### Doktorandenseminar des WIAS

## FG3: Numerische Mathematik und Wissenschaftliches
Rechnen /

*RG3: Numerical Mathematics
and Scientific Computing*/

LG5: Numerik für innovative
Halbleiter-Bauteile

*LG5: Numerics for
innovative semiconductor devices*

## Seminar Numerische Mathematik / *Numerical mathematics seminars*

aktuelles Programm / *current program*

_{Archiv}

Hybrid-/Online-Vorträge finden über ''Zoom'' statt. Der Zoom-Link wird jeweils ca. 15 Minuten vor Beginn des Gesprächs versendet. /

*The zoom link will be send about 15 minutes before the start of the talk. People who are not members of the research group 3 and who are interested in participating should contact christian.merdon@wias-berlin.de to obtain the zoom login details.*

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Donnerstag, 06. 06. 2024, 14:00 Uhr (WIAS-ESH)
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Steffen Maass & Tim Siebert(WIAS/TU Berlin)

... tbd ...... tbd ...

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Dienstag, 04. 06. 2024, 13:30 Uhr (WIAS-ESH)
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Jürgen Fuhrmann(WIAS)

... tbd ...... tbd ...

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Dienstag, 28. 05. 2024, 13:30 Uhr (WIAS-406)
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Sarah Katz & Francesco Romor(WIAS)

... tbd ...... tbd ...

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Donnerstag, 23. 05. 2024, 14:00 Uhr (WIAS-ESH)
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Alfonso Caiazzo(WIAS)

... tbd ...... tbd ...

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Dienstag, 21. 05. 2024, 13:30 Uhr (WIAS-ESH)
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Christian Merdon(WIAS)

... tbd ...... tbd ...

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Donnerstag, 16.05.2024, 14:00 Uhr (WIAS-ESH)
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Cristian Cárcamo & Medine Demir(WIAS)

... tbd ...... tbd ...

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Dienstag, 14. 05. 2024, 13:30 Uhr (WIAS-ESH)
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Daniel Runge & Marwa Zainelabdeen(WIAS)

... tbd ...... tbd ...

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Dienstag, 07. 05. 2024, 13:30 Uhr (WIAS-ESH)
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Dilara Abdel(WIAS)

Modeling and simulation of vacancy-assisted charge transport in innovative semiconductor devicesIn response to the climate crisis, there is a need for technological innovations to reduce the escalating CO$_2$ emissions. Two promising semiconductor technologies in this regard, perovskite-based solar cells and memristive devices based on two-dimensional layered transition metal dichalcogenide (TMDC), can potentially contribute to the expansion of renewable energy sources and the development of energy-efficient computing hardware. Within perovskite and TMDC materials, ions dislocate from their ideal position in the semiconductor crystal and leave void spaces. So far, the precise influence of these vacancies and their dynamics on device performance remain underexplored.

Therefore, this talk is dedicated to comprehensively examining the impact of vacancy-assisted charge transport in innovative semiconductor devices through a theoretical approach by modeling and simulating systems of partial differential equations. We start by deriving drift-diffusion equations using thermodynamic principles. Furthermore, we formulate drift-diffusion models to describe charge transport in perovskite solar cells and TMDC memristors. We discretize the transport equations via the finite volume method and establish the existence of discrete solutions. Our study concludes with simulations conducted with an open source software tool developed in the programming language Julia. These simulations explore the influence of volume exclusion effects on charge transport in perovskite solar cells and compare our simulation results with experimental measurements found in literature for TMDC-based memristive devices.

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Dienstag, 07. 05. 2024, 14:00 Uhr (WIAS-ESH)
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Timo Streckenbach(WIAS)

tbdtbd

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Donnerstag, 02. 05. 2024, 14:00 Uhr (WIAS-ESH)
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Baptiste Moreau(WIAS)

... tbd ...... tbd ...

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Donnerstag, 02. 05. 2024, 14:30 Uhr (WIAS-ESH)
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Ondřej Pártl(WIAS)

Optimization of geothermal energy production from fracture-controlled reservoirs via 3D numerical modeling and simulationWe develop a computation framework from scratch that allows us to conduct 3D numerical simulations of groundwater flow and heat transport in hot fractured-controlled reservoirs to find optimal placements of injection and production wells that sustainably optimize geothermal energy production.

We model the reservoirs as geologically consistent randomly generated discrete fracture networks (DFN) in which the fractures are 2D manifolds with polygonal boundary embedded in a 3D porous medium. The wells are modeled as line sources and sinks. The flow and heat transport in the DFN-matrix system are modeled by solving the balance equations for mass, momentum, and energy. The fully developed computational framework combines the finite element method with semi-implicit time-stepping and algebraic flux correction. To perform the optimization, we use various gradient-free algorithms.

I will present our latest results for several geologically and physically realistic scenarios.

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Dienstag, 30. 04. 2024, 13:30 Uhr (WIAS-ESH)
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Patricio Farrell(WIAS)

Numerical methods for innovative semiconductor devicesThe Leibniz group NUMSEMIC develops and numerically solves nonlinear PDE models. These models are often inspired by charge transport in innovative semiconductor devices. In particular, applications include perovskite solar cells, memristors, nanowires, quantum wells, lasers as well as doping reconstruction. To translate these applications into mathematical models, we rely on nonlinear drift-diffusion, hyperelastic material models, inverse PDE problems, localized landscape theory and atomistic coupling. Our methodologies include physics-preserving finite volume methods, data-driven techniques as well as meshfree methods.

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Donnerstag, 25. 04. 2024, 14:00 Uhr (WIAS-ESH)
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Holger Stephan & trainees(WIAS)

... tbd ...... tbd ...

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Dienstag, 23. 04. 2024, 13:30 Uhr (WIAS-ESH)
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Zeina Amer & Yiannis Hadjimichael(WIAS)

... tbd ...... tbd ...

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Donnerstag, 18. 04. 2024, 14:00 Uhr (WIAS-ESH)
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Volker John(WIAS/FU Berlin)

Finite element methods respecting the discrete maximum principle for convection-diffusion equationsConvection-diffusion-reaction equations model the conservation of scalar quantities. From the analytic point of view, solution 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 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 a 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 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. Methods based on algebraic stabilization, nonlinear and linear ones, are currently as well 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 situation.

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Donnerstag, 14. 12. 2023, 14:00 Uhr (WIAS-ESH)
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Christos P. Papanikas(WIAS/University of Cyprus/Erasmus+)

From medical data to 3D modeling: simulation of cancer growth in mouse liverThe goal of this talk is to present in detail the different steps to construct a realistic 3D cancer growth model from medical data. In the first part, the type of available data will be discussed (MRI of mouse liver - courtesy of the MRE Group of the Charité), presenting how a tetrahedral mesh can be constructed using open-source software. In the second part, the definition of a mathematical model of cancer growth will be discussed, taking into account both biological and mechanical processes. The implementation and preliminary numerical results will also be presented. This work has been done in the context of my Erasmus+ stay at WIAS.

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Dienstag, 12. 12. 2023, 13:30 Uhr (WIAS-ESH)
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Jan Philipp Thiele(WIAS)

Error-controlled space-time finite elements, algorithms and implementations for nonstationary problemsRecently, many advances have been done in the field of space-time finite element discretizations for nonstationary partial differential equations. The temporal dimension leads to additional costs in solving the equations, so adaptive finite element methods can lead to sufficiently accurate solutions at a better efficiency compared to uniform refinement. Additionally, the solution to the equation itself is often not of direct interest, but instead some derived quantity has to be calculated accurately. For this, the dual-weighted residual (DWR) method offers a practical way to calculate estimators for the temporal and spatial error parts with respect to the unknown exact value of that quantity. This talk introduces the extension of the partition-of-unity localization (PU) for non-stationary problems, which provides error indicators for use in marking strategies. For the tensor-product space-time approach we developed the library ideal.II which extends the library deal.II to simplify the implementation of non-stationary problems. We brieftly discuss some of the design decisions and technical details of this library. Both the library and the PU-DWR method are validated by several numerical studies with linear and nonlinear PDEs.

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Donnerstag, 23. 11. 2023, 14:00 Uhr (WIAS-406)
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Grégoire Pourtier(TU Berlin / WIAS)

A Julia-based universal solver for system of one-dimensional nonlinear PDEsIn this talk, we introduce a new Julia library designed for solving universal systems of one-dimensional elliptic and parabolic partial differential equations. This library is based on the API from the MATLAB solver pdepe, and the spatial discretization follows the finite element method described in the paper of Skeel and Berzins [1]. The library enables the use of time integration methods from the DifferentialEquations.jl package [2]. Its primary focus is on achieving high performance by minimizing allocations within the core code loops and using Julia’s optimization tools to ensure code stability. We present various tools employed in the library’s implementation, including the concept of Automatic Differentiation and its application to sparse operator. We also provide explanations and recommendations for specific choices made during development. Additionally, we include convergence tests for both the spatial and time discretization as well as a test case on the modeling of Lithium Ion Batteries [3] and benchmark results comparing the Julia library with the MATLAB solver pdepe. Finally, we introduce an extension designed to address 1D+1D problems for multi-scale modeling.

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Donnerstag, 16. 11. 2023, 14:00 Uhr (WIAS-ESH)
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Francesco Romor(WIAS)

Efficiently solving parametric PDEs on solution manifolds parametrized by neural networksA slow decaying Kolmogorov n-width of the solution manifold of a parametric partial differential equation precludes the realization of efficient linear projection-based reduced-order models. This is due to the high dimensionality of the reduced space needed to approximate with sufficient accuracy the solution manifold. To solve this problem, neural networks, in the form of different architectures, have been employed to build accurate nonlinear regressions of the solution manifolds. However, the majority of the implementations are non-intrusive black-box surrogate models, and only a part of them perform dimension reduction from the number of degrees of freedom of the discretized parametric models to a latent dimension. We present a new intrusive and explicable methodology for reduced-order modelling that employs neural networks for solution manifold approximation but that does not discard the physical and numerical models underneath in the predictive/online stage. We will focus on autoencoders used to compress further the dimensionality of linear approximants of solution manifolds, achieving in the end a nonlinear dimension reduction. After having obtained an accurate nonlinear approximant, we seek for the solutions on the latent manifold with the residual-based nonlinear least-squares Petrov-Galerkin method, opportunely hyper-reduced in order to be independent from the number of degrees of freedom.

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Donnerstag, 12. 10. 2023, 14:00 Uhr (WIAS-ESH)
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Hang Si(Cadence Design Systems)

Perspectives of anisotropic Delaunay mesh adaptationAnisotropic meshes are essential to improve the efficiency and accuracy of numerical solutions. Generating anisotropic meshes remains a challenging problem in both theory and applications. In this talk, we address this topic from an application point of view. In particular, we show applications in surface meshing and numerical solutions. We will review the mathematical principles of anisotropic mesh adaptation and explore the connection with quasi-conformal mapping theory. We then discuss practical metric-based Delaunay algorithms to generate anisotropic meshes. We finish with some remarks on software implementation

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Mittwoch, 26. 07. 2023, 13:30 Uhr (WIAS-ESH)
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Stefan Ringe(Korea University)

First-principles multi-scale modeling of electrochemical CO2 reductionElectrochemical CO2 reduction is a viable pathway towards reducing greenhouse gas emissions and establishing a sustainable energy landscape. Bringing this reaction to industrial competitiveness is thus of significant interest and subject of current research. Recent studies have shown that the reaction is very complicated and multi-scale of nature, involving the coupling of reaction kinetics, electric double layer and mass transport. Modeling this from the computational side offers valuable insights about the importance of cell design parameters, such as the electrolyte species, geometric dimensions and components of the electrolyzer, or solvent type. In this presentation, I present the steps from first-principles density functional theory calculations all the way up to multi-scale models which can help to understand experimentally observed trends and give suggestions about optimal parameters.

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Donnerstag, 20. 07. 2023, 14:00 Uhr (WIAS-ESH)
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David Brust(DLR)

Modelling Study of a Photo-Thermal Catalytic Reactor for rWGS Reaction Under Concentrated IrradiationUtilization of CO2 via hydrogenation with H2 from carbon-free sources in the reverse water gas shift (rWGS) reaction offers a fossil-free pathway to the production of synthesis gas, which represents an important and versatile feedstock for the chemical industry. In this talk a photo driven catalytic reactor for the rWGS reaction operating under concentrated solar irradiation is investigated in a multi-physics modelling study. The numerical model covers chemical reactions relevant for rWGS conditions and considers coupled heat and multi-component mass transfer in the catalyst layer and porous support. For thermal evaluation of the porous modelling domain, the local thermal equilibrium (LTE) method is used, describing both the fluid phase and the solid structure by a single temperature field. Different relations for the effective thermal conductivity of the porous structure are compared. Radiation transport onto the irradiated surfaces is accounted for in a simplified manner. Multi-component mass transfer in the porous structure is described with the Dusty-gas Model. The transport equations for thermal energy and species mass together with boundary conditions are discretized in three space dimensions and solved with the finite-volume method via the open-source package VoronoiFVM.jl. The multi-physics model enables the analysis of energy losses and thus supports the optimization of reactor design and operation conditions.

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Dienstag, 18. 07. 2023, 14:30 Uhr (WIAS-ESH)
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Roman Schärer(ZHAW)

A Non-isothermal Cell Performance Model for Organic Flow BatteriesFlow batteries are an emerging technology for large-scale stationary energy storage. Specifically, organic flow batteries (OFBs) have been proposed as a promising and more sustainable alternative to metal-based batteries. The large chemical space available for organic electro-active redox couples allows to optimize the chemical properties to the specific requirements of the battery system. Furthermore, local synthesis of organic electrolytes offers the advantage of greater sustainability and reduced supply chain risks. Electrochemical flow cells represent one of the core building blocks of flow battery stacks, where the conversion between electrical and chemical energy is performed. To analyse and improve the energy efficiency of flow cells, we developed a continuum-scale model for the performance prediction of a single flow cell. The model solves for the spatially resolved stationary solution of the coupled balance equations of mass, charge, momentum and energy densities over the cell domain, which includes the current collectors, porous electrodes and the ion-exchange membrane. The effective macroscopic transport properties of the porous electrodes are modelled using an up-scaling methodology based on the volume averaging method. This approach facilitates an investigation of the impact of the pore-scale geometry and transport characteristics on the macroscopic transport properties, such as the effective diffusivity and reaction rate. The model has been implemented in the Julia programming language and is published as open source software. The discretization of the balance equations is performed with the FV method implemented by the VoronoiFVM.jl package. First validation studies with a laboratory test cell using a fully organic electrolyte have shown promising results. The model will be used to explore the energy efficiency potential of a single flow cell with respect to the cell geometry and operating condition, such as flow rate and electrolyte concentrations.

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Dienstag, 18. 07. 2023, 13:30 Uhr (WIAS-ESH)
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Benoît Gaudeul(Paris-Saclay)

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Dienstag, 11. 07. 2023, 13:30 Uhr (WIAS-ESH)
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Derk Frerichs-Mihov(WIAS)

Using deep learning techniques to improve the discrete solution of convection-dominated convection-diffusion equationsConvection-diffusion equations are a basic model to describe the distribution of a scalar quantity in fluids. In the convection-dominated regime, the solution usually possesses so-called layers, which are small regions where the solution has a steep gradient. Unfortunately, it is well known in the literature that it is challenging for many classical methods to approximate the solution in these layers. The solution is often polluted by unphysical values, so-called spurious oscillations. In this talk, a deep learning-based slope limiter is presented, that automatically detects regions where the solution has unphysical values and locally corrects the solution on the marked cells. It is applied to the DG solution to a standard benchmark problem and compared to classical slope limiters.

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Dienstag, 04. 07. 2023, 13:30 Uhr (WIAS-405)
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Marwa Zainelabdeen(WIAS Berlin)

An optimally Convergent Convection-Stabilized Taylor–Hood Finite Element Method for the Oseen EquationsWe consider the finite element discretization of the Oseen equations. The LSVS convection stabilization proposed in [1], which is motivated by the underlying vorticity equation has the advantage that it leads for the Scott-Vogelius finite element to a pressure-robust method. We extended the LSVS method to the classical Taylor-Hood finite element space which more flexible with respect to the computational grids and less costly due to a smaller pressure ansatz space. On the other hand it is not pressure-robust and does not satisfy the original assumptions for the LSVS convection stabilization. To improve the lack of pressure-robustness, the popular grad-div stabilization is added. A theoretical result in [2] stating that for large grad-div parameter \(\gamma\) the Taylor- Hood method converges to the Scott–Vogelius method is carried over to the LSVS-grad-div stabilized scheme. In addition, by utilizing the already proved \(O(h^{k+1/2})\) Scott-Vogelius error estimate from [1], we proved an error estimate for the velocity of the same order for Taylor-Hood finite elements. Numerical studies are performed to test the method and investigate the optimal choice of the LSVS and the grad-div stabilization parameters.

References

[1] N. Ahmed, G. R. Barrenechea, E. Burman, J. Guzm ́an, A. Linke, C. Mer- don, A pressure-robust discretization of Oseen’s equation using stabilization in the vorticity equation, SIAM J. Numer. Anal. 59(5), pp. 2746–2774, 2021

[2] M. A. Case, V. J. Ervin, A. Linke, L. G. Rebholz, A connection between Scott–Vogelius and grad-div stabilized Taylor–Hood FE approximations of the Navier-Stokes equations, SIAM J. Numer. Anal. 49(4), pp. 1461–1481, 2011

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Donnerstag, 29. 06. 2023, 14:00 Uhr (WIAS-ESH)
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Sarah Katz(WIAS Berlin)

Impact of Turbulence Modeling on Simulations of Aortic Blood FlowDue to the comparatively high flow rates at systole, blood flow in the aorta exhibits turbulent behavior. Simulations therefore require either very high spatial resolution or the use of a suitable turbulence model. This talk will present past work by RG3 in cooperation with the Charité's Institute of Computer-assisted Cardiovascular Medicine on the latter choice, comparing several relevant models, as well as an outlook on current efforts in applying machine learning to augment such simulations.

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Donnerstag, 22. 06. 2023, 14:00 Uhr (WIAS-ESH)
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Cristian Cárcamo Sánchez(WIAS Berlin)

A stabilized finite element method for the Stokes–Temperature coupled problemIn 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 [1]. 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 [2].

References

[1] M. Alvarez, G.N. Gatica, R. Ruiz–Baier, An augmented mixed-primal finite element method for a coupled flow-transport problem, ESAIM: M2AN 49 (2015) 1399–1427, https://doi.org/10.1051/m2an/2015015..

[2] P.E. van Keken, C. Currie, S.D. King, M.D. Behn, A. Cagnioncle, J. He, R.F. Katz, S.C. Lin, E.M. Parmentier, M. Spiegelman, K. Wang, A community bench- mark for subduction zone modeling, Phys. Earth Planet. Inter. 171 (2008) 187–197, https://doi.org/10.1016/j.pepi.2008.04.015.

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Donnerstag, 15. 06. 2023, 14:00 Uhr (WIAS-ESH)
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Medine Demir(WIAS Berlin)

Vorticity based stabilization method for Fluid-Fluid interaction problemFluid-fluid interaction problem is of considerable interest especially in computational fluid dynamics simulations for different industrial and engineering applications. One of the common problem can be observed in atmosphere-ocean (AO) interactions. This work considers the approximate solutions of the two coupled fluid-fluid interaction problem in AO through nonlinear interface condition. In this study, we propose a different variant of SAV method essentially based on an effective VMS stabilization acts on only the small scales for the numerical simulation of the AO problem. Instead of using projection operator for VMS stabilization, the stabilization is dealt with vorticity term and grad-div stabilization in the viscous term. Geometric averaging will be used for decoupling. An important aspect of this consideration is not only conforming mixed finite element approximation is produced, but also with the use of vorticity extra storage requirement is reduced. In the case of small viscosity, the use of only three variables instead of nine variables enables to improve the solution of the system.

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Dienstag, 16. 05. 2023, 13:30 Uhr (WIAS-ESH)
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Camilla Belponer(WIAS Berlin & Uni Augsburg)

Reduced Lagrange multiplier approach for the non-matching coupled problems in multiscale elasticityThe interest in efficient simulation of vascularized tissues is motivated by the solution of inverse problems in the context of tissue imaging, where available medical data (such as those obtained via Magnetic Resonance Elastography) have a limited resolution, typically at the scale of an effective - macro scale - tissue, and cannot resolve the microscale of quantities of interests related, for instance, to the tissue vasculature. Our model is based on a geometrical multiscale 3D (elastic) -1D (fluid) formulation combined with an immersed method. At the elastic-fluid immersed boundary \(\Gamma\) we impose a trace-averaged boundary condition whose goal is to impose only a local Dirichlet boundary condition on the tissue-vessel interface allowing the enforcement of a pure normal displacement at the fictional vessel boundary. In order to decouple the discretization of the elastic tissue from the vessel boundary, the boundary condition on \(\Gamma\) is imposed via a Lagrange multiplier, modeling the fluid vessels as immersed singular sources for the elasticity equation. Next, to efficiently handle the multiscale nature of the problem, the problem is formulated as a mixed-dimensional PDE using the recently proposed framework of reduced Lagrange multipliers on a space of co-dimension 2. In this talk, we present the numerical analysis of the obtained formulation and we discuss accuracy properties and convergences of the method, validating it in several numerical examples. Finally, we present perspectives for the coupling with a one-dimensional flow model defined on the vascular network and for the numerical upscaling of the tissue model.

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Donnerstag, 11. 05. 2023, 14:00 Uhr (WIAS-ESH/Hybrid Event)
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Théo Belin(Centrale Supélec)

An entropic finite volumes scheme for a fully non-linear forward-backward parabolic equationWe investigate a fully non-linear forward-backward parabolic equation, stemming from a thermodynamical model of phase change in nylon. The non-monotonicity of the non-linearity yields an ill-posed problem in any standard framework for parabolic equations. To overcome this difficulty one can introduce a singular perturbation of the problem called the Sobolev approximation. In the singular limit, entropy solutions are naturally unveiled and have first been studied by Plotnikov and then later in restricted settings by Evans and Porthilheiro, Corrado and others. These entropy solutions exhibit a free discontinuity and a hysteretic behaviour which we describe. While existence of such solutions is guaranteed by time-discrete approximations, the uniqueness of these entropy solutions has yet to be precisely studied in higher dimensions. Because of the ill-posed nature of the problem, the convergence of numerical approximations of the original equation is not yet attained. It is an open problem to decide wether any of these direct schemes can select an entropy solution in the limit. This is why we present an entropy-preserving finite volume scheme which tracks the hysteresis behaviour through a phase indicator function. The estimates proven so far do not directly allow for the identification of the limit as an entropy solution. A missing link to the convergence is the \(L^1\) compactness either for the solution or for the phase indicator function. Numerical simulations are shown.

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Donnerstag, 23. 03. 2023, 14:00 Uhr (WIAS-ESH/Hybrid Event)
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Gabriel Barrenechea(University of Strathclyde)

Positivity-preserving discretisations in general meshesThe quest for physical consistency in the discretisation of PDEs started as soon as the numerical methods started being proposed. By physical consistency we mean a discretisation that by design satisfies a property also satisfied by the continuous PDE. This property might be positivity of the discrete solution, or preservation of some bounds (e.g., concentrations should belong to the interval [0, 1]), or can also be energy preservation, or exactly divergence-free velocities for incompressible fluids. Regarding positivity preservation, this topic has been around since the pioneering work by Ph. Ciarlet in the late 1960s and early 1970s. In the context of finite element methods, it was shown in those early works (and not significantly improved since), that in order for a finite element method to preserve positivity the mesh needs to satisfy certain geometrical restrictions, e.g., in two space dimensions with simplicial elements the triangulation needs to be of Delaunay type (in higher dimensions or quadrilateral meshes the restrictions are more involved). Throughout the years several conclusions have been reached in this topic, but in the context of finite element methods the discretisations tend to be of first order in space. So, many important problems still remain open. In particular, one open problem is how to build a discretisation that will lead to a positive solution regarless of the geometry of the mesh and the order of the finite element method. In this talk I will review recent results addressing the last question posed in the last paragraph. More precisely, I will present a method that enforces bound-preservation (at the degrees of freedom) of the discrete solution. The method is built by first defining an algebraic projection onto the convex closed set of finite element functions that satisfy the bounds given by the solution of the PDE. Then, this projection is hardwired into the definition of the method by writing a discrete problem posed for this projected part of the solution. Since this process is done independently of the shape of the basis functions, and no result on the resulting finite element matrix is used, then the outcome is a finite element function that satisfies the bounds at the degrees of freedom. Another important observation to make is that this approach is related to variational inequalities, and this fact will be exploited in the error analysis. The core of the talk will be devoted to explaining the main idea in the context of linear (and nonlinear) reaction-diffusion equations. Then, I will explain the main difficulties encountered when extending this method to convection-diffusion equations, and, more importantly, to a finite element method defined in polytopal meshes. The results in this talk have been carried out in collaboration with Abdolreza Amiri (Strathclyde, UK), Emmanuil Geourgoulis (Heriot-Watt, UK and Athens, Greece), Tristan Pryer (Bath, UK), and Andreas Veeser (Milan, Italy).

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Donnerstag, 19. 01. 2023, 14:00 Uhr (WIAS-ESH/Hybrid Event)
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Michele Pütz(BTU Cottbus-Senftenberg)

Quadrature-Based Moment Methods for the Solution of Population Balance EquationsParticulate systems of many kinds are mathematically described by a multivariate number density function (NDF) whose evolution is governed by a so-called population balance equation (PBE). Typically, such equations are characterized by a high dimensionality and, accordingly, high computational costs for the numerical solution. Quadrature-based moment methods (QBMMs) aim to efficiently approximate solutions to PBEs by solving only for a set of moments instead of the NDF and closing the PBE using Gaussian quadrature rules. The upcoming presentation will give an introduction to QBMMs, elaborate on limitations and challenges associated with non-smooth terms in the moment equations using physical examples and provide some results related to the performance and accuracy of QBMMs.

For zoom login details please contact Christian Merdon christian.merdon@wias-berlin.de