430--438.

K. Gärtner, Existence of bounded discrete steady state solutions of the van Roosbroeck system on boundary conforming Delaunay grids, SIAM Journal on Scientific Computing, 31 (2009), pp. 1347--1362.
Abstract
The classic van Roosbroeck system describes the carrier transport in semiconductors in a drift diffusion approximation. Its analytic steady state solutions fulfill bounds for some mobility and recombination/generation models. The main goal of this paper is to establish the identical bounds for discrete in space, steady state solutions on 3d boundary conforming Delaunay grids and the classical Scharfetter-Gummel-scheme. Together with a uniqueness proof for small applied voltages and the known dissipativity (continuous as well as space and time discrete) these discretization techniques carry over the essential analytic properties to the discrete case. The proofs are of interest for deriving averaging schemes for space or state dependent material parameters, which are preserving these qualitative properties, too. To illustrate the properties of the scheme 1, 4, 16 elementary cells of a modified CoolMOS like structure are depleted by increasing the applied voltage until steady state avalanche breakdown occurs.

A. Glitzky, Energy estimates for electro-reaction-diffusion systems with partly fast kinetics, Discrete and Continuous Dynamical Systems, 25 (2009), pp. 159--174.
Abstract
We start from a basic model for the transport of charged species in heterostructures containing the mechanisms diffusion, drift and reactions in the domain and at its boundary. Considering limit cases of partly fast kinetics we derive reduced models. This reduction can be interpreted as some kind of projection scheme for the weak formulation of the basic electro--reaction--diffusion system. We verify assertions concerning invariants and steady states and prove the monotone and exponential decay of the free energy along solutions to the reduced problem and to its fully implicit discrete-time version by means of the results of the basic problem. Moreover we make a comparison of prolongated quantities with the solutions to the basic model.

A. Glitzky, K. Gärtner, Energy estimates for continuous and discretized electro-reaction-diffusion systems, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 70 (2009), pp. 788--805.
Abstract
We consider electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistic relations.
We investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. Here the essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly.
The same properties are shown for an implicit time discretized version of the problem. Moreover, we provide a space discretized scheme for the electro-reaction-diffusion system which is dissipative (the free energy decays monotonously). On a fixed grid we use for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species.

V. John, T. Mitkova, M. Roland, K. Sundmacher, L. Tobiska, A. Voigt, Simulations of population balance systems with one internal coordinate using finite element methods, Chemical Engineering Sciences, 64 (2009), pp. 733--741.

A. Mielke, T. Roubíček, Numerical approaches to rate-independent processes and applications in inelasticity, ESAIM: Mathematical Modelling and Numerical Analysis, 43 (2009), pp. 399--429.
Abstract
A general abstract approximation scheme for rate-independent processes in the energetic formulation is proposed and its convergence is proved under various rather mild data qualifications. The abstract theory is illustrated on several examples: plasticity with isotropic hardening, damage, debonding, magnetostriction, and two models of martensitic transformation in shape-memory alloys.

H. Stephan, Modeling of drift-diffusion systems, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 60 (2009), pp. 33--53.
Abstract
We derive drift-diffusion systems describing transport processes starting from free energy and equilibrium solutions by a unique method. We include several statistics, heterostructures and cross diffusion. The resulting systems of nonlinear partial differential equations conserve mass and positivity, and have a Lyapunov function (free energy). Using the inverse Hessian as mobility, non-degenerate diffusivity matrices turn out to be diagonal, or - in the case of cross diffusion - even constant.

S. Amiranashvili, A.G. Vladimirov, U. Bandelow, Solitary-wave solutions for few-cycle optical pulses, Physical Review A, 77 (2008), pp. 063821/1--063821/7.

M. Pietrzyk, I. Kanattšikow, U. Bandelow, On the propagation of vector ultra-short pulses, Journal of Nonlinear Mathematical Physics, 15 (2008), pp. 162-170.

R. Čiegis, M. Radziunas, M. Lichtner, Numerical algorithms for simulation of multisection lasers by using traveling wave model, IEEE J. Select. Topics Quantum Electron., 13 (2008), pp. 327-348.

D. Turaev, M. Radziunas, A.G. Vladimirov, Chaotic soliton walk in periodically modulated media, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 77 (2008), pp. 06520/1--06520/4.

A. Glitzky, Exponential decay of the free energy for discretized electro-reaction-diffusion systems, Nonlinearity, 21 (2008), pp. 1989--2009.
Abstract
Our focus are electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistical relations. We introduce a discretization scheme (in space and fully implicit in time) using a fixed grid but for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species. This scheme has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For the discretized electro-reaction-diffusion system we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. The essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly.

O. Minet, H. Gajewski, J.A. Griepentrog, J. Beuthan, The analysis of laser light scattering during rheumatoid arthritis by image segmentation, Laser Physics Letters, 4 (2007), pp. 604--610.

M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A.G. Vladimirov, M. Taki, Control and removing of modulational instabilities in low dispersion photonic crystal fiber cavities, , 32 (2007), pp. 662-664.

D. Turaev, A.G. Vladimirov, S. Zelik, Chaotic bound state of localized structures in the complex Ginzburg--Landau equation, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 75 (2007), pp. 045601/1-045601/4.

M. Lichtner, M. Radziunas, L. Recke, Well-posedness, smooth dependence and center manifold reduction for a semilinear hyperbolic system from laser dynamics, Mathematical Methods in the Applied Sciences, 30 (2007), pp. 931--960.

A.G. Vladimirov, D.V. Skryabin, G. Kozyreff, P. Mandel, M. Tlidi, Bragg localized structures in a passive cavity with transverse modulation of the refractive index and the pump, Optics Express, 14 (2006), pp. 1--6.

H. Gajewski, J.A. Griepentrog, A descent method for the free energy of multicomponent systems, Discrete and Continuous Dynamical Systems, 15 (2006), pp. 505--528.

M. Kočvara, A. Mielke, T. Roubíček, A rate-independent approach to the delamination problem, Mathematics and Mechanics of Solids, 11 (2006), pp. 423--447.

U. Bandelow, M. Radziunas, A. Vladimirov, B. Hüttl, R. Kaiser, 40 GHz Mode-locked semiconductor lasers: Theory, simulations and experiment, Optical and Quantum Electronics, 38 (2006), pp. 495--512.

M. Radziunas, F. Ivanauskas, The convergence and stability of splitting finite difference schemes for nonlinear evolutionary type equations, Optical and Quantum Electronics, 45 (2005), pp. 334--352.

P. Evans, A. Münch, Interaction of advancing fronts and meniscus profiles formed by surface-tension-gradient-driven liquid films, SIAM Journal on Applied Mathematics, 66 (2006), pp. 1610-1631.

A. Münch, Dewetting rates of thin liquid films, Physics of Fluids, 17 (2005), pp. S309--S318.

N. Nefedov, M. Radziunas, K.R. Schneider, A. Vasil'eva, Change of the type of contrast structures in parabolic Neumann problems, Computational Mathematics and Mathematical Physics, 45 (2005), pp. 37--51.

W. Dreyer, B. Wagner, Sharp-interface model for eutectic alloys. Part I: Concentration dependent surface tension, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 7 (2005), pp. 199--227.

N. Nefedov, M. Radziunas, K.R. Schneider, Analytic-numerical investigation of delayed exchange of stabilities in singularly perturbed parabolic problems, Computational Mathematics and Mathematical Physics, 44 (2004), pp. 1213-1220.

A. Rathsfeld, R. Schneider, On a quadrature algorithm for the piecewise linear wavelet collocation applied to boundary integral equations, Mathematical Methods in the Applied Sciences, 26 (2003), pp. 937--979.

A.M. Krasnosel'skii, D.I. Rachinskii, K.R. Schneider, Hopf bifurcations in resonance 2:1, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 52A (2003), pp. 943--960.

G. Mastroianni, C. Frammartino, A. Rathsfeld, On polynomial collocation for second kind integral equations with fixed singularities of Mellin type, Numerische Mathematik, 94 (2003), pp. 333--365.

N.N. Nefedov, K.R. Schneider, Delay of exchange of stabilities in singularly perturbed parabolic problems, Proceedings of the Steklov Institute of Mathematics, (2003), pp. S144-S154.

M. Tlidi, A.G. Vladimirov, P. Mandel, Interaction and stability of periodic and localized structures in optical bistable systems, IEEE J. Quantum Electron., 39 (2003), pp. 197--205.

H. Gajewski, K. Gärtner, Domain separation by means of sign changing eigenfunctions of $p$-Laplacians, Applicable Analysis. An International Journal, 79 (2001), pp. 483--501.

W. Dreyer, W.H. Müller, A study of the coarsening in tin/lead solders, International Journal of Solids and Structures, 37 (2000), pp. 3841--3871.

A. Glitzky, On a drift-diffusion model for perovskite solar cells, in: 94th Annual Meeting 2024 of the International Association of Applied Mathematics and Mechanics (GAMM), 24 of Proc. Appl. Math. Mech. (Special Issue), Wiley-VCH Verlag, Weinheim, 2024, pp. 17/1--e202400017/8, DOI 10.1002/pamm.202400017 .
Abstract
We introduce a vacancy-assisted charge transport model for perovskite solar cells. This instationary drift-diffusion system describes the motion of electrons, holes, and ionic vacancies and takes into account Fermi--Dirac statistics for electrons and holes and the Fermi--Dirac integral of order -1 for the mobile ionic vacancies in the perovskite. The free energy functional we work with corresponds to that choice of the statistical relations. To verify the existence of weak solutions, we consider a problem with regularized state equations and reaction terms on any arbitrarily chosen finite time interval. We motivate its solvability by time discretization and passage to the time-continuous limit. A priori estimates for the chemical potentials that are independent of the regularization level ensure the existence of solutions to the original problem. These types of estimates rely on Moser iteration techniques and can also be obtained for solutions to the original problem.

A.H. Erhardt, K. Tsaneva-Atanasova, G.T. Lines, E.A. Martens, Editorial: Dynamical systems, PDEs and networks for biomedical applications: Mathematical modeling, analysis and simulations, 10 of Front. Phys., Sec. Statistical and Computational Physics, Frontiers, Lausanne, Switzerland, 2023, pp. 01--03, DOI 10.3389/fphy.2022.1101756 .

M. Heida, Finite volumes for simulation of large molecules, in: Finite Volumes for Complex Applications X -- Volume 1, Elliptic and Parabolic Problems: FVCA10, Strasbourg, France, October 30, 2023 -- November 03, 2023, Invited Contributions, E. Franck, J. Fuhrmann, V. Michel-Dansac, L. Navoret, eds., 432 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham, 2023, pp. 305--313, DOI 10.1007/978-3-031-40864-9_25 .
Abstract
We study a finite volume scheme for simulating the evolution of large molecules within their reduced state space. The finite volume scheme under consideration is the SQRA scheme developed by Lie, Weber and Fackeldey. We study convergence of a more general family of FV schemes in up to 3 dimensions and provide a convergence result for the SQRA-scheme in arbitrary space dimensions.

M. O'Donovan, R. Finn, S. Schulz, Th. Koprucki, Atomistic study of Urbach tail energies in (Al,Ga)N quantum well systems, in: 23nd International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2023), P. Bardella, A. Tibaldi, eds., IEEE, 2023, pp. 79--80, DOI 10.1109/NUSOD59562.2023.10273479 .
Abstract
Aluminium gallium nitride is a system of interest for developing ultraviolet (UV) optoelectronic devices. Here Urbach tails induced by carrier localization effects play a key role in determining device behaviour. We study the electronic structure of Al x Ga 1-x N/Al y Ga 1-y N single quantum wells using an atomistic framework. Results show that the density of states exhibits a tail at low energies due to disorder in the alloy microstructure. Our analysis allows for insight into the orbital character of the states forming the Urbach tails, which can affect light polarization characteristics, and important quantity for deep UV light emitters.

C. Chainais-Hillairet, R. Eymard, J. Fuhrmann, An approximate two-point Dirichlet flux for quasilinear convection diffusion equations, in: Finite Volumes for Complex Applications X -- Volume 1, Elliptic and Parabolic Problems: FVCA10, Strasbourg, France, October 30, 2023 -- November 03, 2023, Invited Contributions, E. Franck, J. Fuhrmann, V. Michel-Dansac, L. Navoret, eds., 432 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham, 2023, pp. 225--233, DOI 10.1007/978-3-031-40864-9_17 .

S. Matera, Ch. Merdon, D. Runge, Reduced basis approach for convection-diffusion equations with non-linear boundary reaction conditions, in: Finite Volumes for Complex Applications X -- Volume 1, Elliptic and Parabolic Problems: FVCA10, Strasbourg, France, October 30, 2023 -- November 03, 2023, Invited Contributions, E. Franck, J. Fuhrmann, V. Michel-Dansac, L. Navoret, eds., 432 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham, 2023, pp. 335--343, DOI 10.1007/978-3-031-40864-9_28 .
Abstract
This paper aims at an efficient strategy to solve drift-diffusion problems with non-linear boundary conditions as they appear, e.g., in heterogeneous catalysis. Since the non-linearity only involves the degrees of freedom along (a part of) the boundary, a reduced basis ansatz is suggested that computes discrete Green?s-like functions for the present drift-diffusion operator such that the global non-linear problem reduces to a smaller non-linear problem for a boundary method. The computed basis functions are completely independent of the non-linearities. Thus, they can be reused for problems with the same differential operator and geometry. Corresponding scenarios might be inverse problems in heterogeneous catalysis but also modeling the effect of different catalysts in the same reaction chamber. The strategy is explained for a mass-conservative finite volume method and demonstrated on a simple numerical example for catalytic CO oxidation.

J. Fuhrmann, B. Gaudeul, Ch. Keller, Two entropic finite volume schemes for a Nernst--Planck--Poisson system with ion volume constraints, in: Finite Volumes for Complex Applications X -- Volume 1, Elliptic and Parabolic Problems: FVCA10, Strasbourg, France, October 30, 2023 -- November 03, 2023, Invited Contributions, E. Franck , J. Fuhrmann, V. Michel-Dansac, L. Navoret, eds., Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham, 2023, pp. 285--294, DOI 10.1007/978-3-031-40864-9_23 .

D. Abdel, N. Courtier, P. Farrell, Volume exclusion effects in perovskite charge transport modeling, in: 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2022), Turin, Italy, 2022, J. Piprek, P. Bardella, eds., IEEE, 2022, pp. 107--108, DOI 10.1109/NUSOD54938.2022.9894826 .

Y. Hadjimichael, O. Marquardt, Ch. Merdon, P. Farrell, Band structures in highly strained 3D nanowires, in: 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2022), Turin, Italy, 2022, J. Piprek, Bardella Paolo, eds., IEEE, 2022, pp. 119--120, DOI 10.1109/NUSOD54938.2022.9894837 .

O. Marquardt, M. O'Donovan, S. Schulz, O. Brandt, Th. Koprucki, Influence of random alloy fluctuations on the electronic properties of axial (In,Ga)N/GaN nanowire heterostructures, in: 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), Turin, Italy, 2022, J. Piprek, P. Bardella, eds., IEEE, 2022, pp. 117--118, DOI 10.1109/NUSOD54938.2022.9894777 .
Abstract
Compound semiconductor heterostructures such as quantum dots, nanowires, or thin films, are commonly subject to randomly fluctuating alloy compositions if they contain ternary and quaternary alloys. These effects are obviously of an atomistic nature and thus rarely considered in heterostructure designs that require simulations on a continuum level for theory-guided design or interpretation of observations. In the following, we present a systematic approach to the treatment of alloy fluctuations in (In,Ga)N/GaN thin films and axial nanowire heterostructures. We demonstrate to what extent random alloy fluctuations can be treated in a continuum picture and discuss the impact of alloy fluctuations on the electronic properties of planar and nano wire-based (In,Ga)N/GaN heterostructures.

S. Piani, W. Lei, L. Heltai, N. Rotundo, P. Farrell, Data-driven doping reconstruction, in: 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2022), Turin, Italy, 2022, J. Piprek, P. Bardella, eds., IEEE, 2022, pp. 109--110, DOI 10.1109/NUSOD54938.2022.9894774 .

J. Moatti, P. Farrell, Comparison of flux discretizations for varying band-edge energies, in: 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2022), Turin, Italy, 2022, J. Piprek, P. Bardella, eds., IEEE, 2022, pp. 103--104, DOI 10.1109/NUSOD54938.2022.9894742 .

M. O'Donovan, P. Farrell, T. Streckenbach, Th. Koprucki, S. Schulz, Carrier transport in (In,Ga)N quantum well systems: Connecting atomistic tight-binding electronic structure theory to drift-diffusion simulations, in: 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), Turin, Italy, 2022, J. Piprek, P. Bardella, eds., IEEE, 2022, pp. 97--98, DOI 10.1109/NUSOD54938.2022.9894745 .

M. Thomas, M. Heida, GENERIC for dissipative solids with bulk-interface interaction, in: Research in the Mathematics of Materials Science, M.I. Espanõl, M. Lewicka, L. Scardia, A. Schlömkemper, eds., 31 of Association for Women in Mathematics Series, Springer, Cham, 2022, pp. 333--364, DOI 10.1007/978-3-031-04496-0_15 .
Abstract
The modeling framework of GENERIC was originally introduced by Grmela and Öttinger for thermodynamically closed systems. It is phrased with the aid of the energy and entropy as driving functionals for reversible and dissipative processes and suitable geometric structures. Based on the definition functional derivatives we propose a GENERIC framework for systems with bulk-interface interaction and apply it to discuss the GENERIC structure of models for delamination processes.

K. Hopf, Global existence analysis of energy-reaction-diffusion systems, in: Report 29: Variational Methods for Evolution (hybrid meeting), A. Mielke, M. Peletier, D. Slepcev, eds., 17 of Oberwolfach Reports, European Mathematical Society Publishing House, Zurich, 2021, pp. 1418--1421, DOI 10.4171/OWR/2020/29 .

D. Abdel, P. Farrell, J. Fuhrmann, Comparison of Scharfetter--Gummel schemes for (non-)degenerate semiconductor device simulation, in: Proceedings of the 20th International Conference on Numerical Simulation of Optoelectronic Devices -- NUSOD 2020, J. Piprek, K. Hinzer, eds., IEEE Conference Publications Management Group, Piscataway, 2020, pp. 107--108.

S. Kayser, N. Rotundo, J. Fuhrmann, N. Dropka, P. Farrell, The lateral photovoltage scanning method (LPS): Understanding doping variations in silicon crystals, in: Proceedings of the 20th International Conference on Numerical Simulation of Optoelectronic Devices -- NUSOD 2020, J. Piprek, K. Hinzer, eds., IEEE Conference Publications Management Group, Piscataway, 2020, pp. 49--50.

A. Linke, Ch. Merdon, On the significance of pressure-robustness for the space discretization of incompressible high Reynolds number flows, in: Finite Volumes for Complex Applications IX -- Methods, Theoretical Aspects, Examples -- FVCA 9, Bergen, June 2020, R. Klöfkorn, E. Keilegavlen, A.F. Radu, J. Fuhrmann, eds., 323 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2020, pp. 103--112.

A. Linke, Ch. Merdon, Well-balanced discretisation for the compressible Stokes problem by gradient-robustness, in: Finite Volumes for Complex Applications IX -- Methods, Theoretical Aspects, Examples -- FVCA 9, Bergen, June 2020, R. Klöfkorn, E. Keilegavlen, A.F. Radu, J. Fuhrmann, eds., 323 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2020, pp. 113--121.

C. Cancès, C. Chainais-Hillairet, J. Fuhrmann, B. Gaudeul, On four numerical schemes for a unipolar degenerate drift-diffusion model, in: Finite Volumes for Complex Applications IX -- Methods, Theoretical Aspects, Examples -- FVCA 9, Bergen, June 2020, R. Klöfkorn, F. Radu, E. Keijgavlen, J. Fuhrmann, eds., Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2020, pp. 163--171, DOI 10.1007/978-3-030-43651-3_13 .

A. Jha, V. John, On basic iteration schemes for nonlinear AFC discretizations, in: Boundary and Interior Layers, Computational and Asymptotic Methods, BAIL 2018, G.N. Barrenechea, J. Mackenzie, eds., 135 of Lecture Notes in Computational Science and Engineering, Springer, Cham, 2020, pp. 113--128, DOI https://doi.org/10.1007/978-3-030-41800-7_7 .
Abstract
Algebraic flux correction (AFC) finite element discretizations of steady-state convection-diffusion-reaction equations lead to a nonlinear problem. This paper presents first steps of a systematic study of solvers for these problems. Two basic fixed point iterations and a formal Newton method are considered. It turns out that the fixed point iterations behave often quite differently. Using a sparse direct solver for the linear problems, one of them exploits the fact that only one matrix factorization is needed to become very efficient in the case of convergence. For the behavior of the formal Newton method, a clear picture is not yet obtained.

S. Schulz, D. Chaudhuri, M. O'Donovan, S. Patra, T. Streckenbach, P. Farrell, O. Marquardt, Th. Koprucki, Multi-scale modeling of electronic, optical, and transport properties of III-N alloys and heterostructures, in: Physics and Simulation of Optoelectronic Devices XXVIII, B. Witzigmann, M. Osiński, Y. Arakawa, eds., 11274 of Proceedings of SPIE, San Francisco, 2020, pp. 1127416/1--1127416/10, DOI 10.1117/12.2551055 .

J. Fuhrmann, D.H. Doan, A. Glitzky, M. Liero, G. Nika, Unipolar drift-diffusion simulation of S-shaped current-voltage relations for organic semiconductor devices, in: Finite Volumes for Complex Applications IX -- Methods, Theoretical Aspects, Examples -- FVCA 9, Bergen, June 2020, R. Klöfkorn, E. Keilegavlen, F.A. Radu, J. Fuhrmann, eds., 323 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2020, pp. 625--633, DOI 10.1007/978-3-030-43651-3_59 .
Abstract
We discretize a unipolar electrothermal drift-diffusion model for organic semiconductor devices with Gauss--Fermi statistics and charge carrier mobilities having positive temperature feedback. We apply temperature dependent Ohmic contact boundary conditions for the electrostatic potential and use a finite volume based generalized Scharfetter-Gummel scheme. Applying path-following techniques we demonstrate that the model exhibits S-shaped current-voltage curves with regions of negative differential resistance, only recently observed experimentally.

M. Kantner, Th. Koprucki, Non-isothermal Scharfetter--Gummel scheme for electro-thermal transport simulation in degenerate semiconductors, in: Finite Volumes for Complex Applications IX -- Methods, Theoretical Aspects, Examples -- FVCA 9, Bergen, June 2020, R. Klöfkorn, E. Keilegavlen, F.A. Radu, J. Fuhrmann, eds., 323 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2020, pp. 173--182, DOI 10.1007/978-3-030-43651-3_14 .
Abstract
Electro-thermal transport phenomena in semiconductors are described by the non-isothermal drift-diffusion system. The equations take a remarkably simple form when assuming the Kelvin formula for the thermopower. We present a novel, non-isothermal generalization of the Scharfetter--Gummel finite volume discretization for degenerate semiconductors obeying Fermi--Dirac statistics, which preserves numerous structural properties of the continuous model on the discrete level. The approach is demonstrated by 2D simulations of a heterojunction bipolar transistor.

M. Kantner, Th. Koprucki, H.-J. Wünsche, U. Bandelow, Simulation of quantum dot based single-photon sources using the Schrödinger--Poisson-Drift-Diffusion-Lindblad system, in: Proceedings of the 24th International Conference on Simulation of Semiconductor Processes and Devices (SISPAD 2019), F. Driussi, ed., 2019, pp. 355--358, DOI 10.1007/978-3-030-22116-0_11 .

M. Kantner, A. Mielke, M. Mittnenzweig, N. Rotundo, Mathematical modeling of semiconductors: From quantum mechanics to devices, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 269--293, DOI 10.1007/978-3-030-33116-0 .
Abstract
We discuss recent progress in the mathematical modeling of semiconductor devices. The central result of this paper is a combined quantum-classical model that self-consistently couples van Roosbroeck's drift-diffusion system for classical charge transport with a Lindblad-type quantum master equation. The coupling is shown to obey fundamental principles of non-equilibrium thermodynamics. The appealing thermodynamic properties are shown to arise from the underlying mathematical structure of a damped Hamitlonian system, which is an isothermal version of socalled GENERIC systems. The evolution is governed by a Hamiltonian part and a gradient part involving a Poisson operator and an Onsager operator as geoemtric structures, respectively. Both parts are driven by the conjugate forces given in terms of the derivatives of a suitable free energy.

M. Kantner, A generalized Scharfetter--Gummel scheme for degenerate and non-isothermal semiconductors, in: Proceedings of the 19th International Conference on Numerical Simulation of Optoelectronic Devices -- NUSOD 2019, J. Piprek, K. Hinzer, eds., IEEE Conference Publications Management Group, Piscataway, 2019, pp. 7--8, DOI 10.1109/NUSOD.2019.8806839 .
Abstract
We present a highly accurate generalization of the Scharfetter--Gummel scheme for the discretization of the currentdensities in degenerate semiconductors under non-isothermalconditions. The underlying model relies on the Kelvin formula forthe Seebeck coefficient, which has the intriguing property that the?T-term in the electrical current density expressions vanishesexactly when passing to the drift-diffusion form ? even thoughthe thermoelectric cross-coupling is fully taken into account.

M. Kantner, Hybrid modeling of quantum light emitting diodes: Self-consistent coupling of drift-diffusion, Schrödinger--Poisson, and quantum master equations, in: Proc. SPIE 10912, B. Witzigmann, M. Osiński, Y. Arakawa, eds., Physics and Simulation of Optoelectronic Devices XXVII, SPIE Digital Library, Bellingham, 2019, pp. 10912OU/1--10912OU/8, DOI 10.1117/12.2515209 .
Abstract
The device-scale simulation of electrically driven solid state quantum light emitters, such as single-photon sources and nanolasers based on semiconductor quantum dots, requires a comprehensive modeling approach, that combines classical device physics with cavity quantum electrodynamics. In a previous work, we have self-consistently coupled the semi-classical drift-diffusion system with a Markovian quantum master equation in Lindblad form to describe (i) the spatially resolved current injection into a quantum dot embedded within a semiconductor device and (ii) the fully quantum mechanical light-matter interaction in the coupled quantum dot-photon system out of one box. In this paper, we extend our hybrid quantum-classical modeling approach by including a Schroedinger?Poisson problem to account for energy shifts of the quantum dot carriers in response to modifications of its macroscopic environment (e.g., quantum confined Stark effect due to the diode's internal electric field and plasma screening). The approach is demonstrated by simulations of a single-photon emitting diode.

M. Kantner, Simulation of quantum light sources using the self-consistently coupled Schrödinger--Poisson-Drift-Diffusion-Lindblad system, in: Proceedings of the 19th International Conference on Numerical Simulation of Optoelectronic Devices -- NUSOD 2019, J. Piprek, K. Hinzer, eds., IEEE Conference Publications Management Group, Piscataway, 2019, pp. 15--16, DOI 10.1109/NUSOD.2019.8806839 .
Abstract
The device-scale simulation of electrically drivenquantum light sources based on semiconductor quantum dotsrequires a combination of the (classical) semiconductor deviceequations with cavity quantum electrodynamics. In this paper, weextend our previously developed hybrid quantum-classical modelsystem ? where we have coupled the drift-diffusion system witha Lindblad-type quantum master equation ? by including a self-consistent Schrödinger?Poisson problem. The latter describes the(quasi-)bound states of the quantum dot carriers. The extendedmodel allows to describe the bias-dependency of the emissionspectrum due to the quantum confined Stark effect

M. Lübbering, J. Kunkel, P. Farrell, What company does my news article refer to? Tackling multiclass problems with topic modeling, in: CEUR-WS.org/Vol-2454/paper_41, CEUR-WS.org, Sun SITE Central Europe RWTH Aachen, Aachen, 2019, pp. 353--364, DOI 10.20347/WIAS.PREPRINT.2621 .
Abstract
While it is technically trivial to search for the company name to predict the company a new article refers to, it often leads to incorrect results. In this article, we compare the two approaches bag-of-words with k-nearest neighbors and Latent Dirichlet Allocation with k-nearest neighbor by assessing their applicability for predicting the S&P 500 company which is mentioned in a business news article or press release. Both approaches are evaluated on a corpus of 13k documents containing 84% news articles and 16% press releases. While the bag-of-words approach yields accurate predictions, it is highly inefficient due to its gigantic feature space. The Latent Dirichlet Allocation approach, on the other hand, manages to achieve roughly the same prediction accuracy (0.58 instead of 0.62) but reduces the feature space by a factor of seven.

J.H.M. Ten Thije Boonkkamp, N. Kumar, B. Koren, D.A.M. caps">caps">van der Woude, A. Linke, Nonlinear flux approximation scheme for Burgers equation derived from a local BVP, in: Numerical Mathematics and Advanced Applications 2017 -- ENUMATH 2017, F.A. Radu, K. Kumar, I. Berre, J.M. Nordbotten, I.S. Pop, eds., 126 of Lecture Notes Comput. Sci. Engrg., Springer Nature Switzerland AG, Cham, 2019, pp. 1015--1023, DOI 10.1007/978-3-319-96415-7_96 .

J. Fuhrmann, C. Guhlke, A. Linke, Ch. Merdon, R. Müller, Models and numerical methods for electrolyte flows, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 183--209.

J. Fuhrmann, C. Guhlke, A. Linke, Ch. Merdon, R. Müller, Voronoi finite volumes and pressure robust finite elements for electrolyte models with finite ion sizes, in: Numerical Geometry, Grid Generation and Scientific Computing. Proceedings of the 9th International Conference, NUMGRID 2018 / Voronoi 150, V.A. Garanzha, L. Kamenski, H. Si, eds., 131 of Lecture Notes in Computational Science and Engineering, Springer Nature Switzerland AG, Cham, 2019, pp. 73--83, DOI 10.1007/978-3-030-23436-2 .

M. Kantner, M. Mittnenzweig, Th. Koprucki, A hybrid quantum-classical modeling approach for electrically driven quantum dot devices, in: Proc. SPIE 10526, Physics and Simulation of Optoelectronic Devices XXVI, B. Witzigmann, M. Osiński, Y. Arakawa, eds., SPIE Digital Library, 2018, pp. 1052603/1--1052603/6, DOI 10.1117/12.2289185 .
Abstract
The design of electrically driven quantum light sources based on semiconductor quantum dots, such as singlephoton emitters and nanolasers, asks for modeling approaches combining classical device physics with cavity quantum electrodynamics. In particular, one has to connect the well-established fields of semi-classical semiconductor transport theory and the theory of open quantum systems. We present a first step in this direction by coupling the van Roosbroeck system with a Markovian quantum master equation in Lindblad form. The resulting hybrid quantum-classical system obeys the fundamental laws of non-equilibrium thermodynamics and provides a comprehensive description of quantum dot devices on multiple scales: It enables the calculation of quantum optical figures of merit (e.g. the second order intensity correlation function) together with the spatially resolved simulation of the current flow in realistic semiconductor device geometries in a unified way.

M. Kantner, M. Mittnenzweig, Th. Koprucki, Modeling and simulation of electrically driven quantum light sources: From classical device physics to open quantum systems, in: 14th International Conference on Nonlinear Optics and Excitation Kinetics in Semiconductors, September 23--27, 2018, Berlin, Germany (Conference Program), 2018, pp. 135.

S. Bartels, M. Milicevic, M. Thomas, Numerical approach to a model for quasistatic damage with spatial $BV$-regularization, in: Proceedings of the INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin, eds., 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 179--203, DOI 10.1007/978-3-319-75940-1_9 .
Abstract
We address a model for rate-independent, partial, isotropic damage in quasistatic small strain linear elasticity, featuring a damage variable with spatial BV-regularization. Discrete solutions are obtained using an alternate time-discrete scheme and the Variable-ADMM algorithm to solve the constrained nonsmooth optimization problem that determines the damage variable at each time step. We prove convergence of the method and show that discrete solutions approximate a semistable energetic solution of the rate-independent system. Moreover, we present our numerical results for two benchmark problems.

M. Patriarca, P. Farrell, J. Fuhrmann, Th. Koprucki, M. Auf DER Maur, Highly accurate discretizations for non-Boltzmann charge transport in semiconductors, in: Proceedings of the 18th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2018), A. Djurišić, J. Piprek, eds., IEEE Conference Publications Management Group, Piscataway, NJ, 2018, pp. 53--54.

M. Liero, J. Fuhrmann, A. Glitzky, Th. Koprucki, A. Fischer, S. Reineke, Modeling and simulation of electrothermal feedback in large-area organic LEDs, in: Proceedings of the 17th International Conference on Numerical Simulation of Optoelectronic Devices -- NUSOD 2017, J. Piprek, M. Willatzen, eds., IEEE Conference Publications Management Group, Piscataway, 2017, pp. 105--106, DOI 10.1109/NUSOD.2017.8010013 .

N. Ahmed, A. Linke, Ch. Merdon, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, in: Finite Volumes for Complex Applications VIII -- Methods and Theoretical Aspects, FVCA 8, Lille, France, June 2017, C. Cancès, P. Omnes, eds., 199 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing AG, Cham, 2017, pp. 351--359.

P. Farrell, Th. Koprucki, J. Fuhrmann, Comparison of consistent flux discretizations for drift diffusion beyond Boltzmann statistics, in: Proceedings of the 17th International Conference on Numerical Simulation of Optoelectronic Devices -- NUSOD 2017, J. Piprek, M. Willatzen, eds., IEEE Conference Publications Management Group, Piscataway, 2017, pp. 219--220, DOI 10.1109/NUSOD.2017.8010070 .

J. Fuhrmann, A. Glitzky, M. Liero, Hybrid finite-volume/finite-element schemes for p(x)-Laplace thermistor models, in: Finite Volumes for Complex Applications VIII -- Hyperbolic, Elliptic and Parabolic Problems, FVCA 8, Lille, France, June 2017, C. Cancès, P. Omnes, eds., 200 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing AG, Cham, 2017, pp. 397--405, DOI 10.1007/978-3-319-57394-6_42 .

J. Fuhrmann, C. Guhlke, A finite volume scheme for Nernst--Planck--Poisson systems with Ion size and solvation effects, in: Finite Volumes for Complex Applications VIII -- Hyperbolic, Elliptic and Parabolic Problems, FVCA 8, Lille, France, June 2017, C. Cancès, P. Omnes, eds., 200 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing AG, Cham, 2017, pp. 497--505, DOI 10.1007/978-3-319-57394-6_52 .

M. Kantner, U. Bandelow, Th. Koprucki, H.-J. Wünsche, Multi-scale modelling and simulation of single-photon sources on a device level, in: Euro-TMCS II -- Theory, Modelling & Computational Methods for Semiconductors, 7th -- 9th December 2016, Tyndall National Institute, University College Cork, Ireland, E. O'Reilly, S. Schulz, S. Tomic, eds., Tyndall National Institute, 2016, pp. 65.

S. Ganesan, V. John, G. Matthies, R. Meesala, A. Shamim, U. Wilbrandt, An object oriented parallel finite element scheme for computations of PDEs: Design and implementation, in: 2016 IEEE 23rd International Conference on High Performance Computing Workshops (PDF only), pp. 106--115, DOI 10.1109/HiPCW.2016.19 .

G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics, in: MURPHYS-HSFS-2014: 7th International Workshop on MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012009/1--012009/20.
Abstract
This note deals with the analysis of a model for partial damage, where the rate-independent, unidirectional flow rule for the damage variable is coupled with the rate-dependent heat equation, and with the momentum balance featuring inertia and viscosity according to Kelvin-Voigt rheology. The results presented here combine the approach from [Roubíček M2AS'09, SIAM'10] with the methods from Lazzaroni/Rossi/Thomas/Toader [WIAS Preprint 2025]. The present analysis encompasses, differently from [Roubíček SIAM'10], the monotonicity in time of damage and the dependence of the viscous tensor on damage and temperature, and, unlike [WIAS Preprint 2025], a nonconstant heat capacity and a time-dependent Dirichlet loading.

A. Caiazzo, J. Mura, A two-scale homogenization approach for the estimation of porosity in elastic media subject area, in: Trends in Differential Equations and Applications, F.O. Gallego, M.V. Redondo Neble, J.R.R. Galván, eds., 8 of SEMA SIMAI Springer Series, Springer International Publishing Switzerland, Cham, 2016, pp. 89--105.

A. Mielke, Free energy, free entropy, and a gradient structure for thermoplasticity, in: Innovative Numerical Approaches for Multi-Field and Multi-Scale Problems. In Honor of Michael Ortiz's 60th Birthday, K. Weinberg, A. Pandolfi, eds., 81 of Lecture Notes in Applied and Computational Mechanics, Springer International Publishing Switzerland, Cham, 2016, pp. 135--160.
Abstract
In the modeling of solids the free energy, the energy, and the entropy play a central role. We show that the free entropy, which is defined as the negative of the free energy divided by the temperature, is similarly important. The derivatives of the free energy are suitable thermodynamical driving forces for reversible (i.e. Hamiltonian) parts of the dynamics, while for the dissipative parts the derivatives of the free entropy are the correct driving forces. This difference does not matter for isothermal cases nor for local materials, but it is relevant in the non-isothermal case if the densities also depend on gradients, as is the case in gradient thermoplasticity.

Using the total entropy as a driving functional, we develop gradient structures for quasistatic thermoplasticity, which again features the role of the free entropy. The big advantage of the gradient structure is the possibility of deriving time-incremental minimization procedures, where the entropy-production potential minus the total entropy is minimized with respect to the internal variables and the temperature.

We also highlight that the usage of an auxiliary temperature as an integrating factor in Yang/Stainier/Ortiz "A variational formulation of the coupled thermomechanical boundary-value problem for general dissipative solids" (J. Mech. Physics Solids, 54, 401-424, 2006) serves exactly the purpose to transform the reversible driving forces, obtained from the free energy, into the needed irreversible driving forces, which should have been derived from the free entropy. This reconfirms the fact that only the usage of the free entropy as driving functional for dissipative processes allows us to derive a proper variational formulation.

D. Peschka, Numerics of contact line motion for thin films, in: 8th Vienna International Conference on Mathematical Modelling -- MATHMOD 2015, 48 of IFAC-PapersOnLine, Elsevier, 2015, pp. 390--393.

W. Huang, L. Kamenski, J. Lang, Stability of explicit Runge--Kutta methods for high order finite element approximation of linear parabolic equations, in: Numerical Mathematics and Advanced Applications -- ENUMATH 2013, A. Abdulle, S. Deparis, D. Kressner, F. Nobile, M. Picasso, eds., 103 of Lecture Notes in Computational Science and Engineering, Springer International Publishing, Cham [et al.], 2015, pp. 165--173.
Abstract
We study the stability of explicit Runge-Kutta methods for high order Lagrangian finite element approximation of linear parabolic equations and establish bounds on the largest eigenvalue of the system matrix which determines the largest permissible time step. A bound expressed in terms of the ratio of the diagonal entries of the stiffness and mass matrices is shown to be tight within a small factor which depends only on the dimension and the choice of the reference element and basis functions but is independent of the mesh or the coefficients of the initial-boundary value problem under consideration. Another bound, which is less tight and expressed in terms of mesh geometry, depends only on the number of mesh elements and the alignment of the mesh with the diffusion matrix. The results provide an insight into how the interplay between the mesh geometry and the diffusion matrix affects the stability of explicit integration schemes when applied to a high order finite element approximation of linear parabolic equations on general nonuniform meshes.

R.M. Arkhipov, M. Radziunas, A.G. Vladimirov, Theoretical analysis of the influence of external periodic forcing on nonlinear dynamics of passively mode-locked semiconductor lasers, in: Proceedings of the XIV School Seminar Wave Phenomena in Inhomogeneous Media (Waves 2014), Section 9, Nonlinear dynamics and information systems (in electronic form and in Russian), 2014, pp. 3--6.

R.M. Arkhipov, M.V. Arkhipov, Mode-locking in two section and single section lasers due to coherent interaction of light and matter in the gain and absorbing media, in: Proceedings of the XIV School Seminar Wave Phenomena in Inhomogeneous Media (Waves 2014), Section 3, Nonlinear and coherent optics (in electronic form and in Russian), 2014, pp. 43--45.

TH. Koprucki, M. Kantner, J. Fuhrmann, K. Gärtner, On modifications of the Scharfetter--Gummel scheme for drift-diffusion equations with Fermi-like statistical distribution functions, in: Proceedings of the 14th International Conference on Numerical Simulation of Optoelectronic Devices, NUSOD 2014, 1--4 September 2014, J. Piprek, J. Javaloyes, eds., IEEE Conference Publications Management Group, Piscataway, NJ, USA, 2014, pp. 155--156.

A. Glitzky, A. Mielke, L. Recke, M. Wolfrum, S. Yanchuk, D2 -- Mathematics for optoelectronic devices, in: MATHEON -- Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 243--256.

J. Fuhrmann, A. Linke, Ch. Merdon, Coupling of fluid flow and solute transport using a divergence-free reconstruction of the Crouzeix--Raviart element, in: Finite Volumes for Complex Applications VII -- Elliptic, Parabolic and Hyperbolic Problems -- FVCA 7, Berlin, June 2014, J. Fuhrmann, M. Ohlberger, Ch. Rohde, eds., 78 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2014, pp. 587--595.

K. Götze, Free fall of a rigid body in a viscoelastic fluid, in: Geophysical Fluid Dynamics, Workshop, February 18--22, 2013, 10 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2013, pp. 554--556.

R.M. Arkhipov, M. Radziunas, A.G. Vladimirov, Numerical simulation of passively mode-locked semiconductor lasers under dual mode optical injection regime, in: Proceedings of the Conference ICONO/LAT 2013 (Technical Digest on CD ROM), Section LAT-01: Solid-State Lasers, Materials and Applications, Russian Academy of Sciences, Moscow, 2013, pp. 111--112.

R.M. Arkhipov, I. Babushkin, M.V. Arkhipov , Y.A. Tolmachev, Spectral and temporal characteristics of radiation from a periodic resonant medium excited at the superluminal velocity, in: Proceedings of the Conference ICONO/LAT 2013 (Technical Digest on CD ROM), Section LAT-04: Diffractive Optics and Nanophotonics, Russian Academy of Sciences, Moscow, 2013, pp. 14--15.

TH. Koprucki, K. Gärtner, Generalization of the Scharfetter--Gummel scheme, in: Proceedings of the 13th International Conference on Numerical Simulation of Optoelectronic Devices, NUSOD 2013, 19--22 August 2013, J. Piprek, L. Chrostowski, eds., IEEE Conference Publications Management Group, Piscataway, NJ, USA, 2013, pp. 85--86.

C. Carstensen, C. Merdon, J. Neumann, Aspects of guaranteed error control in CPDE, in: Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications, O.P. Iliev, S.D. Margenov, P.D. Minev, P.S. Vassilevski, L.T. Zikatanov, eds., 45 of Springer Proceedings in Mathematics & Statistics, Springer, New York, 2013, pp. 103--119.

A. Mielke, Gradient structures and dissipation distances for reaction-diffusion systems, in: Material Theory, Workshop, Dezember 16--20, 2013, A. Desimone, S. Luckhaus, L. Truskinovsky, eds., 10 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2013, pp. 3455--3458.

TH. Koprucki, K. Gärtner, Discretization scheme for drift-diffusion equations with strong diffusion enhancement, in: Proceedings of the 12th International Conference on Numerical Simulation of Optoelectronic Devices, NUSOD'12, J. Piprek, W. Lu, eds., IEEE Conference Publications Management Group, New Jersey, USA, 2012, pp. 103--104.

A. Glitzky, J.A. Griepentrog, On discrete Sobolev--Poincaré inequalities for Voronoi finite volume approximations, in: Finite volumes for complex applications VI: Problems and perspectives, J. Fořt, J. Fürst, J. Halama, R. Herbin, F. Hubert, eds., Springer Proceedings in Mathematics 4, Springer, Heidelberg, 2011, pp. 533--541.

S. Bartels, R. Müller, Die kalte Zunge, in: Besser als Mathe --- Moderne angewandte Mathematik aus dem MATHEON zum Mitmachen, K. Biermann, M. Grötschel, B. Lutz-Westphal, eds., Reihe: Populär, Vieweg+Teubner, Wiesbaden, 2010, pp. 227--235.

M. Jensen, R. Müller, Stable Crank--Nicolson discretisation for incompressible miscible displacement problems of low regularity, in: Numerical Mathematics and Advanced Applications 2009, Part 2, G. Kreiss, P. Lötstedt, A. Målqvist, M. Neytcheva, eds., Springer, Heidelberg et al., pp. 469--477.
Abstract
In this article we study the numerical approximation of incompressible miscible displacement problems with a linearised Crank-Nicolson time discretisation, combined with a mixed finite element and discontinuous Galerkin method. At the heart of the analysis is the proof of convergence under low regularity requirements. Numerical experiments demonstrate that the proposed method exhibits second-order convergence for smooth and robustness for rough problems.

D. Turaev, A.G. Vladimirov, S. Zelik, Strong enhancement of interaction of optical pulses induced by oscillatory instability, in: CLEO/Europe and EQEC 2009 Conference Digest (Optical Society of America, 2009), poster EH.P.13 WED, 2009, pp. 1--1.

J. Fuhrmann, K. Gärtner, Modeling of two-phase flow and catalytic reaction kinetics for DMFCs, in: Device and Materials Modeling in PEM Fuel Cells, S. Paddison, K. Promislow, eds., 113 of Topics in Applied Physics, Springer, Berlin/Heidelberg, 2009, pp. 297--316.

H. Gajewski, J.A. Griepentrog, A. Mielke, J. Beuthan, U. Zabarylo, O. Minet, Image segmentation for the investigation of scattered-light images when laser-optically diagnosing rheumatoid arthritis, in: Mathematics -- Key Technology for the Future, W. Jäger, H.-J. Krebs, eds., Springer, Heidelberg, 2008, pp. 149--161.

M. Ehrhardt, J. Fuhrmann, A. Linke, E. Holzbecher, Mathematical modeling of channel-porous layer interfaces in PEM fuel cells, in: Proceedings of FDFC2008 --- Fundamentals and Developments of Fuel Cell Conference 2008, Nancy, France, December 10--12 (CD), 2008, pp. 8 pages.
Abstract
In proton exchange membrane (PEM) fuel cells, the transport of the fuel to the active zones, and the removal of the reaction products are realized using a combination of channels and porous diffusion layers. In order to improve existing mathematical and numerical models of PEM fuel cells, a deeper understanding of the coupling of the flow processes in the channels and diffusion layers is necessary.
After discussing different mathematical models for PEM fuel cells, the work will focus on the description of the coupling of the free flow in the channel region with the filtration velocity in the porous diffusion layer as well as interface conditions between them.
The difficulty in finding effective coupling conditions at the interface between the channel flow and the membrane lies in the fact that often the orders of the corresponding differential operators are different, e.g., when using stationary (Navier-)Stokes and Darcy's equation. Alternatively, using the Brinkman model for the porous media this difficulty does not occur.
We will review different interface conditions, including the well-known Beavers-Joseph-Saffman boundary condition and its recent improvement by Le Bars and Worster.

A. Mielke, Numerical approximation techniques for rate-independent inelasticity, in: Proceedings of the IUTAM Symposium on Theoretical, Computational and Modelling Aspects of Inelastic Media, B.D. Reddy, ed., 11 of IUTAM Bookseries, Springer, 2008, pp. 53--63.

A. Mussot, M. Tlidi, E. Louvergneaux, G. Kozyref, A.G. Vladimirov, M. Taki, Removing modulational instabilities in low dispersion fiber cavities, in: 2007 European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference (CLEO® / Europe-IQEC) Conference Digest (oral presentation CD-9-WED), IEEE, 2007, pp. 1--1.

A.G. Vladimirov, D.V. Skryabin, M. Tlidi, Localized structures of light in nonlinear devices with intracavity photonic bandgap material, in: 2007 European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference (CLEO®/Europe-IQEC) Conference Digest (oral presentation IG-4-MON), IEEE, 2007, pp. 1--1.

U. Bandelow, H. Gajewski, R. Hünlich, Thermodynamic designed energy model, in: Proceedings of the IEEE/LEOS 3rd International Conference on Numerical Simulation of Semiconductor Optoelectronic Devices (NUSOD'03), J. Piprek, ed., 2003, pp. 35--37.

H. Gajewski, H.-Chr. Kaiser, H. Langmach, R. Nürnberg, R.H. Richter, Mathematical modelling and numerical simulation of semiconductor detectors, in: Mathematics --- Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jäger, H.-J. Krebs, eds., Springer, Berlin [u.a.], 2003, pp. 355--364.

R. Hünlich, G. Albinus, H. Gajewski, A. Glitzky, W. Röpke, J. Knopke, Modelling and simulation of power devices for high-voltage integrated circuits, in: Mathematics --- Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jäger, H.-J. Krebs, eds., Springer, Berlin [u.a.], 2003, pp. 401--412.

H. Gajewski, An application of eigenfunctions of $p$-Laplacians to domain separation, in: Proceedings of Partial Differential Equations and Applications. A conference held in honor of the 70th birthday of Professor Jindřich Nečas, Olomouc, December 13--17, 1999, Š. Nečasová, H. Petzeltová, N. Pokorný, A. Sequeira, eds., 126 of Math. Bohem., Academy of Sciences of the Czech Republic, Mathematical Institute, Prague, 2001, pp. 395--401.

Y. Hadjimichael, O. Brandt, Ch. Merdon, C. Manganelli, P. Farrell, Strain distribution in zincblende and wurtzite GaAs nanowires bent by a one-sided (In, Al)As shell: Consequences for torsion, chirality, and piezoelectricity, Preprint no. 3141, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3141 .
Abstract, PDF (31 MByte)
We present a finite-strain model that is capable of describing the large deformations in bent nanowire heterostructures. The model incorporates a nonlinear strain formulation derived from the first Piola-Kirchhoff stress tensor, coupled with an energy functional that effectively captures the lattice-mismatch-induced strain field. We use the finite element method to solve the resulting partial differential equations and extract cross- sectional maps of the full strain tensor for both zincblende and wurtzite nanowires with lattice-mismatched core and one-sided stressor shell. In either case, we show that the bending is essentially exclusively determined by $varepsilonzz$. However, the distinct difference in shear strain has important consequences with regard to both the mechanical deformation and the existence of transverse piezoelectric fields in the nanowires.

V. John, M. Matthaiou, M. Zainelabdeen, Bound-preserving PINNs for steady-state convection-diffusion-reaction problems, Preprint no. 3134, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3134 .
Abstract, PDF (1983 kByte)
Numerical approximations of solutions of convection-diffusion-reaction problems should take only physically admissible values. Provided that bounds for the admissible values are known, this paper presents several approaches within PINNs and $hp$-VPINNs for preserving these bounds. Numerical simulations are performed for convection-dominated problems. One of the proposed approaches turned out to be superior to the other ones with respect to the accuracy of the computed solutions.

A. Alphonse, C. Christof, M. Hintermüller, I. Papadopoulos, A globalized inexact semismooth Newton method for nonsmooth fixed--point equations involving variational inequalities, Preprint no. 3132, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3132 .
Abstract, PDF (23 MByte)
We develop a semismooth Newton framework for the numerical solution of fixed-point equations that are posed in Banach spaces. The framework is motivated by applications in the field of obstacle-type quasi-variational inequalities and implicit obstacle problems. It is discussed in a general functional analytic setting and allows for inexact function evaluations and Newton steps. Moreover, if a certain contraction assumption holds, we show that it is possible to globalize the algorithm by means of the Banach fixed-point theorem and to ensure q-superlinear convergence to the problem solution for arbitrary starting values. By means of a localization technique, our Newton method can also be used to determine solutions of fixed-point equations that are only locally contractive and not uniquely solvable. We apply our algorithm to a quasi-variational inequality which arises in thermoforming and which not only involves the obstacle problem as a source of nonsmoothness but also a semilinear PDE containing a nondifferentiable Nemytskii operator. Our analysis is accompanied by numerical experiments that illustrate the mesh-independence and q -superlinear convergence of the developed solution algorithm.

TH. Eiter, L. Schmeller, Weak solutions to a model for phase separation coupled with finite-strain viscoelasticity subject to external distortion, Preprint no. 3130, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3130 .
Abstract, PDF (376 kByte)
We study the coupling of a viscoelastic deformation governed by a Kelvin--Voigt model at equilibrium, based on the concept of second-grade nonsimple materials, with a plastic deformation due to volumetric swelling, described via a phase-field variable subject to a Cahn--Hilliard model expressed in a Lagrangian frame. Such models can be used to describe the time evolution of hydrogels in terms of phase separation within a deformable substrate. The equations are mainly coupled via a multiplicative decomposition of the deformation gradient into both contributions and via a Korteweg term in the Eulerian frame. To treat time-dependent Dirichlet conditions for the deformation, an auxiliary variable with fixed boundary values is introduced, which results in another multiplicative structure. Imposing suitable growth conditions on the elastic and viscous potentials, we construct weak solutions to this quasistatic model as the limit of time-discrete solutions to incremental minimization problems. The limit passage is possible due to additional regularity induced by the hyperelastic and viscous stresses.

A. Alphonse, D. Caetano, Ch.M. Elliott, Ch. Venkataraman, Free boundary limits of coupled bulk--surface models for receptor--ligand interactions on evolving domains, Preprint no. 3122, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3122 .
Abstract, PDF (5947 kByte)
We derive various novel free boundary problems as limits of a coupled bulk-surface reaction-diffusion system modelling ligand-receptor dynamics on evolving domains. These limiting free boundary problems may be formulated as Stefan-type problems on an evolving hypersurface. Our results are new even in the setting where there is no domain evolution. The models are of particular relevance to a number of applications in cell biology. The analysis utilises L^∞-estimates in the manner of De Giorgi iterations and other technical tools, all in an evolving setting. We also report on numerical simulations.

C. Cárcamo, A. Caiazzo, F. Galarce, J. Mura, A stabilized total pressure-formulation of the Biot's poroelasticity equations in frequency domain: Numerical analysis and applications, Preprint no. 3101, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3101 .
Abstract, PDF (7379 kByte)
This work focuses on the numerical solution of the dynamics of a poroelastic material in the frequency domain. We provide a detailed stability analysis based on the application of the Fredholm alternative in the continuous case, considering a total pressure formulation of the Biot's equations. In the discrete setting, we propose a stabilized equal order finite element method complemented by an additional pressure stabilization to enhance the robustness of the numerical scheme with respect to the fluid permeability. Utilizing the Fredholm alternative, we extend the well-posedness results to the discrete setting, obtaining theoretical optimal convergence for the case of linear finite elements. We present different numerical experiments to validate the proposed method. First, we consider model problems with known analytic solutions in two and three dimensions. As next, we show that the method is robust for a wide range of permeabilities, including the case of discontinuous coefficients. Lastly, we show the application for the simulation of brain elastography on a realistic brain geometry obtained from medical imaging.

L. Ermoneit, B. Schmidt, Th. Koprucki, J. Fuhrmann, T. Breiten, A. Sala, N. Ciroth, R. Xue, L.R. Schreiber, M. Kantner, Optimal control of conveyor-mode spin-qubit shuttling in a Si/SiGe quantum bus in the presence of charged defects, Preprint no. 3082, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3082 .
PDF (9473 kByte)

CH. Keller, J. Fuhrmann, M. Landstorfer, B. Wagner, A model framework for ion channels with selectivity filters based on continuum non-equilibrium thermodynamics, Preprint no. 3072, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3072 .
Abstract, PDF (7287 kByte)
A mathematical model framework to describe ion transport in nanopores is presented. The model is based on non-equilibrium thermodynamics and considers finite size effects, solvation phenomena as well as the electrical charges of membrane surfaces and channel proteins. Par- ticular emphasis is placed on the consistent modelling of the selectivity filter in the pore. It is treated as an embedded domain in which the constituents can change their chemical properties. The diffusion process through the filter is governed by an independent diffusion coefficient and at the interfaces, de- and resolvation reactions are introduced as Neumann interface conditions. The evolution of the molar densities is described by drift-diffusion equations, where the fluxes depend on the gradient of the chemical potentials and the electric force. The chemical potentials depend on the molar fractions and on the pressure in the electrolyte and accounts for solvation effects. The framework allows the calculation of current-voltage relations for a variety of chan- nel properties and ion concentrations. We compare our model framework to experimental results for calcium-selective ion channels and show the general validity of our approach. Our parameter studies show that calcium and sodium currents are proportional to the surface charge in the se- lectivity filter and to the diffusion coefficients of the ions. Moreover, they show that the negative charges inside the pore have a decisive influence on the selectivity of divalent over monovalent ions.

Y. Hadjimichael, Ch. Merdon, M. Liero, P. Farrell, An energy-based finite-strain model for 3D heterostructured materials and its validation by curvature analysis, Preprint no. 3064, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3064 .
Abstract, PDF (6517 kByte)
This paper presents a comprehensive study of the intrinsic strain response of 3D het- erostructures arising from lattice mismatch. Combining materials with different lattice constants induces strain, leading to the bending of these heterostructures. We propose a model for nonlinear elastic heterostructures such as bimetallic beams or nanowires that takes into account local prestrain within each distinct material region. The resulting system of partial differential equations (PDEs) in Lagrangian coordinates incorporates a nonlinear strain and a linear stress-strain relationship governed by Hooke?s law. To validate our model, we apply it to bimetallic beams and hexagonal hetero-nanowires and perform numerical simulations using finite element methods (FEM). Our simulations ex- amine how these structures undergo bending under varying material compositions and cross-sectional geometries. In order to assess the fidelity of the model and the accuracy of simulations, we compare the calculated curvature with analytically derived formula- tions. We derive these analytical expressions through an energy-based approach as well as a kinetic framework, adeptly accounting for the lattice constant mismatch present at each compound material of the heterostructures. The outcomes of our study yield valuable insights into the behavior of strained bent heterostructures. This is particularly significant as the strain has the potential to influence the electronic band structure, piezoelectricity, and the dynamics of charge carriers.

M. Heida, On the computation of high dimensional Voronoi diagrams, Preprint no. 3041, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3041 .
Abstract, PDF (553 kByte)
We investigate a recently implemented new algorithm for the computation of a Voronoi diagram in high dimensions and generalize it to N nodes in general or non-general position using a geometric characterization of edges merging in a given vertex. We provide a mathematical proof that the algorithm is exact, convergent and has computational costs of O(E*nn(N)), where E is the number of edges and nn(N) is the computational cost to calculate the nearest neighbor among N points. We also provide data from performance tests in the recently developed Julia package ,,HighVoronoi.jl”.

K. Hopf, A. Jüngel, Convergence of a finite volume scheme and dissipative measure-valued--strong stability for a hyperbolic-parabolic cross-diffusion system, Preprint no. 3006, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3006 .
Abstract, PDF (444 kByte)
This article is concerned with the approximation of hyperbolic-parabolic cross-diffusion systems modeling segregation phenomena for populations by a fully discrete finite-volume scheme. It is proved that the numerical scheme converges to a dissipative measure-valued solution of the PDE system and that, whenever the latter possesses a strong solution, the convergence holds in the strong sense. Furthermore, the “parabolic density part” of the limiting measure-valued solution is atomic and converges to its constant state for long times. The results are based on Young measure theory and a weak-strong stability estimate combining Shannon and Rao entropies. The convergence of the numerical scheme is achieved by means of discrete entropy dissipation inequalities and an artificial diffusion, which vanishes in the continuum limit.

M. Heida, A. Sikorski, M. Weber, Consistency and order 1 convergence of cell-centered finite volume discretizations of degenerate elliptic problems in any space dimension, Preprint no. 2913, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2913 .
Abstract, PDF (601 kByte)
We study consistency of cell-centered finite difference methods for elliptic equations with degenerate coefficients in any space dimension $dgeq2$. This results in order of convergence estimates in the natural weighted energy norm and in the weighted discrete $L^2$-norm on admissible meshes. The cells of meshes under consideration may be very irregular in size. We particularly allow the size of certain cells to remain bounded from below even in the asymptotic limit. For uniform meshes we show that the order of convergence is at least 1 in the energy semi-norm, provided the discrete and continuous solutions exist and the continuous solution has $H^2$ regularity.

Y. Hadjimichael, D.I. Ketcheson, L. Lóczi, Positivity preservation of implicit discretizations of the advection equation, Preprint no. 2846, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2846 .
Abstract, PDF (1121 kByte)
We analyze, from the viewpoint of positivity preservation, certain discretizations of a fundamental partial differential equation, the one-dimensional advection equation with periodic boundary condition. The full discretization is obtained by coupling a finite difference spatial semidiscretization (the second- and some higher-order centered difference schemes, or the Fourier spectral collocation method) with an arbitrary θ-method in time (including the forward and backward Euler methods, and a second-order method by choosing  θ ∈ [0, 1] suitably). The full discretization generates a two-parameter family of circulant matrices M ∈ ℝ ^mxm , where each matrix entry is a rational function in θ and ν . Here, ν denotes the CFL number, being proportional to the ratio between the temporal and spatial discretization step sizes. The entrywise non-negativity of the matrix M---which is equivalent to the positivity preservation of the fully discrete scheme---is investigated via discrete Fourier analysis and also by solving some low-order parametric linear recursions. We find that positivity preservation of the fully discrete system is impossible if the number of spatial grid points m is even. However, it turns out that positivity preservation of the fully discrete system is recovered for odd values of m provided that θ ≥ 1/2 and ν are chosen suitably. These results are interesting since the systems of ordinary differential equations obtained via the spatial semi-discretizations studied are not positivity preserving.

M.H. Farshbaf Shaker, M. Thomas, Analysis of a compressible Stokes-flow with degenerating and singular viscosity, Preprint no. 2786, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2786 .
Abstract, PDF (744 kByte)
In this paper we show the existence of a weak solution for a compressible single-phase Stokes flow with mass transport accounting for the degeneracy and the singular behavior of a density-dependent viscosity. The analysis is based on an implicit time-discrete scheme and a Galerkin-approximation in space. Convergence of the discrete solutions is obtained thanks to a diffusive regularization of p-Laplacian type in the transport equation that allows for refined compactness arguments on subdomains.

K.M. Gambaryan, O. Marquardt, T. Boeck, A. Trampert, Micro- and nano-scale engineering and structures shape architecture at nucleation from In-As-Sb-P composition liquid phase on an InAs(100) surface, Preprint no. 2775, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2775 .
Abstract, PDF (4287 kByte)
In this review paper we present results of the growth, characterization and electronic properties of In(As,Sb,P) composition strain-induced micro- and nanostructures. Nucleation is performed from In-As-Sb-P quaternary composition liquid phase in Stranski--Krastanow growth mode using steady-state liquid phase epitaxy. Growth features and the shape transformation of pyramidal islands, lens-shape and ellipsoidal type-II quantum dots (QDs), quantum rings and QD-molecules are under consideration. It is shown that the application of a quaternary In(As,Sb,P) composition wetting layer allows not only more flexible control of lattice-mismatch between the wetting layer and an InAs(100) substrate, but also opens up new possibilities for nanoscale engineering and nanoarchitecture of several types of nanostructures. HR-SEM, AFM, TEM and STM are used for nanostructure characterization. Optoelectronic properties of the grown structures are investigated by FTIR and photoresponse spectra measurements. Using an eight-band $mathbfkcdotmathbfp$ model taking strain and built-in electrostatic potentials into account, the electronic properties of a wide range of InAs$_1-x-y$Sb$_x$P$_y$ QDs and QD-molecules are computed. Two types of QDs mid-infrared photodetectors are fabricated and investigated. It is shown that the incorporation of QDs allows to improve some output device characteristics, in particularly sensitivity, and to broaden the spectral range.

M.G. Hennessy, G.L. Celora, A. Münch, S.L. Waters, B. Wagner, Asymptotic study of the electric double layer at the interface of a polyelectrolyte gel and solvent bath, Preprint no. 2751, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2751 .
Abstract, PDF (2265 kByte)
An asymptotic framework is developed to study electric double layers that form at the inter-face between a solvent bath and a polyelectrolyte gel that can undergo phase separation. The kinetic model for the gel accounts for the finite strain of polyelectrolyte chains, free energy ofinternal interfaces, and Stefan?Maxwell diffusion. By assuming that the thickness of the doublelayer is small compared to the typical size of the gel, matched asymptotic expansions are used toderive electroneutral models with consistent jump conditions across the gel-bath interface in two-dimensional plane-strain as well as fully three-dimensional settings. The asymptotic frameworkis then applied to cylindrical gels that undergo volume phase transitions. The analysis indicatesthat Maxwell stresses are responsible for generating large compressive hoop stresses in the double layer of the gel when it is in the collapsed state, potentially leading to localised mechanicalinstabilities that cannot occur when the gel is in the swollen state. When the energy cost of in-ternal interfaces is sufficiently weak, a sharp transition between electrically neutral and chargedregions of the gel can occur. This transition truncates the double layer and causes it to have finitethickness. Moreover, phase separation within the double layer can occur. Both of these featuresare suppressed if the energy cost of internal interfaces is sufficiently high. Thus, interfacial freeenergy plays a critical role in controlling the structure of the double layer in the gel.

G.L. Celora, M.G. Hennessy, A. Münch, S.L. Waters, B. Wagner, Spinodal decomposition and collapse of a polyelectrolyte gel, Preprint no. 2731, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2731 .
Abstract, PDF (2259 kByte)
The collapse of a polyelectrolyte gel in a (monovalent) salt solution is analysed using a new model that includes interfacial gradient energy to account for phase separation in the gel, finite elasticity and multicomponent transport. We carry out a linear stability analysis to determine the stable and unstable spatially homogeneous equilibrium states and how they phase separate into localized regions that eventually coarsen to a new stable state. We then investigate the problem of a collapsing gel as a response to increasing the salt concentration in the bath. A phase space analysis reveals that the collapse is obtained by a front moving through the gel that eventually ends in a new stable equilibrium. For some parameter ranges, these two routes to gel shrinking occur together.

S. Bartels, M. Milicevic, M. Thomas, N. Weber, Fully discrete approximation of rate-independent damage models with gradient regularization, Preprint no. 2707, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2707 .
Abstract, PDF (3444 kByte)
This work provides a convergence analysis of a time-discrete scheme coupled with a finite-element approximation in space for a model for partial, rate-independent damage featuring a gradient regularization as well as a non-smooth constraint to account for the unidirectionality of the damage evolution. The numerical algorithm to solve the coupled problem of quasistatic small strain linear elasticity with rate-independent gradient damage is based on a Variable ADMM-method to approximate the nonsmooth contribution. Space-discretization is based on P1 finite elements and the algorithm directly couples the time-step size with the spatial grid size h. For a wide class of gradient regularizations, which allows both for Sobolev functions of integrability exponent r ∈ (1, ∞) and for BV-functions, it is shown that solutions obtained with the algorithm approximate as h → 0 a semistable energetic solution of the original problem. The latter is characterized by a minimality property for the displacements, a semistability inequality for the damage variable and an energy dissipation estimate. Numerical benchmark experiments confirm the stability of the method.

M. Lübbering, J. Kunkel, P. Farrell, What company does my news article refer to? Tackling multiclass problems with topic modeling, Preprint no. 2621, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2621 .
Abstract, PDF (354 kByte)
While it is technically trivial to search for the company name to predict the company a new article refers to, it often leads to incorrect results. In this article, we compare the two approaches bag-of-words with k-nearest neighbors and Latent Dirichlet Allocation with k-nearest neighbor by assessing their applicability for predicting the S&P 500 company which is mentioned in a business news article or press release. Both approaches are evaluated on a corpus of 13k documents containing 84% news articles and 16% press releases. While the bag-of-words approach yields accurate predictions, it is highly inefficient due to its gigantic feature space. The Latent Dirichlet Allocation approach, on the other hand, manages to achieve roughly the same prediction accuracy (0.58 instead of 0.62) but reduces the feature space by a factor of seven.

R. Schlundt, A multilevel Schur complement preconditioner with ILU factorization for complex symmetric matrices, Preprint no. 2556, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2556 .
Abstract, PDF (318 kByte)
This paper describes a multilevel preconditioning technique for solving complex symmetric sparse linear systems. The coefficient matrix is first decoupled by domain decomposition and then an approximate inverse of the original matrix is computed level by level. This approximate inverse is based on low rank approximations of the local Schur complements. For this, a symmetric singular value decomposition of a complex symmetric matix is used. The block-diagonal matrices are decomposed by an incomplete LDL^T factorization with the Bunch-Kaufman pivoting method. Using the example of Maxwell's equations the generality of the approach is demonstrated.

P.-É. Druet, Analysis of improved Nernst--Planck--Poisson models of isothermal compressible electrolytes subject to chemical reactions: The case of a degenerate mobility matrix, Preprint no. 2321, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2321 .
Abstract, PDF (387 kByte)
We continue our investigations of the improved Nernst-Planck-Poisson model introduced by Dreyer, Guhlke and Müller 2013. In the paper by Dreyer, Druet, Gajewski and Guhlke 2016, the analysis relies on the hypothesis that the mobility matrix has maximal rank under the constraint of mass conservation (rank N-1 for the mixture of N species). In this paper we allow for the case that the positive eigenvalues of the mobility matrix tend to zero along with the partial mass densities of certain species. In this approach the mobility matrix has a variable rank between zero and N-1 according to the number of locally available species. We set up a concept of weak solution able to deal with this scenario, showing in particular how to extend the fundamental notion of emphdifferences of chemical potentials that supports the modelling and the analysis in Dreyer, Druet, Gajewski and Guhlke 2016. We prove the global-in-time existence in this solution class.

W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Existence of weak solutions for improved Nernst--Planck--Poisson models of compressible reacting electrolytes, Preprint no. 2291, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2291 .
Abstract, PDF (638 kByte)

J.A. Griepentrog, On regularity, positivity and long-time behavior of solutions to an evolution system of nonlocally interacting particles, Preprint no. 1932, WIAS, Berlin, 2014, DOI 10.20347/WIAS.PREPRINT.1932 .
Abstract, PDF (1279 kByte)
An analytical model for multicomponent systems of nonlocally interacting particles is presented. Its derivation is based on the principle of minimization of free energy under the constraint of conservation of particle number and justified by methods established in statistical mechanics. In contrast to the classical Cahn-Hilliard theory with higher order terms, the nonlocal theory leads to an evolution system of second order parabolic equations for the particle densities, weakly coupled by nonlinear and nonlocal drift terms, and state equations which involve both chemical and interaction potential differences. Applying fixed-point arguments and comparison principles we prove the existence of variational solutions in suitable Hilbert spaces for evolution systems. Moreover, using maximal regularity for nonsmooth parabolic boundary value problems in Sobolev-Morrey spaces and comparison principles, we show uniqueness, global regularity and uniform positivity of solutions under minimal assumptions on the regularity of interaction. Applying a refined version of the Łojasiewicz-Simon gradient inequality, this paves the way to the convergence of solutions to equilibrium states. We conclude our considerations with the presentation of simulation results for a phase separation process in ternary systems.

W. Dreyer, C. Guhlke, R. Müller, Rational modeling of electrochemical double-layers and derivation of Butler--Volmer equations, Preprint no. 1860, WIAS, Berlin, 2013, DOI 10.20347/WIAS.PREPRINT.1860 .
Abstract, PDF (443 kByte)
We derive the boundary conditions for the contact between an electrolyte and a solid electrode. At first we revisit the thermodynamic consistent complete model that resolves the actual electrode--electrolyte interface and its adjacent boundary layers. The width of these layers is controlled by the Debye length that is typically very small, leading to strongly different length scales in the system. We apply the method of asymptotic analysis to derive a simpler reduced model that does not resolve the boundary layers but instead incorporates the electrochemical properties of the layers into a set of new boundary conditions. This approach fully determines the relation of bulk quantities to the boundary conditions of the reduced model. In particular, the Butler-Volmer equations for electrochemical reactions, which are still under discussion in the literature, are rational consequences of our approach. For illustration and to compare with the literature, we consider a simple generic reaction.

C. Carstensen, Ch. Merdon, J. Neumann, Aspects of guaranteed error control in CPDEs, Preprint no. 1775, WIAS, Berlin, 2013, DOI 10.20347/WIAS.PREPRINT.1775 .
Abstract, PDF (337 kByte)
Whenever numerical algorithms are employed for a reliable computational forecast, they need to allow for an error control in the final quantity of interest. The discretisation error control is of some particular importance in computational PDEs (CPDEs) where guaranteed upper error bounds (GUB) are of vital relevance. After a quick overview over energy norm error control in second-order elliptic PDEs, this paper focuses on three particular aspects. First, the variational crimes from a nonconforming finite element discretisation and guaranteed error bounds in the discrete norm with improved postprocessing of the GUB. Second, the reliable approximation of the discretisation error on curved boundaries and, finally, the reliable bounds of the error with respect to some goal-functional, namely, the error in the approximation of the directional derivative at a given point.

R. Huth, Numerical convergence for semilinear parabolic equations, Preprint no. 1747, WIAS, Berlin, 2012, DOI 10.20347/WIAS.PREPRINT.1747 .
Abstract, PDF (198 kByte)
We present a convergence result for finite element discretisations of semilinear parabolic equations, in which the evaluation of the nonlinearity requires some high order of regularity of the solution. For example a coefficient might depend on derivatives or pointevaluation of the solution. We do not rely on high regularity of the exact solution itself and as a payoff we can not deduce convergence rates. As an example the convergence result is applied to a nonlinear Fokker--Planck type battery model.

D. Turaev, A.G. Vladimirov, S. Zelik, Strong synchronization of weakly interacting oscillons, Preprint no. 1659, WIAS, Berlin, 2011, DOI 10.20347/WIAS.PREPRINT.1659 .
Abstract, Postscript (23 MByte), PDF (7332 kByte)
We study interaction of well-separated oscillating localized structures (oscillons). We show that oscillons emit weakly decaying dispersive waves, which leads to formation of bound states due to subharmonic synchronization. We also show that in optical applications the Andronov-Hopf bifurcation of stationary localized structures leads to a drastic increase in their interaction strength.

H. Stephan, A mathematical framework for general classical systems and time irreversibility as its consequence, Preprint no. 1629, WIAS, Berlin, 2011, DOI 10.20347/WIAS.PREPRINT.1629 .
Abstract, Postscript (3232 kByte), PDF (431 kByte)
It is well known that important models in statistical physics like the Fokker-Planck equation satisfy an H-theorem, i.e., have a decreasing Lyapunov function (or increasing entropy). This illustrates a symmetry break in time and reflects the second law of thermodynamics. In this paper, we show that any physically reasonable classical system has to have this property. For this purpose, we develop an abstract mathematical framework based on the theory of compact topological spaces and convex analysis. Precisely, we show:
1) Any statistical state space can be described as the convex hull of the image of the canonical embedding of the bidual space of its deterministic state space (a compact topological Hausdorff space).
2) The change of any statistical state is effected by the adjoint of a Markov operator acting in the space of observables.
3) Any Markov operator satisfies a wide class of inequalities, generated by arbitrary convex functions. As a corollary, these inequalities imply a time monotone behavior of the solution of the corresponding evolution equations.
Moreover, due to the general abstract setting, the proof of the underlying inequalities is very simple and therefore illustrates, where time symmetry breaks: A model is time reversible for any states if and only if the corresponding Markov operator is a deterministic one with dense range.
In addition, the proposed framework provides information about the structure of microscopic evolution equations, the choice of the best function spaces for their analysis and the derivation of macroscopic evolution equations.

L. Paoli, A. Petrov, Global existence result for thermoviscoelastic problems with hysteresis, Preprint no. 1616, WIAS, Berlin, 2011, DOI 10.20347/WIAS.PREPRINT.1616 .
Abstract, Postscript (426 kByte), PDF (243 kByte)
We consider viscoelastic solids undergoing thermal expansion and exhibiting hysteresis effects due to plasticity or phase transformations. Within the framework of generalized standard solids, the problem is described in a 3D setting by the momentum equilibrium equation, the flow rule describing the dependence of the stress on the strain history, and the heat transfer equation. Under appropriate regularity assumptions on the data, a local existence result for this thermodynamically consistent system is established, by combining existence results for ordinary differential equations in Banach spaces with a fixed-point argument. Then global estimates are obtained by using both the classical energy estimate and more specific techniques for the heat equation introduced by Boccardo and Gallouet. Finally a global existence result is derived.

A. Petrov, On the numerical approximation of a viscoelastodynamic problem with unilateral constraints, Preprint no. 1523, WIAS, Berlin, 2010, DOI 10.20347/WIAS.PREPRINT.1523 .
Abstract, Postscript (591 kByte), PDF (236 kByte)
The present work is dedicated to the study of numerical schemes for a viscoelastic bar vibrating longitudinally and having its motion limited by rigid obstacles at the both ends. Finite elements and finite difference schemes are presented and their convergence is proved. Finally, some numerical examples are reported and analyzed.

M. Ehrhardt, J. Fuhrmann, A. Linke, A model of an electrochemical flow cell with porous layer, Preprint no. 1437, WIAS, Berlin, 2009, DOI 10.20347/WIAS.PREPRINT.1437 .
Abstract, Postscript (6623 kByte), PDF (586 kByte)
In this paper we discuss three different mathematical models for fluid-porous interfaces in a simple channel geometry that appears e.g. in thin-layer channel flow cells. Here the difficulties arise from the possibly different orders of the corresponding differential operators in the different domains. A finite volume discretization of this model allows to calculate the limiting current of the H_2 oxidation in a porous electrode with platinum catalyst particles.

Z. Amer, Numerical methods for coupled drift-diffusion and Helmholtz models for laser applications, Leibniz MMS Days 2024, April 10 - 12, 2024, Leibniz Network "Mathematical Modeling and Simulation", Leibniz Institut für Verbundwerkstoffe GmbH (IVW), Kaiserslautern, April 11, 2024.

CH. Keller, A drift-diffusion model to describe ion channel dynamics, Applied Mathematics and Simulation for Semiconductor Devices (AMaSiS 2024), Berlin, September 10 - 13, 2024.

CH. Keller, A model framework for calcium ion channels: Consistent modeling of selectivity filters, The European Conference on Mathematical and Theoretical Biology (ECMTB 2024), July 22 - 26, 2024, University of Castilla La Mancha, Toledo, Spain, July 25, 2024.

A. Erhardt, Mathematical modeling of cell-hydrogel interactions, The XXIII Symposium on Trends in Applications of Mathematics to Mechanics (STAMM), April 3 - 5, 2024, Universität Würzburg, April 3, 2024.

L. Ermoneit, Simulation and optimal control of single-electron shuttling in a SiGe quantum bus, Applied Mathematics and Simulation for Semiconductor Devices (AMaSiS 2024), September 10 - 13, 2024.

M. O'Donovan, Multi-scale simulation of electronic and transport properties in (Al,Ga)N quantum well systems for UV-C emission, Applied Mathematics and Simulation for Semiconductor Devices (AMaSiS 2024), September 10 - 13, 2024, WIAS Berlin, September 11, 2024.

M. O'Donovan, Simulation of the alloy fluctuations on luminescence and transport in AIGaN-based UV-LEDs, XXXV. Heimbach Workshop, September 23 - 27, 2024, Technische Universität Berlin, Mansfeld, September 26, 2024.

M. O'Donovan, Theoretical investigations on different scales towards novel III-N materials and devices, Rundgespräch des SPP 2477 ``Nitrides4Future'', Magdeburg, September 24 - 25, 2024.

I. Papadopoulos, A semismooth Newton method for obstacle--type quasivariational inequalities, Firedrake 2024, September 16 - 18, 2024, University of Oxford, UK, September 18, 2024.

D. Peschka, Dissipative processes in thin film flows, Liquid Thin Films, August 26 - 30, 2024, (Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, August 27, 2024.

D. Peschka, Wetting of soft deformable substrate - Phase fields for fluid structure interaction with moving contact lines, Colloquium on ``Interfaces, Complex Structures, and Singular Limits in Continuum Mechanics - Analysis and Numerics'', Universität Regensburg, Fakultät für Mathematik, May 24, 2024.

A. Thayil, Optimization of valley splitting in Si/SiGe spin qubits, Applied Mathematics and Simulation for Semiconductor Devices (AMaSiS 2024), September 10 - 13, 2024, WIAS Berlin, September 12, 2024.

A. Mielke, Asymptotic self-similar behaviour in reaction-diffusion systems on Rd, Dynamical Systems Approaches towards Nonlinear PDEs, August 28 - 30, 2024, Universität Stuttgart, August 29, 2024.

A. Mielke, Balanced-viscosity solutions for generalized gradient systems in mechanics, Frontiers of the Calculus of Variations, September 16 - 20, 2024, University of the Aegean, Karlovasi, Greece, September 17, 2024.

M. Demir, Pressure-robust approximation of the Navier--Stokes equations with coriolis force, 9th European Congress of Mathematics (9ECM), July 15 - 19, 2024, Congress of the European Mathematical Society, School of Engineering of the University of Seville, Spain, July 15, 2024.

M. Demir, Pressure-robust approximation of the incompressable Navier--Stokes equationsin a rotating frame of reference, Exploring Scientific Research: Workshop for Early Career Researchers, November 11 - 12, 2024, Gulf University for Science and Technology, Center for Applied Mathematics and Bioinformatics, Kuwait-Stadt, Kuwait, November 11, 2024.

M. Demir, Time filtered second order backward Euler method for EMAC formulation of Navier--Stokes equations, 20th Annual Workshop on Numerical Methods for Problems with Layer Phenomena, May 23 - 24, 2024, University of Cyprus, Department of Mathematics and Statistics, Protaras, Cyprus, May 24, 2024.

A. Glitzky, Analysis of a drift-diffusion model for perovskite solar cells, 94th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2024), Session 14.07 ``Various topics in Applied Analysis'', March 18 - 22, 2024, Otto-von-Guericke-Universität Magdeburg, March 21, 2024.

M. Thomas, Analysis of a model for visco-elastoplastic two-phase flows in geodynamics, 23rd Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2024), April 3 - 5, 2024, Julius-Maximilians-Universität Würzburg, April 5, 2024.

M. Thomas, Analysis of a model for visco-elastoplastic two-phase flows in geodynamics, 9th European Congress of Mathematics (9ECM), Minisymposium 27 ``New Trends in Calculus of Variations'', July 15 - 19, 2024, Universidad de Sevilla, Spain, July 16, 2024.

M. Thomas, Analysis of a model for visco-elastoplastic two-phase flows in geodynamics, Seminar on Nonlinear Partial Differential Equations, Texas A&M University, Department of Mathematics, College Station, USA, March 19, 2024.

M. Zainelabdeen, Augmenting the grad-div stabilization for Taylor--Hood finite elements with a vorticity stabilization, The Chemnitz Finite Element Symposium 2024, September 9 - 11, 2024, Technische Universität Chemnitz, September 11, 2024.

M. Zainelabdeen, Physics-informed neural networks for convection-dominated convection-diffusion problems, 20th Annual Workshop on Numerical Methods for Problems with Layer Phenomena, May 22 - 25, 2024, Department of Mathematics and Statistics, University of Cyprus, Protaras, May 24, 2024.

M. Zainelabdeen, Physics-informed neural networks for convection-dominated convection-diffusion problems, International Conference on Boundary and Interior Layers, BAIL 2024, June 10 - 14, 2024, University of A Coruña, Department of Mathematic, Spain, June 11, 2024.

A. Caiazzo, Validation of an open-source lattice Boltzmann solver (OpenLB) for the simulation of airflow over diary building, Leibniz MMS Days 2024, April 10 - 12, 2024, Leibniz Network "Mathematical Modeling and Simulation", Leibniz Institut für Verbundwerkstoffe GmbH (IVW), Kaiserslautern, April 11, 2024.

C. Cárcamo, Frequency-domain formulation and convergence analysis of Biot's poroelasticity equations based on total pressure, Computational Techniques and Applications Conference (CTAC 2024), November 19 - 22, 2024, Monash University, School of Mathematics, Melbourne, Australia, November 20, 2024.

C. Cárcamo, Frequency-domain formulation and convergence analysis of Biots poroelasticity equations based on total pressure, The Chemnitz Finite Element Symposium 2024, September 9 - 11, 2024, Technische Universität Chemnitz, September 9, 2024.

C. Cárcamo, Total pressure-based frequency-domain formulation and convergence analysis of Biot's poroelasticity equations with a new finite element stabilization, Minisymposium "Full and reduced-order modeling of multiphysics problems", WONAPDE 2024: Seventh Chilean Workshop on Numerical Analysis of Partial Differential Equations, Concepción, January 15 - 19, 2024, Universidad de Concepción, Barrio Universitario s/n, Region of Bío-Bío, Chile, January 16, 2024.

P. Farrell, Charge transport in perovskites solar cells: modeling, analysis and simulations, Inria-ECDF Partnership Kick-Off, Robert-Koch-Forum, Wilhelmstraße 67, Berlin, June 7, 2024.

J. Fuhrmann, Ch. Keller, M. Landstorfer, B. Wagner, Development of an ion-channel model-framework for in-vitro assisted interpretation of current voltage relations, MATH+ Day, Urania Berlin, October 18, 2024.

J. Fuhrmann, S. Maass, S. Ringe, Monolithic coupling of a CatMAP based microkinetic model for heterogeneous electrocatalysis and ion transport with finite ion sizes, Applied Mathematics and Simulation for Semiconductor Devices (AMaSiS 2024), Berlin, September 10 - 13, 2024.

J. Fuhrmann, Development of numerical methods and tools for drift-diffusion simulations, Applied Mathematics and Simulation for Semiconductor Devices (AMaSiS 2024), Berlin, September 10 - 13, 2024.

J. Fuhrmann, VoronoiFVM.jl, Workshop: Finite Volumes and Optimal Transport, November 19 - 21, 2024, Université Paris-Saclay, Institut de Mathématiques d'Orsay, France, November 19, 2024.

J. Fuhrmann, What's new with VoronoiFVM.jl, JuliaCon 2024, July 9 - 13, 2024, JuliCon.org with TU and PyData Eindhoven, Netherlands, July 11, 2024.

M. Heida, Voronoi diagrams and finite volume methods in any dimension, 94th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2024), Session 18.01 ``Discontinuous Galerkin and Software'', March 18 - 22, 2024, Otto-von-Guericke-Universität Magdeburg, March 19, 2024.

G. Heinze, Graph-based nonlocal gradient systems and their local limits, Aggregation-Diffusion Equations & Collective Behavior: Analysis, Numerics and Applications, Marseille, France, April 8 - 12, 2024.

M. Hintermüller, A hybrid physics-informed neural network based multiscale solver as a PDE constrained optimization problem, ISMP 2024 -- 25th International Symposium on Mathematical Programming, Session TA90 ``Nonsmooth PDE Constrained Optimization'', July 21 - 26, 2024, Montreal, Canada, July 23, 2024.

V. John, Finite element methods respecting the discrete maximum principle for convection-diffusion equations, International Conference on 'Latest Advances in Computational and Applied Mathematics' (LACAM) 2024, February 21 - 24, 2024, Indian Institute of Science Education and Research Thiruvananthapuram, Kerala, India, February 21, 2024.

V. John, Finite element methods respecting the discrete maximum principle for convection-diffusion equations, Mathematical Fluid Mechanics In 2024, August 19 - 23, 2024, Czech Academy of Sciences, Institute of Mathematics, Prague, Czech Republic, August 21, 2024.

V. John, Finite element methods respecting the discrete maximum principle for convection-diffusion equations, Trends in Scientific Computing - 30 Jahre Wissenschaftliches Rechnen in Dortmund, May 21 - 22, 2024, TU Dortmund, Fakultät für Mathematik, LSIII, May 21, 2024.

V. John, On using machine learning techniques for the numerical solution of convection-diffusion problems, ALGORITMY 2024, Central-European Conference on Scientific Computing, Minisymposium ``Numerical methods for convection-dominated problems'', March 16 - 20, 2024, Department of Mathematics and Descriptive Geometry, Slovak University of Technology in Bratislava, High Tatra Mountains, Podbanske, Slovakia, March 19, 2024.

V. John, On using machine learning techniques for the numerical solution of convection-diffusion problems, Seminar-talk, Prof. Sashikumaar Ganesan, Indian Institute of Science Bangalore, Department of Computational and Data Sciences, Bangalore, India, February 13, 2024.

CH. Merdon, Mass-conservative reduced basis approach for heterogeneous catalysis, Leibniz MMS Days 2024, Kaiserslautern, April 10 - 12, 2024.

CH. Merdon, Mass-conservative reduced basis approach for heterogeneous catalysis, Leibniz MMS Days 2024, April 10 - 12, 2024, Leibniz Network "Mathematical Modeling and Simulation", Leibniz Institut für Verbundwerkstoffe GmbH (IVW), Kaiserslautern.

CH. Merdon, Pressure-robustness in Navier--Stokes finite element simulations, 10th International Conference on Computational Methods in Applied Mathematics (CMAM-10), June 10 - 14, 2024, Universität Bonn, Institut für Numerische Simulation, June 11, 2024.

CH. Merdon, Pressure-robustness in the context of the weakly compressible Navier--Stokes equations, The Chemnitz Finite Element Symposium 2024, September 9 - 11, 2024, Technische Universität Chemnitz, September 9, 2024.

O. Pártl, Fracture-controlled reservoir performance optimization via 3D numerical modeling and simulation, Leibniz MMS Days 2024, April 10 - 12, 2024, Leibniz Network "Mathematical Modeling and Simulation", Leibniz Institut für Verbundwerkstoffe GmbH (IVW), Kaiserslautern, April 11, 2024.

D. Runge, Mass-conservative reduced basis approach for heterogeneous catalysis, Leibniz MMS Days 2024, Kaiserslautern, April 10 - 12, 2024.

D. Runge, Mass-conservative reduced basis approach for heterogeneous catalysis, Leibniz MMS Days 2024, April 10 - 12, 2024, Leibniz Network "Mathematical Modeling and Simulation", Leibniz Institut für Verbundwerkstoffe GmbH (IVW), Kaiserslautern.

T. Siebert, The general purpose algorithmic differentiation wrapper ADOLC.jl, JuliaCon 2024, July 9 - 13, 2024, JuliCon.org with TU and PyData Eindhoven, Netherlands, July 12, 2024.

J.P. Thiele, RSE / PostDoc am WIAS, PhoenixD Research School für Promovierende, Exzellenzcluster PhoenixD, Leibniz Universität Hannover, April 4, 2024.

J.P. Thiele, RSE training and professional development BoF, Research Software Engineering Conference, RSECon24, September 3 - 5, 2024, Society of Research Software Engineering (SocRSE), a charitable incorporated organisation based in the UK, Newcastle, UK, September 5, 2024.

J.P. Thiele, RSE und RDM: Code and perish?! How about publishing your software (and data)?, Oberseminar Numerik und Optimierung, Institut für Angewandte Mathematik, Leibniz Universität Hannover, May 2, 2024.

J.P. Thiele, The Research Software Engineer (RSE): Who is that? And what skills do they have to help you?, European Trilinos & Kokkos User Group Meeting 2024 (EuroTUG 2024), June 24 - 26, 2024, Helmut-Schmidt-Universität, Universität der Bundeswehr Hamburg, June 24, 2024.

D. Abdel, N.E. Courtier, P. Farrell, Modelling and simulation of charge transport in Perovskite solar cells, SIAM Conference on Computational Science and Engineering (CSE23), Amsterdam, Netherlands, February 26 - March 3, 2023.

CH. Keller, Continuum-based modeling of calcium-selective ion channels, Conference on Mathematical Life Sciences, April 17 - 20, 2023, Universität Bonn, April 20, 2023.

CH. Keller, Mathematical modeling of biological ion channels, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), May 29 - June 2, 2023, Technische Universität Dresden, May 31, 2023.

Y. Hadjimichael, Efficient implementation of implicit Runge--Kutta methods with downwind-biased operators., SIAM Conference on Computational Science and Engineering (CSE23), Minisymposium MS419 ``Structure-Preserving Time-Stepping Methods for Differential Equations", February 26 - March 3, 2023, Society for Industrial and Applied Mathematic, Amsterdam, Netherlands, March 3, 2023.

Y. Hadjimichael, An energy-based finite-strain constitutive model for bent heterostructured materials, GAMM 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics, May 30 - June 2, 2023, Technische Universität Dresden, June 2, 2023.

M. Heida, Finite volumes for simulation of large molecules, Finite Volumes for Complex Applications (FVCA10), Strasbourg, France, October 30 - November 3, 2023.

M. O'Donovan, Atomistic study of Urbach tail energies in (Al,Ga)N quantum well systems, 23rd International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2023), September 18 - 21, 2023, Politecnico di Torino, Italy, September 21, 2023.

M. O'Donovan, Impact of alloy disorder on carrier transport and recombination in (Al,Ga)N-based UV-C emitters, The 6th International Workshop on Ultraviolet Materials and Devices (IWUMD 2023), June 5 - 8, 2023, Metz Congrès Robert Schuman, France, June 7, 2023.

M. O'Donovan, Modeling random alloy fluctuations in carrier transport simulations of III-N based light emitting diodes - Connecting atomistic tight-binding to drift-diffusion, 15th Annual Meeting Photonic Devices, March 29 - 31, 2023, Zuse-Institut Berlin, March 31, 2023.

D. Peschka, Multiscale limits of thin-film models with moving support, In Search of Model Structures for Non-equilibrium Systems, April 24 - 28, 2023, Westfälische Wilhelms-Universität Münster, April 27, 2023.

D. Peschka, Multiscale limits of thin-film models with moving support, Kolloquium des SFB 1114, Freie Universität Berlin, April 20, 2023.

D. Peschka, Sharp-interface limit of models with mechanics and contact lines, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Session 00247 ``Interfaces and Free Boundaries in Fluid Mechanics and Materials Science'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 24, 2023.

D. Runge, Reduced basis approach for convection-diffusion equations with non-linear boundary reaction conditions, Finite Volumes for Complex Applications 10 (FVCA10), Université de Strasbourg, France, November 2, 2023.

D. Runge, Mass-conservative reduced basis approach for convection-diffusion equations with non-linear boundary reaction conditions, Leibniz MMS Days 2023, Leibniz Network "Mathematical Modeling and Simulation", Leibniz Institute for Agricultural Engineering and Bioeconomy Potsdam (ATB), Potsdam, April 18, 2023.

L. Schütz, M. Heida, M. Thomas, Materials with discontinuities on many scales, SCCS Days 2023 of the Collaborative Research Center -- CRC 1114 ``Scaling Cascades in Complex Systems'', November 13 - 15, 2023.

M. Kniely, A thermodynamically correct framework for electro-energy-reaction-diffusion systems, 22nd ECMI Conference on Industrial and Applied Mathematics, June 26 - 30, 2023, Wrocław University of Science and Technology, Poland, June 30, 2023.

M. Kniely, On a thermodynamically consistent electro-energy-reaction-diffusion system, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), Session 14 ``Applied Analysis'', May 30 - June 2, 2023, Technische Universität Dresden, June 1, 2023.

M. Demir, Subgrid artificial viscosity modelling based defect-deferred correction method for fluid-fluid interaction, 2023 International CMMSE conference and the Second conference on high performance computing (CHPC), July 3 - 8, 2023, Universidad de Cádiz, Spain, July 6, 2023.

M. Demir, Subgrid artificial viscosity modelling based defect-deferred correction method for fluid-fluid interaction, European Conference on Numerical Mathematics and Advanced Applications (ENUMATH), September 4 - 8, 2023, Instituto Superior Técnico, Lisboa, Portugal, September 7, 2023.

K. Hopf, Normal form and the Cauchy problem for cross-diffusive mixtures, Workshop ``Variational Methods for Evolution'', December 3 - 8, 2023, Mathematisches Forschungsinstitut Oberwolfach, December 4, 2023.

K. Hopf, On cross-diffusive coupling of hyperbolic-parabolic type, Variational and Geometric Structures for Evolution, October 9 - 13, 2023, Centro Internazionale per la Ricerca Matematica (CIRM), Levico Terme, Italy, October 13, 2023.

K. Hopf, Structure and approximation of cross-diffusive mixtures with incomplete diffusion, Universität Kassel, Fachbereich Mathematik und Naturwissenschaften, September 21, 2023.

K. Hopf, Structure, dynamics, and approximation of cross-diffusive mixtures with incomplete diffusion, Universität Hamburg, Fachbereich Mathematik, May 10, 2023.

K. Hopf, The Cauchy problem for multi-component systems with strong cross-diffusion, Johannes Gutenberg-Universität Mainz, Fachbereich Physik, Mathematik und Informatik, January 11, 2023.

M. Thomas, Damage in viscoelastic materials at finite strains, Workshop ``Variational Methods for Evolution'', December 3 - 8, 2023, Mathematisches Forschungsinstitut Oberwolfach, December 7, 2023.

M. Thomas, Some aspects of damage in nonlinearly elastic materials: From damage to delamination in nonlinearly elastic materials, Variational and Geometric Structures for Evolution, October 9 - 13, 2023, Università Commerciale Luigi Bocconi, Levico Terme, Italy, October 10, 2023.

M. Thomas, Approximating dynamic phase-field fracture with a first-order formulation for velocity and stress, Nonlinear PDEs: Recent Trends in the Analysis of Continuum Mechanics, July 17 - 21, 2023, Universität Bonn, Hausdorff School for Advanced Studies in Mathematics, July 17, 2023.

M. Thomas, Approximating dynamic phase-field fracture with a first-order formulation for velocity and stress, Seminar für Angewandte Mathematik, Technische Universität Dresden, June 5, 2023.

M. Thomas, Nonlinear fracture dynamics: Modeling, analysis, approximation, and applications, Presentation of project proposals in SPP 2256 ``Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials'', Bad Honnef, March 27, 2023.

A. Alphonse, Analysis of a quasi-variational contact problem arising in thermoelasticity, European Conference on Computational Optimization (EUCCO), Session ``Non-smooth Optimization'', September 25 - 27, 2023, Universität Heidelberg, September 25, 2023.

U. Bandelow, Laserdynamik -- mathematische Modellierung, MBI-Technikerschulung 2023, October 16, 2023, Max-Born-Institut, Wandlitz.

U. Bandelow, Modeling and simulation of semiconductor devices: From high-power lasers to quantum technologies, Winter School on III-Sb Applications: Non-Volatile Memories -- A Modelling Perspective, February 27 - March 3, 2023, Technische Universität Berlin, February 27, 2023.

U. Bandelow, Ultrashort solitons in the regime of event horizons in nonlinear optical media, Extreme Waves 2023, August 28 - September 1, 2023, Max-Planck-Institut für Physik komplexer Systeme, Dresden, August 29, 2023.

U. Bandelow, Unusual scenarios in the context of the modulation instability, Dissipative Solitons, Turbulence and Extreme Events in Nonlinear Photonics, September 6 - 8, 2023, International Solvay Institutes, Brussels, Belgium, September 7, 2023.

U. Bandelow, Unusual wave-mixing processes in the context of the modulation instability, XIX International Workshop on Instabilities and Nonequilibrium Structures -- INES 2023, December 4 - 8, 2023, Pontificia Universidad Catolica de Valparaiso, Chile, December 7, 2023.

A. Caiazzo, Multiscale and reduced-order modeling for poroelasticity, European Conference on Numerical Mathematics and Advanced Applications (ENUMATH), September 4 - 8, 2023, Instituto Superior Técnico, Lisboa, Portugal.

P. Farrell, Modeling and numerical simulation of two-dimensional TMDC memristive devices, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Tokyo, Japan, August 20 - 25, 2023.

P. Farrell, Modeling and numerical simulation of two-dimensional memristive devices, 22nd European Consortium for Mathematics in Industry (ECMI) Conference on Industrial and Applied Mathematics, June 26 - 30, 2023, Wrocław University of Science and Technology, Poland.

P. Farrell, Device physics characterization and interpretation in perovskite and organic materials (DEPERO), October 3 - 5, 2023, Eidgenössische Technische Hochshcule Zürich, nanoGe, Switzerland.

J. Fuhrmann, Thermodynamically consistent finite volume schemes for electrolyte simulations, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 22, 2023.

J. Fuhrmann, Two entropic finite volume schemes for a Nernst--Planck--Poisson system with ion volume constraints, Finite Volumes for Complex Applications 10 (FVCA10), Université de Strasbourg, France, November 1, 2023.

J. Fuhrmann, Voronoi finite volume methods for complex applications in Julia, International Conference on Numerical Analysis of Partial Differential Equations (ANEDP 2023), October 16 - 18, 2023, Moulay Ismail University, Faculty of Sciences, Meknes, Morocco.

V. John, On slope limiters in discontinuous Galerkin discretizations of convection-diffusion problems, European Conference on Numerical Mathematics and Advanced Applications (ENUMATH), Minisymposium MS06 ' 'Theoretical and computational aspects of the discontinuous Galerkin method' ', September 4 - 8, 2023, Instituto Superior Técnico, Lisboa, Portugal, September 5, 2023.

V. John, Finite element methods respecting the discrete maximum principle for convection-diffusion equations I, 19th Workshop on Numerical Methods for Problems with Layer Phenomena, Charles University, Faculty of Mathematics and Physics, Department of Numerical Mathematics, Prague, Czech Republic, May 26, 2023.

V. John, On recent topics in the finite element analysis of convection-diffusion problems (online talk), Numerical Analysis Seminar (Hybrid Event), University of Waterloo, Applied Mathematics, Canada, April 11, 2023.

M. Liero, Variational modeling of biomechanical systems, 10th International Conference on Computational Bioengineering (ICCB 2023), Minisymposium 22-3 ``Continuum Biomechanics of Active Biological Systems'', September 20 - 22, 2023, Technische Universität Wien, Austria, September 22, 2023.

CH. Merdon, Gradient-robust hybrid discontinuous Galerkin discretizations for the compressible Stokes equations, Forschungsseminar von Prof. Carsten Carstensen, Humboldt-Universität zu Berlin, Institut fuer Mathematik, October 17, 2023.

CH. Merdon, Raviart--Thomas enriched Scott--Vogelius finite element methods for the Navier--Stokes equations (online talk), City University of Hong Kong, Department of Mathematics, Hong Kong, January 18, 2023.

CH. Merdon, Raviart--Thomas enriched Scott--Vogelius FEM for the Navier--Stokes equations, Capita Selecta Seminar, SACS - Systems, Analysis and Computational Sciences, Department of Mathematics University of Twente (DAMUT), Enschede, Netherlands, May 10, 2023.

CH. Merdon, Raviart--Thomas enriched Scott--Vogelius finite element methods for the Navier--Stokes equations, GAMM 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics, May 30 - June 2, 2023, Technische Universität Dresden, June 2, 2023.

CH. Merdon, Raviart--Thomas-enriched Scott--Vogelius finite element methods for the Stokes equations on general meshes, The 29th Biennial Numerical Analysis Conference 2023, June 27 - 30, 2023, University of Strathclyde, Department of Mathematics and Statistics, Glasgow, UK, June 27, 2023.

O. Pártl, A computational framework for sustainable geothermal energy production in fracture-controlled reservoir based on well placement optimization, Leibniz MMS Days 2023, Potsdam, April 17, 2023.

O. Pártl, Finite element methods respecting the discrete maximum principle for convection-diffusion equations III, 19th Workshop on Numerical Methods for Problems with Layer Phenomena, Charles University, Faculty of Mathematics and Physics, Department of Numerical Mathematics, Prague, Czech Republic, May 26, 2023.

O. Pártl, Reconstruction of flow domain boundaries from velocity data via multi-step optimization of distributed resistance, European Conference on Numerical Mathematics and Advanced Applications (ENUMATH), September 4 - 8, 2023, Instituto Superior Técnico, Lisboa, Portugal.

A. Stephan, On time-splitting methods for gradient flows with two dissipation mechanisms, In Search of Model Structures for Non-equilibrium Systems, April 24 - 28, 2023, Westfälische Wilhelms-Universität Münster, April 28, 2023.

CH. Keller, J. Fuhrmann, M. Landstorfer, B. Wagner, Development of an ion-channel model-framework for in-vitro assisted interpretation of current voltage relations, MATH+-Day 2022, Technische Universität Berlin, November 18, 2022.

A. Maltsi, Symmetries in TEM images of strained crystals, ``European Women in Mathematics'' General Meeting 2022, Espoo, Finland, August 22 - 26, 2022.

A. Maltsi, Symmetries in TEM images of strained crystals, BMS-BGSMath Junior Meeting, September 5 - 7, 2022, Universidad de Barcelona, Spain, September 6, 2022.

A. Alphonse, Directional differentiability and optimal control for quasi-variational inequalities (online talk), ``Partial Differential Equations and their Applications'' Seminar, University of Warwick, Mathematics Institute, UK, January 25, 2022.

R. Müller, Non-equilibrium thermodynamics modeling of polycrystalline electrode liquid electrolyte interface, 31st Topical Meeting of the International Society of Electrochemistry, Meeting topic: ``Theory and Computation in Electrochemistry: Seeking Synergies in Methods, Materials and Systems'', Session 2: ``Theory and Computation of Interfacial and Nanoscale Phenomena'', May 15 - 19, 2022, Rheinisch-Westfälische Technische Hochschule Aachen, May 17, 2022.

D. Peschka, Discretization of compressible Stokes flow using Hamiltonian and Onsager structures, Workshop on Numerical Methods and Analysis in CFD, July 5 - 8, 2022, WIAS Berlin, July 5, 2022.

D. Peschka, Gradient flows coupling order parameters and mechanics (online talk), Colloquium of the SPP 2171 (Online Event), Westfälische Wilhelms-Universität Münster, October 21, 2022.

D. Peschka, Steering the pattern formation of dewetting liquids (online talk), SFB 910 Symposium ``Pattern formation and coherent structure in dissipative systems'' (Online Event), Technische Universität Berlin, January 14, 2022.

A. Selahi, Homogenization in PDE-based battery models and recovery of ageing dynamics via Bayesian inference (online talk), Second Conference of Young Applied Mathematicians (Hybrid Event), September 18 - 22, 2022, Arenzano, Italy, September 21, 2022.

P. Vágner, Capacitance of the blocking YSZ I Au electrode, 18th Symposium on Modeling and Experimental Validation of Electrochemical Energy Technologies, March 14 - 16, 2022, DLR Institut für Technische Thermodynamik, Hohenkammer, March 16, 2022.

J. Sprekels, Deep quench approach and sparsity in the optimal control of a phase field model for tumor growth, PHAse field MEthods in applied sciences (PHAME 2022), May 23 - 27, 2022, Istituto Nazionale di Alta Matematica, Rome, Italy, May 27, 2022.

K. Hopf, Relative entropies and stability in strongly coupled parabolic systems (online talk), SIAM Conference on Analysis of Partial Differential Equations (PD22) (Online Event), Minisymposium ``Variational Evolution: Analysis and Multi-Scale Aspects'', March 14 - 18, 2022, March 16, 2022.

M. Thomas, First-order formulation for dynamic phase-field fracture in visco-elastic materials, PHAse field MEthods in applied sciences (PHAME 2022), May 23 - 27, 2022, Istituto Nazionale di Alta Matematica, Rome, Italy, May 25, 2022.

M. Thomas, First-order formulation for dynamic phase-field fracture in visco-elastic materials, Beyond Elasticity: Advances and Research Challenges, May 16 - 20, 2022, Centre International de Rencontres Mathématiques, Marseille, France, May 16, 2022.

TH. Eiter, Energy-variational solutions for a viscoelastoplastic fluid model (online talk), SIAM Conference on Analysis of Partial Differential Equations (PD22) (Online Event), Minisymposium ``Generalized Solvability Concepts for Evolutionary PDEs and their Properties'', March 14 - 18, 2022, March 16, 2022.

P. Farrell, Challenges for non-Boltzmann drift-diffusion charge transport simulations in semiconductors, International Conference on Boundary and Interior Layers, November 28 - December 2, 2022, Universidad de Buenos Aires, Argentina, December 1, 2022.

P. Farrell, Data-driven solutions of ill-posed inverse problems arising from doping reconstruction in semiconductors, AI4Science 2022 : AI for Science, NASSMA Workshop, Rabat, Morocco, December 12 - 16, 2022.

M. Landstorfer, A. Selahi, M. Heida, M. Eigel, Recovery of battery ageing dynamics with multiple timescales, MATH+-Day 2022, Technische Universität Berlin, November 18, 2022.

M. Liero, From diffusion to reaction-diffusion in thin structures via EDP-convergence (online talk), SIAM Conference on Analysis of Partial Differential Equations (PD22) (Online Event), Minisymposium ``Bridging Gradient Flows, Hypocoercivity and Reaction-Diffusion Systems'', March 14 - 18, 2022, March 14, 2022.

M. Liero, Modeling, analysis, and simulation of electrothermal feedback in organic devices, Audit 2022, September 22 - 23, 2022, Weierstraß-Institut Berlin, September 22, 2022.

M. Liero, The impact of modeling, analysis, and simulation on organic semiconductor development (online talk), ERCOM Meeting 2022 (Hybrid Event), March 25 - 26, 2022, European Research Centers on Mathematics, Bilbao, Spain, March 26, 2022.

A. Mielke, On the existence and longtime behavior of solutions to a degenerate parabolic system (online talk), SIAM Conference on Analysis of Partial Differential Equations (PD22) (Online Event), Minisymposium MS43: ``Nonlinear Parabolic Equations and Systems'', March 14 - 18, 2022, March 16, 2022.

D. Abdel, P. Vágner, J. Fuhrmann, P. Farrell, Modeling and simulation of charge transport in perovskite solar cells, AMaSiS 2021: Applied Mathematics and Simulation for Semiconductors and Electrochemical Systems (Online Event), September 6 - 9, 2021.

D. Abdel, P. Vágner, J. Fuhrmann, P. Farrell, Modelling charge transport in perovskite solar cells: Potential-based and limiting ion depletion, SIAM Conference on Computational Science and Engineering -- CSE21 (Online Event), Texas, USA, March 1 - 5, 2021.

A. Maltsi, Model-based geometry reconstruction of TEM images, MATH+ Day 2021 (Online Event), Technische Universität Berlin, November 5, 2021.

A. Maltsi, On the Darwin--Howie--Whelan equations in TEM imaging (online talk), Young Women in PDEs and Applications (Online Event), September 20 - 22, 2021, Universität Bonn, September 22, 2021.

A. Maltsi, Quantum dots and TEM images from a mathematician's perspective, Women in Mathematics Webinar (Online Event), UK, February 11 - 12, 2021.

M.H. Farshbaf Shaker, D. Peschka, M. Thomas, B. Wagner, Variational methods for viscoelastic flows and gelation, MATH+ Day 2021 (Online Event), Technische Universität Berlin, November 5, 2021.

M. Landstorfer, Modeling of concentration and electric field dependent susceptibilities in electrolytes (online talk), AA2 -- Materials, Light, Devices, Freie Universität Berlin, Humboldt-Universität zu Berlin, WIAS Berlin, February 26, 2021.

O. Marquardt, Modelling the electronic properties of semiconductor nanowire heterostructures (online talk), PDI-Seminar, Paul-Drude-Institut für Festkörperelektronik, August 23, 2021.

L. Mertenskötter, M. Kantner, H. Wenzel , U. Bandelow, Modeling and optimization of semiconductor lasers for quantum metrology applications, MATH+ Day 2021 (Online Event), Technische Universität Berlin, November 5, 2021.

R. Müller, Modeling polycrystalline electrode-electrolyte interfaces: The differential capacitance (online talk), 14th Virtual Congress WCCM & ECCOMAS 2020, January 11 - 15, 2021, January 11, 2021.

D. Peschka, Mathematical modeling and simulation of substrate-flow interaction using generalized gradient flows, Conference ``Dynamic Wetting of Flexible, Adaptive, and Switchable Substrates'' of the SPP 2171, Freiburg, November 8 - 10, 2021.

D. Peschka, Mathematical modeling and simulation of substrate-flow interaction using generalized gradient flows, SPP 2171 Conference ``Dynamic Wetting of Flexible, Adaptive, and Switchable Substrates'', November 8 - 10, 2021, Albert-Ludwigs-Universität Freiburg, November 10, 2021.

A. Zafferi, Coupling of thermoviscoelastic solids and reactive flows via GENERIC (online talk), CRC 1114 Conference 2021 (Online Event), MSDI4: ``Modeling and Analysis of Geological Fluid Flows'', March 1 - 3, 2021, Freie Universität Berlin, March 2, 2021.

A. Zafferi, Dynamics of rock dehydration on multiple scales, CRC 1114 Conference 2021 (Online Event), March 1 - 3, 2021.

A. Zafferi, Thermodynamics of reaction-diffusion-induced rock dehydration processes (online talk), 16thJoint European Thermodynamics Conference (Hybrid Event), June 14 - 18, 2021, Charles University Prague, Czech Republic, June 16, 2021.

B. Gaudeul, J. Fuhrmann, Two entropic finite volume schemes for a Nernst--Planck--Poisson system with ion volume constraints, AMaSiS 2021: Applied Mathematics and Simulation for Semiconductors and Electrochemical Systems (Online Event), September 6 - 9, 2021.

A. Glitzky, A coarse-grained electrothermal model for organic semiconductor devices (online talk), DMV-ÖMG Jahrestagung 2021 (Online Event), September 27 - October 1, 2021, Universität Passau, September 29, 2021.

M. Thomas, Convergence analysis for fully discretized damage and phase-field fracture models (online talk), 15th International Conference on Free Boundary Problems: Theory and Applications 2021 (FBP 2021, Online Event), Minisymposium ``Phase Field Models'', September 13 - 17, 2021, WIAS, Berlin, September 14, 2021.

M. Thomas, GENERIC structures with bulk-interface interaction (online talk), 16th Joint European Thermodynamics Conference (Hybrid Event), June 14 - 18, 2021, Charles University Prague, Czech Republic, June 17, 2021.

U. Bandelow, Modeling and simulation of the dynamics in semiconductor lasers (online talk), 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics, MS1: ``Computational Photonics''' (Online Event), March 15 - 19, 2021, Universität Kassel, March 16, 2021.

U. Bandelow , Ultrashort solitons in the regime of event horizons in nonlinear dispersive optical media, Solvay Workshop on Dissipative Solitons and Optical Frequency Comb Generation, September 15 - 16, 2021, International Solvay Institutes, Brussels, Belgium, September 16, 2021.

P.-E. Druet, Modeling and analysis for multicomponent incompressible fluids (online talk), 8th European Congress of Mathematics (8ECM), Minisymposium ID 51 ``Partial Differential Equations describing Far-from-Equilibrium Open Systems'' (Online Event), June 20 - 26, 2021, Portorož, Slovenia, June 23, 2021.

P.-E. Druet, Well-posedness results for mixed-type systems modelling pressure-driven multicomponent fluid flows (online talk), 8th European Congress of Mathematics (8ECM), Minisymposium ID 42 ``Multicomponent Diffusion in Porous Media'' (Online Event), June 20 - 26, 2021, Portorož, Slovenia, June 22, 2021.

M. Hintermüller, Semi-smooth Newton methods: Theory, numerical algorithms and applications I (online talk), International Forum on Frontiers of Intelligent Medical Image Analysis and Computing 2021 (Online Forum), Xidian University, Southeastern University, and Hong Kong Baptist University, China, July 19, 2021.

M. Hintermüller, Semi-smooth Newton methods: Theory, numerical algorithms and applications II (online talk), International Forum on Frontiers of Intelligent Medical Image Analysis and Computing 2021 (Online Forum), Xidian University, Southeastern University, and Hong Kong Baptist University, China, July 26, 2021.

O. Marquardt, Data-driven electronic structure calculations for semiconductor nanostructures, Efficient Algorithms for Numerical Problems, January 17, 2020, WIAS Berlin, January 17, 2020.

G. Nika, An existence result for a class of electrothermal drift-diffusion models with Fermi--Gauss statistics for organic semiconductors, Joint Mathematics Meeting, January 15 - 18, 2020, American Mathematical Society/ Mathematical Association of America, Denver, USA, January 15, 2020.

D. Peschka, Mathematical modeling and simulation of flows and the interaction with a substrate using energetic variational methods, CRC 1194 ``Interaction between Transport and Wetting Processes'', Technische Universität Darmstadt, January 22, 2020.

A. Zafferi, K. Huber, C09 - Dynamics of rock dehydration (online talk), SCCS Days 2020 of the Collaborative Research Center - CRC 1114 ``Scaling Cascades in Complex Systems'', December 2 - 4, 2020, Freie Universität Berlin, December 3, 2020.

A. Zafferi, K. Huber, Dynamics of rock dehydration, CRC 1114: Scaling Cascades in Complex Systems (SCCS Days) (Online Event), December 2 - 4, 2020.

N. Dropka , P. Farrell, S. Kayser, N. Rotundo, Numerics for innovative semiconductor devices -- An outlook, German Conference on Crystal Growth, Munich, March 11 - 13, 2020.

M.H. Farshbaf Shaker, D. Peschka, M. Thomas, Modeling and analysis of suspension flows, MATH+ Day 2020 (Online Event), Berlin, November 6, 2020.

C. Cancès, C. Chainais-Hillairet, J. Fuhrmann, B. Gaudeul, On four numerical schemes for a unipolar degenerate drift-diffusion model, Finite Volumes for Complex Applications IX (Online Event), Bergen, Norway, June 15 - 19, 2020.

K. Hopf, Global existence analysis of energy-reaction-diffusion systems, Workshop ``Variational Methods for Evolution'', September 13 - 19, 2020, Mathematisches Forschungsinstitut Oberwolfach, September 15, 2020.

M. Thomas, Nonlinear fracture dynamics: Modeling, analysis, approximation, and applications, Presentation of project proposals in SPP 2256 ``Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials'', Bad Honnef, January 30, 2020.

M. Thomas, Thermodynamical modelling via energy and entropy functionals (online talks), Thematic Einstein Semester on Energy-based Mathematical Methods for Reactive Multiphase Flows: Student Compact Course ``Variational Methods for Fluids and Solids'' (Online Event), October 12 - 23, 2020, WIAS Berlin.

M. Thomas, Weierstraß-Gruppe "Volumen-Grenzschicht-Prozesse", Sitzung des Wissenschaftlichen Beirats, WIAS Berlin, September 18, 2020.

B. Wagner, Pattern formation in dewetting films (online talk), Workshop ``Mathematical Modeling and Scientific Computing: Focus on Complex Processes and Systems'' (Online Event), November 19 - 20, 2020, Technische Universität München, November 19, 2020.

B. Wagner, Phase-field models of the lithiation/delithiation cycle of thin-film electrodes (online talk), Oxford Battery Modelling Symposium (Online Event), March 16 - 17, 2020, University of Oxford, UK, March 16, 2020.

P. Farrell, D. Peschka, Challenges in drift-diffusion semiconductor simulations, Finite Volumes for Complex Applications IX (Online Event), Bergen, Norway, June 15 - 19, 2020.

J. Fuhrmann, D.H. Doan, A. Glitzky, M. Liero, G. Nika, Unipolar drift-diffusion simulation of S-shaped current-voltage relations for organic semiconductor devices, Finite Volumes for Complex Applications IX (Online Event), Bergen, Norway, June 15 - 19, 2020.

J. Fuhrmann, C. Guhlke, M. Landstorfer, A. Linke, Ch. Merdon, R. Müller, Quality preserving numerical methods for electroosmotic flow, Einstein Semester on Energy-based Mathematical Methods for Reactive Multiphase Flows: Kick-off Conference (Online Event), October 26 - 30, 2020.

M. Kantner, Non-isothermal Scharfetter--Gummel scheme for electro-thermal transport simulation in degenerate semiconductors, Finite Volumes for Complex Applications IX (Online Event), June 15 - 19, 2020, University of Bergen, Bergen, Norway, June 16, 2020, DOI 10.1007/978-3-030-43651-3_14 .

M. Kantner, Non-isothermal Scharfetter--Gummel scheme for electro-thermal transport simulation in degenerate semiconductors (online talk), Finite Volumes for Complex Applications IX (Online Event), June 15 - 19, 2020, NORCE - Norwegian Research Centre, Bergen, Norway, June 16, 2020.

TH. Koprucki, K. Tabelow, T. Streckenbach, T. Niermann, A. Maltsi, Model-based geometry reconstruction of TEM images, MATH+ Day 2020 (Online Event), Berlin, November 6, 2020.

R. Lasarzik, Dissipative solutions in the context of the numerical approximation of nematic electrolytes (online talk), Oberseminar Numerik, Universität Bielefeld, Fakultät für Mathematik, June 23, 2020.

M. Liero, Drift-diffusion simulation of S-shaped current-voltage relations for organic semiconductor devices (online talk), SimOEP 2020: International Conference on Simulation of Organic Electronics and Photovoltaics (Online Event), August 31 - September 2, 2020, Zürcher Hochschule für Angewandte Wissenschaften, Switzerland, September 1, 2020.

A. Linke, Ch. Merdon, On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond, Einstein Semester on Energy-based Mathematical Methods for Reactive Multiphase Flows: Kick-off Conference (Online Event), Berlin, October 26 - 30, 2020.

A. Linke, Ch. Merdon, On the significance of pressure-robustness for the space discretization of incompressible high Reynolds number flows, Finite Volumes for Complex Applications IX (Online Event), Bergen, Norway, June 15 - 19, 2020.

H. Si, Adaptive exponential time integration of the Navier--Stokes equations, 3rd AIAA Sonic Boom Prediction Workshop, January 5 - 10, 2020, American Institute of Aeronautics and Astronautics SciTech Forum, Orlando, Florida, USA, January 10, 2020.

A. Maltsi, Th. Koprucki, T. Streckenbach, K. Tabelow, J. Polzehl, Model-based geometry reconstruction of quantum dots from TEM, Microscopy Conference 2019, Poster session IM 4, Berlin, September 1 - 5, 2019.

A. Maltsi, Th. Koprucki, T. Streckenbach, K. Tabelow, J. Polzehl, Model-based geometry reconstruction of quantum dots from TEM, BMS Summer School 2019: Mathematics of Deep Learning, Berlin, August 19 - 30, 2019.

A. Alphonse, Directional differentiability for elliptic quasi-variational inequalities, Workshop ``Surface, Bulk, and Geometric Partial Differential Equations: Interfacial, Stochastic, Non-local and Discrete Structures'', January 20 - 26, 2019, Mathematisches Forschungsinstitut Oberwolfach, January 25, 2019.

A. Alphonse, Directional differentiability for elliptic quasi-variational inequalities, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Best Paper Session, August 5 - 8, 2019, Berlin, August 5, 2019.

M. Heida, Convergences of the squareroot approximation scheme to the Fokker--Planck operator, ``9th International Congress on Industrial and Applied Mathematics" (ICIAM 2019), July 15 - 19, 2019, Universitat de València, Spain, July 17, 2019.

M. Kantner, A generalized Scharfetter--Gummel scheme for degenerate and non-isothermal semiconductors, 19th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), July 8 - 12, 2019, University of Ottawa, Canada, July 8, 2019.

M. Kantner, Device-scale simulation of quantum light emitting diodes, International Symposium ,,Semiconductor Nanophotonics'', November 4 - 5, 2019, Technische Universität Berlin, November 4, 2019.

M. Kantner, Hybrid modeling of quantum light emitting diodes: Self-consistent coupling of drift-diffusion, Schrödinger--Poisson, and quantum master equations, SPIE Photonics West, February 5 - 7, 2019, San Francisco, USA, February 6, 2019, DOI 10.1117/12.2515209 .

M. Kantner, Simulation of quantum dot based single-photon sources using the Schrödinger--Poisson-Drift-Diffusion-Lindblad system, International Conference on Simulation of Semiconductor Processes and Devices (SISPAD 2019), September 4 - 6, 2019, Università degli Studi di Udine, Italy, September 6, 2019.

M. Kantner, Simulation of quantum light sources using the self-consistently coupled Schrödinger--Poisson-Drift-Diffusion-Lindblad system, 19th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), July 8 - 12, 2019, University of Ottawa, Canada, July 8, 2019.

T. Keil, Optimal control of a coupled Cahn--Hilliard--Navier--Stokes system with variable fluid densities, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Session ``Optimal Control of Phase Field Models'', August 5 - 8, 2019, Berlin, August 5, 2019.

R. Müller, Transport of solvated ions in nanopores: Modeling, asymptotics and simulation, Conference to celebrate the 80th jubilee of Miroslav Grmela, May 18 - 19, 2019, Czech Technical University, Faculty of Nuclear Sciences and Physical Engineering, Prag, Czech Republic, May 18, 2019.

R. Müller, Transport phenomena in electrolyte within a battery cell, Battery Colloquium, Technische Universität Berlin, April 18, 2019.

G. Nika, An existence result for a class of electrothermal drift-diffusion models with Gauss--Fermi statistics for organic semiconductors, DMV-Jahrestagung 2019, September 23 - 26, 2019, KIT - Karlsruher Institut für Technologie.

G. Nika, Homogenization for a multi-scale model of magnetorheological suspension, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Minisymposium MS ME-1-3 1 ``Emerging Problems in the Homogenization of Partial Differential Equations'', July 15 - 19, 2019, Valencia, Spain, July 15, 2019.

D. Peschka, Dynamic contact angles via generalized gradient flows, Modelling of Thin Liquid Films -- Asymptotic Approach vs. Gradient Dynamics, April 28 - May 3, 2019, Banff International Research Station for Mathematical Information and Discovery, Canada, April 30, 2019.

D. Peschka, Dynamic contact angles via gradient flows, 694. WE-Heraeus-Seminar ``Wetting on Soft or Microstructured Surfaces'', Bad Honnef, April 10 - 13, 2019.

D. Peschka, Gradient formulations with flow maps -- Mathematical and numerical approaches to free boundary problems, Kolloquium des Graduiertenkollegs 2339 ``Interfaces, Complex Structures, and Singular Limits'', Universität Regensburg, May 24, 2019.

D. Peschka, Gradient structures for flows of concentrated suspensions - jamming and free boundaries, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S11 ``Interfacial Flows", February 18 - 22, 2019, Technische Universität Wien, Austria, February 20, 2019.

D. Peschka, Mathematical modeling of fluid flows using gradient systems, Seminar in PDE and Applications, Delft University of Technology, Netherlands, May 28, 2019.

D. Peschka, Steering pattern formation of viscous flows, DMV-Jahrestagung 2019, Sektion ``Differentialgleichungen und Anwendungen'', September 23 - 26, 2019, KIT - Karlsruher Institut für Technologie, September 23, 2019.

D. Peschka, ``Numerical methods for charge transport in semiconductors: FEM vs FV", 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), July 15 - 19, 2019, Valencia, Spain, July 17, 2019.

A. Zafferi, An approach to multi-phase flows in geosciences, MURPHYS-HSFS 2019 Summer School on Multi-Rate Processes, Slow-Fast Systems and Hysteresis, Turin, Italy, June 17 - 21, 2019.

A. Glitzky, Drift-diffusion problems with Gauss--Fermi statistics and field-dependent mobility for organic semiconductor devices, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis'', February 18 - 22, 2019, Universität Wien, Technische Universität Wien, Austria, February 22, 2019.

K. Hopf, On the singularity formation and relaxation to equilibrium in 1D Fokker--Planck model with superlinear drift, Gradient Flows and Variational Methods in PDEs, November 25 - 29, 2019, Universität Ulm, November 25, 2019.

M. Thomas, Analysis for the discrete approximation of gradient-regularized damage models, Mathematics Seminar Brescia, Università degli Studi di Brescia, Italy, March 13, 2019.

M. Thomas, Analysis for the discrete approximation of gradient-regularized damage models, PDE Afternoon, Universität Wien, Austria, April 10, 2019.

M. Thomas, Analytical and numerical aspects for the approximation of gradient-regularized damage models, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Thematic Minisymposium MS A3-2-26 ``Phase-Field Models in Simulation and Optimization'', July 15 - 19, 2019, Valencia, Spain, July 17, 2019.

M. Thomas, Analytical and numerical aspects of rate-independent gradient-regularized damage models, Conference ``Dynamics, Equations and Applications (DEA 2019)'', Session D444 ``Topics in the Mathematical Modelling of Solids'', September 16 - 20, 2019, AGH University of Science and Technology, Kraków, Poland, September 19, 2019.

M. Thomas, Coupling of rate-independent and rate-dependent systems, MURPHYS-HSFS 2019 Summer School on Multi-Rate Processes, Slow-Fast Systems and Hysteresis, June 17 - 19, 2019, Politecnico di Torino, Turin, Italy.

M. Thomas, Dynamics of rock dehydration on multiple scales, SCCS Days 2019 of the Collaborative Research Center - CRC 1114, May 20 - 22, 2019, Freie Universität Berlin, Zeuthen, May 21, 2019.

M. Thomas, GENERIC structures with bulk-interface interaction, SFB 910 Symposium ``Energy Based Modeling, Simulation and Control'', October 25, 2019, Technische Universität Berlin, October 25, 2019.

A. Caiazzo, Data assimilation in one-dimensional hemodynamics, European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2019), Minisymposium 36 ``Data-Driven Computational Fluid Dynamics (Part 2)'', September 30 - October 4, 2019, Eindhoven University of Technology, Netherlands, October 1, 2019.

A. Caiazzo, Geothermal reservoir: Modeling, simulation and optimization for district heating in hot sedimentary acquires, Leibniz MMS Days 2019, March 20 - 22, 2019, Universität Rostock , Leibniz-Institut für Atmosphärenphysik, Kühlungsborn, March 22, 2019.

A. Caiazzo, Multiscale hybrid modeling and simulation of cancer growth within a 3D heterogeneous tissue, Canada-Germany Workshop Mathematical Biology and Numerics, June 24 - 26, 2019, Universität Heidelberg, June 26, 2019.

J. Fuhrmann, A. Linke, Ch. Merdon, R. Müller, Induced charge electroosmotic flow including finite ion size effects, 13th International Symposium on Electrokinetics (ELKIN), Cambridge, USA, June 12 - 14, 2019.

A. Mielke, Thermodynamical modeling via GENERIC: From quantum mechanics to semiconductor devices, Institute of Thermomechanics's Seminar, Czech Academy of Sciences, Prague, March 21, 2019.

A. Maltsi, Th. Koprucki, T. Niermann, T. Streckenbach, K. Tabelow, J. Polzehl, Computing TEM images of semiconductor nanostructures, Applied Mathematics and Simulation for Semiconductors (AMaSiS 2018), WIAS Berlin, October 8 - 10, 2018.

N. Rotundo, On a thermodynamically consistent coupling of quantum system and device equations, The 20th European Conference on Mathematics for Industry (ECMI 2018), Minisymposium ``Mathematical Modeling of Charge Transport in Graphene and Low Dimensional Structures'', August 18 - June 22, 2018, Budapest, Hungary, June 19, 2018.

A. Alphonse, Directional differentiability for elliptic QVIs of obstacle type, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2018), Session PP07 ``DFG Priority Program 1962'', March 19 - 23, 2018, Technische Universität München, March 20, 2018.

A. Alphonse, Directional differentiability for elliptic quasi-variational inequalities, Workshop ``Challenges in Optimal Control of Nonlinear PDE-Systems'', April 8 - 14, 2018, Mathematisches Forschungsinstitut Oberwolfach, April 12, 2018.

A. Alphonse, Optimal Control of Elliptic and Parabolic Quasi-Variational Inequalities, Annual Meeting of the DFG Priority Programme 1962, October 1 - 3, 2018, Kremmen (Sommerfeld), October 3, 2018.

A. Alphonse, Parabolic quasi-variational inequalities: Existence and sensitivity analysis, 4th Central European Set-Valued and Variational Analysis Meeting (CESVVAM 2018), November 24, 2018, Philipps-Universität Marburg, November 24, 2018.

D.H. Doan, J. Fuhrmann, A. Glitzky, Th. Koprucki, M. Liero, On van Roosbroeck systems with Gauss--Fermi statistics, Applied Mathematics and Simulation for Semiconductors (AMaSiS 2018), WIAS Berlin, October 8 - 10, 2018.

M. Heida, On G-convergence and stochastic two-scale convergences of the square root approximation scheme to the Fokker--Planck operator, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2018), Section S14 ``Applied Analysis'', March 19 - 23, 2018, Technische Universität München, March 21, 2018.

M. Heida, On convergence of the square root approximation scheme to the Fokker--Planck operator, Technische Universität Berlin, Institut für Mathematik, May 14, 2018.

M. Heida, On convergence of the square root approximation scheme to the Fokker--Planck operator, Oberseminar ``Optimierung'', Humboldt-Universität zu Berlin, Institut für Mathematik, May 29, 2018.

M. Kantner, M. Mittnenzweig, Th. Koprucki, A hybrid quantum-classical modeling approach for electrically driven quantum dot devices, SPIE Photonics West 2018: Physics and Simulation of Optoelectronic Devices XXVI, January 29 - February 1, 2018, The Moscone Center, San Francisco, USA, January 29, 2018.

M. Kantner, Hybrid quantum-classical modeling of quantum dot based single-photon emitting diodes, Workshop Applied Mathematics and Simulation for Semiconductors, WIAS Berlin, October 10, 2018.

M. Kantner, Modeling and simulation of electrically driven quantum light emitters, Leibniz MMS Days, Leibniz Institut für Oberflächenmodifizierung (IOM), Leipzig, March 2, 2018.

M. Kantner, Thermodynamically consistent modeling of electrically driven quantum dot based light emitters on a device scale, Workshop ,,Nonlinear Dynamics in Semiconductor Lasers (NDSL2018)'', June 18 - 20, 2018, WIAS, Berlin, June 18, 2018.

C. Cancès, C. Chainais-Hillairet, J. Fuhrmann, B. Gaudeul, Numerical schemes for a reduced case of an improved Nernst--Planck--Poisson model, Applied Mathematics and Simulation for Semiconductors (AMaSiS 2018), WIAS Berlin, October 8 - 10, 2018.

P. Farrell, D. Peschka, Challenges for drift-diffusion simulations of semiconductors: A comparative study of different discretization philosophies, Applied Mathematics and Simulation for Semiconductors (AMaSiS 2018), WIAS Berlin, October 8 - 10, 2018.

L. Blank, A robust finite element method for the Brinkman problem, 13th International Workshop on Variational Multiscale and Stabilized Finite Elemements, Weierstraß-Institut, Berlin, December 5, 2018.

L. Blank, An unconditionally stable, low order, and robust finite element method for the numerical simulation of porous media flow, 39th Northern German Colloquium on Applied Analysis and Numerical Mathematics (NoKo 2018), June 1 - 2, 2018, Technische Universität Braunschweig, June 2, 2018.

A. Glitzky, Electrothermal feedback in organic LEDs, Workshop ``Numerical Optimization of the PEM Fuel Cell Bipolar Plate'', March 20, 2018, Zentrum für Solarenergie- und Wasserstoff-Forschung (ZSW), Ulm, March 20, 2018.

M. Thomas, D. Peschka, B. Wagner, V. Mehrmann, M. Rosenau, Modeling and analysis of suspension flows, MATH+ Center Days 2018, October 31 - November 2, 2018, Zuse-Institut Berlin (ZIB), Berlin, October 31, 2018.

M. Thomas, Analysis and simulation for a phase-field fracture model at finite strains based on modified invariants, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2018), Section DFG Priority Programmes PP1748 ``Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis'', March 19 - 23, 2018, Technische Universität München, March 20, 2018.

M. Thomas, Analysis and simulation for a phase-field fracture model at finite strains based on modified invariants, Workshop ``Special Materials and Complex Systems'' (SMACS 2018), June 18 - 22, 2018, University of Milan/University of Pavia, Gargnano, Italy, June 18, 2018.

M. Thomas, Analysis and simulation for a phase-field fracture model at finite strains based on modified invariants, Analysis Seminar, University of Brescia, Department of Mathematics, Italy, May 10, 2018.

M. Thomas, Analysis for the discrete approximation of damage and fracture, Applied Analysis Day, June 28 - 29, 2018, Technische Universität Dresden, Chair of Partial Differential Equations, June 29, 2018.

M. Thomas, Analysis for the discrete approximation of gradient-regularized damage models, Workshop ``Women in Mathematical Materials Science'', November 5 - 6, 2018, Universität Regensburg, Fakultät für Mathematik, November 6, 2018.

M. Thomas, Analytical and numerical approach to a class of damage models, The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 75 ``Mathematics and Materials: Models and Applications'', July 5 - 9, 2018, National Taiwan University, Taipeh, Taiwan, Province Of China, July 6, 2018.

M. Thomas, Analytical and numerical aspects of damage models, Berlin Dresden Prague Würzburg Workshop ``Mathematics of Continuum Mechanics'', November 29 - 30, 2018, Universität Würzburg, Institut für Mathematik, November 30, 2018.

M. Thomas, Gradient structures for flows of concentrated suspensions, The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 18 ``Emergence and Dynamics of Patterns in Nonlinear Partial Differential Equations and Related Fields'', July 5 - 9, 2018, National Taiwan University, Taipeh, Taiwan, Province Of China, July 7, 2018.

M. Thomas, Optimization of the radiative emission for mechanically strained optoelectronic semiconductor devices, 9th International Conference ``Inverse Problems: Modeling and Simulation'' (IPMS 2018), Minisymposium M16 ``Inverse and Control Problems in Mechanics'', May 21 - 25, 2018, The Eurasian Association on Inverse Problems, Malta, May 24, 2018.

M. Thomas, Rate-independent evolution of sets & applications to damage and delamination, PDEs Friends, June 21 - 22, 2018, Politecnico di Torino, Dipartimento di Scienze Matematiche ``Giuseppe Luigi Lagrange'', Italy, June 22, 2018.

W. Dreyer, Non-Newtonian fluids and the 2nd law of thermodynamics, 3rd Leibniz MMS Days 2018, February 28 - March 2, 2018, Wissenschaftszentrum Leipzig, March 1, 2018.

J. Fuhrmann, A. Linke, Ch. Merdon, C. Guhlke, R. Müller, Models and numerical methods for electroosmotic flow including finite ion size effects, Workshop on Ion Exchange Membranes for Energy Applications (EMEA2018), Bad Zwischenahn, June 26 - 28, 2018.

J. Fuhrmann, A. Linke, Ch. Merdon, C. Guhlke, R. Müller, Models and numerical methods for electroosmotic flow including finite ion size effects, Applied Mathematics and Simulation for Semiconductors (AMaSiS 2018), WIAS Berlin, October 8 - 10, 2018.

TH. Koprucki, Highly accurate discretizations for non-Boltzmann charge transport in semiconductors, 18th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2018), Session ``Numerical Methods'', November 5 - 9, 2018, The University of Hong Kong, China, November 6, 2018.

TH. Koprucki, Numerical methods for drift-diffusion equations, sc Matheon 11th Annual Meeting ``Photonic Devices'', February 8 - 9, 2018, Konrad-Zuse-Zentrum für Informationstechnik Berlin, February 8, 2018.

TH. Koprucki, Towards model-based geometry reconstruction of quantum dots from TEM, 18th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2018), Session ``Nanostructures'', November 5 - 9, 2018, The University of Hong Kong, China, November 8, 2018.

M. Liero, Feel the heat---Modeling of electrothermal feedback in organic devices, A Joint Meeting of the Society for Natural Philosophy and the International Society for the Interaction of Mathematics and Mechanics ``Mathematics & Mechanics: Natural Philosophy in the 21st Century'', June 24 - 27, 2018, University of Oxford, Mathematical Institute, UK, June 25, 2018.

O. Marquardt, Data-driven electronic structure calculations in semiconductor nanostructures --- Beyond the eight-band k&cdot&p formalism, 18th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2018), Session ``Numerical Methods'', November 5 - 9, 2018, The University of Hong Kong, China, November 6, 2018.

U. Wilbrandt, Iterative subdomain methods for the Stokes--Darcy coupling, 6th European Conference on Computational Mechanics, 7th European Conference on Computational Fluid Dynamics (ECCM-ECFD 2018), June 11 - 15, 2018, University of Glasgow, UK, June 11, 2018.

A. Alphonse, A coupled bulk-surface reaction-diffusion system on a moving domain, Workshop ``Emerging Developments in Interfaces and Free Boundaries'', January 23 - 28, 2017, Mathematisches Forschungszentrum Oberwolfach, January 25, 2017.

M. Heida, On G-convergence and stochastic two-scale convergences of the squareroot approximation scheme to the Fokker--Planck operator, GAMM-Workshop on Analysis of Partial Differential Equations, September 27 - 29, 2017, Eindhoven University of Technology, Mathematics and Computer Science Department, Netherlands, September 28, 2017.

M. Kantner, Hybrid quantum-classical modeling of electrically driven quantum light sources, Meeting of the MATHEON Scientific Advisory Board 2017, TU Berlin, Institut für Mathematik, November 13, 2017.

M. Kantner, Simulations of quantum dot devices by coupling of quantum master equations and semi-classical transport theory, 17th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD2017), July 24 - 28, 2017, Technical University of Denmark, Copenhagen, July 27, 2017.

M. Liero, Modeling and simulation of electrothermal feedback in large-area organic LEDs, Numerical Simulation of Optoelectronic Devices (NUSOD 2017), session ``Light-Emitting Diodes'', July 24 - 28, 2017, Technical University of Denmark, Lyngby Campus, Kopenhagen, Denmark, July 25, 2017.

CH. Merdon, A novel concept for the discretisation of the coupled Nernst--Planck--Poisson--Navier--Stokes system, 14th Symposium on Fuel Cell Modelling and Experimental Validation (MODVAL 14), March 2 - 3, 2017, Karlsruher Institut für Technologie, Institut für Angewandte Materialien, Karlsruhe, Germany, March 3, 2017.

CH. Merdon, Druckrobuste Finite-Elemente-Methoden für die Navier-Stokes-Gleichungen, Universität Paderborn, Institut für Mathematik, April 25, 2017.

CH. Merdon, Pressure-robustness in mixed finite element discretisations for the Navier--Stokes equations, Universität des Saarlandes, Fakultät für Mathematik und Informatik, July 12, 2017.

D. Peschka, Doping optimization for optoelectronic devices, Numerical Simulation of Optoelectronic Devices (NUSOD 2017), Post-Deadline session, July 27 - 28, 2017, Technical University, Lyngby Campus, Kopenhagen, Denmark, July 28, 2017.

D. Peschka, Mathematical and numerical approaches to moving contact lines, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, December 6, 2017.

D. Peschka, Motion of thin droplets over surfaces, Making a Splash -- Driplets, Jets and Other Singularities, March 20 - 24, 2017, Brown University, Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, USA, March 22, 2017.

D. Peschka, Variational structure of fluid motion with contact lines in thin-film models, Kolloquium Angewandte Mathematik, Universität der Bundeswehr, München, May 31, 2017.

N. Ahmed, Higher-order discontinuous Galerkin time discretizations for the evolutionary Navier--Stokes equations, Technische Universität Dresden, Institut für Numerische Mathematik, March 9, 2017.

N. Ahmed, On really locking-free mixed finite element methods for the transient incompressible Stokes equations, CASM International Conference on Applied Mathematics, May 22 - 24, 2017, Lahore University of Management Sciences, Centre for Advanced Studies in Mathematics, Pakistan, May 22, 2017.

C. Bartsch, ParMooN -- A parallel finite element solver, Part I, Indian Institute of Science, Supercomputer Education and Research Centre, Bangalore, India, March 16, 2017.

A. Caiazzo, Estimation of cardiovascular system parameters from real data, 2nd Leibniz MMS Days 2017, February 22 - 23, 2017, Technische Informationsbibliothek, Hannover, February 22, 2017.

A. Caiazzo, Homogenization methods for weakly compressible elastic materials forward and inverse problem, Workshop on Numerical Inverse and Stochastic Homogenization, February 13 - 17, 2017, Universität Bonn, Hausdorff Research Institute for Mathematics, February 17, 2017.

P.-É. Druet, Analysis of recent Nernst--Planck--Poisson--Navier--Stokes systems of electrolytes, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S14 ``Applied Analysis'', March 6 - 10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 7, 2017.

P.-É. Druet, Existence of weak solutions for improved Nernst--Planck--Poisson models of compressible electrolytes, Seminar EDE, Czech Academy of Sciences, Institute of Mathematics, Department of Evolution Differential Equations (EDE), Prague, Czech Republic, January 10, 2017.

P. Friz, Geometric aspects in pathwise stochastic analysis, High Risk High Gain -- Groundbreaking Research in Berlin, August 31 - September 3, 2017, Technische Universität Berlin, Stabsstelle Presse, September 2, 2017.

J. Fuhrmann, A. Linke, Ch. Merdon, Models and numerical methods for ionic mixtures with volume constraints, 12th International Symposium on Electrokinetics, Dresden, September 10 - 12, 2017.

M. Hintermüller, Generalized Nash equilibrium problems in Banach spaces: Theory, Nikaido--Isoda-based path-following methods, and applications, The Third International Conference on Engineering and Computational Mathematics (ECM2017), Stream 3 ``Computational Optimization'', May 31 - June 2, 2017, The Hong Kong Polytechnic University, China, June 2, 2017.

M. Hintermüller, Non-smooth structures in PDE-constrained optimization, Mathematisches Kolloquium, Universität Duisburg-Essen, Fakultät für Mathematik, Essen, January 11, 2017.

M. Hintermüller, Recent trends in PDE-constrained optimization with non-smooth structures, Fourth Conference on Numerical Analysis and Optimization (NAOIV-2017), January 2 - 5, 2017, Sultan Qaboos University, Muscat, Oman, January 4, 2017.

TH. Koprucki, Comparison of consistent flux discretizations for drift diffusion beyond Boltzmann statistics, Numerical Simulation of Optoelectronic Devices (NUSOD 2017), session ``Numerical Methods'', July 24 - 28, 2017, Technical University of Denmark, Lyngby Campus, Kopenhagen, Denmark, July 27, 2017.

TH. Koprucki, Mathematical models as research data in numerical simulation of opto-electronic devices, Numerical Simulation of Optoelectronic Devices (NUSOD 2017), session ``Model Representation'', July 24 - 28, 2017, Technical University of Denmark, Lyngby Campus, Kopenhagen, Denmark, July 27, 2017.

CH. Merdon, Pressure-robust finite element methods for the Navier--Stokes equations, GAMM Workshop on Numerical Analysis, November 1 - 2, 2017, Rheinisch-Westfälische Technische Hochschule Aachen, November 2, 2017.

CH. Merdon, Pressure-robust mixed finite element methods for the Navier--Stokes equations, scMatheon Workshop RMMM 8 - Berlin 2017, Reliable Methods of Mathematical Modeling, July 31 - August 3, 2017, Humboldt-Universität zu Berlin, August 2, 2017.

A. Mielke, Mathematical modeling of semiconductors: From quantum mechanics to devices, CIM-WIAS Workshop ``Topics in Applied Analysis and Optimisation'', December 6 - 8, 2017, Centro de Matemática, Lisboa, Portugal, December 8, 2017.

U. Wilbrandt, ParMooN -- A parallel finite element solver, Part II, Indian Institute of Science, Supercomputer Education and Research Centre, Bangalore, India, March 16, 2017.

K. Disser, Convergence for gradient systems of slow and fast chemical reactions, Joint Annual Meeting of DMV and GAMM, Session ``Applied Analysis'', March 7 - 11, 2016, Technische Universität Braunschweig, Braunschweig, March 11, 2016.

M. Kantner, Multi-scale modeling and numerical simulation of single-photon emitters, Matheon Workshop--9th Annual Meeting ``Photonic Devices", Zuse Institut, Berlin, March 3, 2016.

M. Kantner, Multi-scale modelling and simulation of single-photon sources on a device level, Euro--TMCS II Theory, Modelling & Computational Methods for Semiconductors, Tyndall National Institute and University College Cork, Cork, Ireland, December 9, 2016.

M. Liero, On $p(x)$-Laplace thermistor models describing eletrothermal feedback in organic semiconductors, The 19th European Conference on Mathematics for Industry (ECMI 2016), Minisymposium 23 ``Charge Transport in Semiconductor Materials: Emerging and Established Mathematical Topics'', June 13 - 17, 2016, Universidade de Santiago de Compostela, Spain, June 15, 2016.

M. Liero, On electrothermal feedback in organic light-emitting diodes, Berlin Dresden Prague Würzburg Workshop ``Mathematics of Continuum Mechanics'', Technische Universität Dresden, Fachbereich Mathematik, December 5, 2016.

CH. Merdon, J. Fuhrmann, A. Linke, A.A. Abd-El-Latif, M. Khodayari, P. Reinsberg, H. Baltruschat, Inverse modelling of thin layer flow cells and RRDEs, The 67th Annual Meeting of the International Society of Electrochemistry, Den Haag, Netherlands, August 21 - 26, 2016.