Publications

Monographs

  • A. Caiazzo, V.-C. Irene E., Mathematical modeling of blood flow in the cardiovascular system, I. Sack, T. Schaeffter, eds., Quantification of Biophysical Parameters in Medical Imaging, Springer, Cham, 2018, pp. 45--70, (Chapter Published), DOI 10.1007/978-3-319-65924-4_3 .

  • 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).

Articles in Refereed Journals

  • W. Dreyer, P. Friz, P. Gajewski, C. Guhlke, M. Maurelli, Stochastic model for LFP-electrodes, Continuum Mechanics and Thermodynamics, published online on 05.02.2018, DOI 10.1007/s00161-018-0629-7 .
    Abstract
    In the framework of non-equilibrium thermodynamics we derive a new model for porous electrodes. The model is applied to LiFePO4 (LFP) electrodes consisting of many LFP particles of nanometer size. The phase transition from a lithium-poor to a lithium-rich phase within LFP electrodes is controlled by surface fluctuations leading to a system of stochastic differential equations. The model is capable to derive an explicit relation between battery voltage and current that is controlled by thermodynamic state variables. This voltage-current relation reveals that in thin LFP electrodes lithium intercalation from the particle surfaces into the LFP particles is the principal rate limiting process. There are only two constant kinetic parameters in the model describing the intercalation rate and the fluctuation strength, respectively. The model correctly predicts several features of LFP electrodes, viz. the phase transition, the observed voltage plateaus, hysteresis and the rate limiting capacity. Moreover we study the impact of both the particle size distribution and the active surface area on the voltagecharge characteristics of the electrode. Finally we carefully discuss the phase transition for varying charging/discharging rates.

  • J. DE Frutos, B. Garc'ia-Archilla, V. John, J. Novo, Analysis of the grad-div stabilization for the time-dependent Navier--Stokes equations with inf-sup stable finite elements, Advances in Computational Mathematics, 44 (2018), pp. 195--225.

  • N. Ahmed, C. Bartsch, V. John, U. Wilbrandt, An assessment of solvers for 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, (2018), published online on 07.02.2018, 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).

  • F. Anker, Ch. Bayer, M. Eigel, M. Ladkau, J. Neumann, J.G.M. Schoenmakers, SDE based regression for random PDEs, SIAM Journal on Scientific Computing, 39 (2017), pp. A1168--A1200.
    Abstract
    A simulation based method for the numerical solution of PDE with random coefficients is presented. By the Feynman-Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to an approximation of the global solution of the random PDE. We provide an initial error and complexity analysis of the proposed method along with numerical examples illustrating its behaviour.

  • F. Anker, Ch. Bayer, M. Eigel, J. Neumann, J.G.M. Schoenmakers, A fully adaptive interpolated stochastic sampling method for linear random PDEs, International Journal for Uncertainty Quantification, 7 (2017), pp. 189--205, DOI 10.1615/Int.J.UncertaintyQuantification.2017019428 .
    Abstract
    A numerical method for the fully adaptive sampling and interpolation of PDE with random data is presented. It is based on the idea that the solution of the PDE with stochastic data can be represented as conditional expectation of a functional of a corresponding stochastic differential equation (SDE). The physical domain is decomposed subject to a non-uniform grid and a classical Euler scheme is employed to approximately solve the SDE at grid vertices. Interpolation with a conforming finite element basis is employed to reconstruct a global solution of the problem. An a posteriori error estimator is introduced which provides a measure of the different error contributions. This facilitates the formulation of an adaptive algorithm to control the overall error by either reducing the stochastic error by locally evaluating more samples, or the approximation error by locally refining the underlying mesh. Numerical examples illustrate the performance of the presented novel method.

  • W. Dreyer, C. Guhlke, Sharp limit of the viscous Cahn--Hilliard equation and thermodynamic consistency, Continuum Mechanics and Thermodynamics, 29 (2017), pp. 913--934.
    Abstract
    Diffuse and sharp interface models represent two alternatives to describe phase transitions with an interface between two coexisting phases. The two model classes can be independently formulated. Thus there arises the problem whether the sharp limit of the diffuse model fits into the setting of a corresponding sharp interface model. We call a diffuse model admissible if its sharp limit produces interfacial jump conditions that are consistent with the balance equations and the 2nd law of thermodynamics for sharp interfaces. We use special cases of the viscous Cahn-Hilliard equation to show that there are admissible as well as non-admissible diffuse interface models.

  • 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.

  • 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 .

  • S. Giere, V. John, Towards physically admissible reduced-order solutions for convection-diffusion problems, Applied Mathematics Letters, 73 (2017), pp. 78--83, DOI 10.1016/j.aml.2017.03.022 .

  • V. Wiedmeyer, F. Anker, C. Bartsch, A. Voigt, V. John, K. Sundmacher, Continuous crystallization in a helically-coiled flow tube: Analysis of flow field, residence time behavior and crystal growth, Industrial and Engineering Chemistry Research, 56 (2017), pp. 3699--3712, DOI 10.1021/acs.iecr.6b04279 .

  • 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 .

  • J. Bulling, V. John, P. Knobloch, Isogeometric analysis for flows around a cylinder, Applied Mathematics Letters, 63 (2017), pp. 65--70.

  • F. Dassi, L. Kamenski, P. Farrell, H. Si, Tetrahedral mesh improvement using moving mesh smoothing, lazy searching flips, and RBF surface reconstruction, Computer-Aided Design, published online on 8.12.2017, urlhttps://doi.org/10.1016/j.cad.2017.11.010, DOI 10.1016/j.cad.2017.11.010 .
    Abstract
    Given a tetrahedral mesh and objective functionals measuring the mesh quality which take into account the shape, size, and orientation of the mesh elements, our aim is to improve the mesh quality as much as possible. In this paper, we combine the moving mesh smoothing, based on the integration of an ordinary differential equation coming from a given functional, with the lazy flip technique, a reversible edge removal algorithm to modify the mesh connectivity. Moreover, we utilize radial basis function (RBF) surface reconstruction to improve tetrahedral meshes with curved boundary surfaces. Numerical tests show that the combination of these techniques into a mesh improvement framework achieves results which are comparable and even better than the previously reported ones.

  • F. Dassi, H. Si, S. Perotto, T. Streckenbach, A priori anisotropic mesh adaptation driven by a higher dimensional embedding, Computer-Aided Design, 85 (2017), pp. 111--122, DOI 10.1016/j.cad.2016.07.012 .
    Abstract
    In this paper we provide a novel anisotropic mesh adaptation technique for adaptive finite element analysis. It is based on the concept of higher dimensional embedding, which was exploited in [1-4] to obtain an anisotropic curvature adapted mesh that fits a complex surface in R^3. In the context of adaptive finite element simulation, the solution (which is an unknown function f : Ω ⊂ R^d → R) is sought by iteratively modifying a finite element mesh according to a mesh sizing field described via a (discrete) metric tensor field that is typically obtained through an error estimator. We proposed to use a higher dimensional embedding, Φ_f(x) := (x_1, …, x_d, s f (x_1, …, x_d), s ∇ f (x_1, …, x_d))^t, instead of the mesh sizing field for the mesh adaption. This embedding contains both informations of the function f itself and its gradient. An isotropic mesh in this embedded space will correspond to an anisotropic mesh in the actual space, where the mesh elements are stretched and aligned according to the features of the function f. To better capture the anisotropy and gradation of the mesh, it is necessary to balance the contribution of the components in this embedding. We have properly adjusted Φ_f(x) for adaptive finite element analysis. To better understand and validate the proposed mesh adaptation strategy, we first provide a series of experimental tests for piecewise linear interpolation of known functions. We then applied this approach in an adaptive finite element solution of partial di erential equations. Both tests are performed on two-dimensional domains in which adaptive triangular meshes are generated. We compared these results with the ones obtained by the software BAMG - a metric-based adaptive mesh generator. The errors measured in the L_2 norm are comparable. Moreover, our meshes captured the anisotropy more accurately than the meshes of BAMG.

  • F. Dassi, P. Farrell, H. Si, A novel surface remeshing scheme via higher dimensional embedding and radial basis functions, SIAM Journal on Scientific Computing, 39 (2017), pp. B522--B547, DOI 10.1137/16M1077015 .
    Abstract
    Many applications heavily rely on piecewise triangular meshes to describe complex surface geometries. High-quality meshes significantly improve numerical simulations. In practice, however, one often has to deal with several challenges. Some regions in the initial mesh may be overrefined, others too coarse. Additionally, the triangles may be too thin or not properly oriented. We present a novel mesh adaptation procedure which greatly improves the problematic input mesh and overcomes all of these drawbacks. By coupling surface reconstruction via radial basis functions with the higher dimensional embedding surface remeshing technique, we can automatically generate anisotropic meshes. Moreover, we are not only able to fill or coarsen certain mesh regions but also align the triangles according to the curvature of the reconstructed surface. This yields an acceptable trade-off between computational complexity and accuracy.

  • W. Huang, L. Kamenski, On the mesh nonsingularity of the moving mesh PDE method, Mathematics of Computation, published online on 02.10.2017, urlhttps://doi.org/10.1090/mcom/3271, 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, pp. published online on 18.11.2017, urlhttps://doi.org/10.1515/cmam-2017-0047, 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.

  • A. Caiazzo, F. Caforio, G. Montecinos, L.O. Müller, P.J. Blanco, E.F. Toro, Assessment of reduced order Kalman filter for parameter identification in one-dimensional blood flow models using experimental data, International Journal of Numerical Methods in Biomedical Engineering, 33 (2017), pp. e2843/1--e2843/26, DOI 10.1002/cnm.2843 .
    Abstract
    This work presents a detailed investigation of a parameter estimation approach based on the reduced order unscented Kalman filter (ROUKF) in the context of one-dimensional blood flow models. In particular, the main aims of this study are (i) to investigate the effect of using real measurements vs. synthetic data (i.e., numerical results of the same in silico model, perturbed with white noise) for the estimation and (ii) to identify potential difficulties and limitations of the approach in clinically realistic applications in order to assess the applicability of the filter to such setups. For these purposes, our numerical study is based on the in vitro model of the arterial network described by [Alastruey et al. 2011, J. Biomech. bf 44], for which experimental flow and pressure measurements are available at few selected locations. In order to mimic clinically relevant situations, we focus on the estimation of terminal resistances and arterial wall parameters related to vessel mechanics (Young's modulus and thickness) using few experimental observations (at most a single pressure or flow measurement per vessel). In all cases, we first perform a theoretical identifiability analysis based on the generalized sensitivity function, comparing then the results obtained with the ROUKF, using either synthetic or experimental data, to results obtained using reference parameters and to available measurements.

  • 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, published online on 13.12.2017, urlhttps://doi.org/10.1017/S0956792517000341, 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, K. Gillow, H. Wendland, Multilevel interpolation of divergence-free vector fields, IMA Journal of Numerical Analysis, 37 (2017), pp. 332--353, DOI 10.1093/imanum/drw006 .
    Abstract
    We introduce a multilevel technique for interpolating scattered data of divergence-free vector fields with the help of matrix-valued compactly supported kernels. The support radius at a given level is linked to the mesh norm of the data set at that level. There are at least three advantages of this method: no grid structure is necessary for the implementation, the multilevel approach is computationally cheaper than solving a large one-shot system and the interpolant is guaranteed to be analytically divergence-free. Furthermore, though we will not pursue this here, our multiscale approach is able to represent multiple scales in the data if present. We will prove convergence of the scheme, stability estimates and give a numerical example.

  • 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.

  • M. Radziunas, A. Zeghuzi, J. Fuhrmann, Th. Koprucki, H.-J. Wünsche, H. Wenzel, U. Bandelow, Efficient coupling of inhomogeneous current spreading and dynamic electro-optical models for broad-area edge-emitting semiconductor devices, Optical and Quantum Electronics, 49 (2017), pp. 332/1--332/8, DOI 10.1007/s11082-017-1168-3 .
    Abstract
    We extend a 2 (space) + 1 (time)-dimensional traveling wave model for broad-area edge-emitting semiconductor lasers by a model for inhomogeneous current spreading from the contact to the active zone of the laser. To speedup the performance of the device simulations, we suggest and discuss several approximations of the inhomogeneous current density in the active zone.

  • H. Si, N. Goerigk, Generalised Bagemihl polyhedra and a tight bound on the number of interior Steiner points, Computer-Aided Design, published online on 19.12.2017, urlhttps://doi.org/10.1016/j.cad.2017.11.009, DOI 10.1016/j.cad.2017.11.009 .

  • H. Si, N. Goerigk, On tetrahedralisations of generalised Chazelle polyhedra with interior Steiner points, Computer-Aided Design, published online on 18.12.2017, urlhttps://doi.org/10.1016/j.cad.2017.11.005, DOI 10.1016/j.cad.2017.11.005 .

Contributions to Collected Editions

  • 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. Kumar, J.H.M. Ten Thije Boonkkamp, B. Koren, A. Linke, A nonlinear flux approximation scheme for the viscous Burgers equation, 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. 457--465.

  • N. Lei, X. Zheng, H. Si, Z. Luo, X. Gu, Generalized regular quadrilateral mesh generation based on surface foliation, in: 26th International Meshing Roundtable, S. Owen, X. Roca, S.A. Mitchell, eds., 203 of Procedia Engineering, Elsevier, Amsterdam, 2017, pp. 336--348, DOI 10.1016/j.proeng.2017.09.818 .

  • M. Ma, X. Yu, N. Lei, H. Si, X. Gu, Guaranteed quality isotropic surface remeshing based on uniformization, in: 26th International Meshing Roundtable, S. Owen, X. Roca, S.A. Mitchell, eds., 203 of Procedia Engineering, Elsevier, Amsterdam, 2017, pp. 297--309, DOI 10.1016/j.proeng.2017.09.811 .

  • 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, A. Linke, Uniform second order convergence of a complete flux scheme on nonuniform 1D grids, 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. 303--310.

  • 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. Radziunas, A. Zeghuzi, J. Fuhrmann, Th. Koprucki, H.-J. Wünsche, H. Wenzel, U. Bandelow, Efficient coupling of inhomogeneous current spreading and electro-optical models for simulation of dynamics in broad-area semiconductor lasers, 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. 231--232.

  • H. Si, Y. Ren, N. Lei, X. Gu, On tetrahedralisations containing knotted and linked line segments, in: 26th International Meshing Roundtable, S. Owen, X. Roca, S.A. Mitchell, eds., 203 of Procedia Engineering, Elsevier, Amsterdam, 2017, pp. 323--335, DOI 10.1016/j.proeng.2017.09.816 .

Preprints, Reports, Technical Reports

  • M. Patriarca, P. Farrell, J. Fuhrmann, Th. Koprucki, Highly accurate quadrature-based Scharfetter--Gummel schemes for charge transport in degenerate semiconductors, Preprint no. 2498, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2498 .
    Abstract, PDF (646 kByte)
    We introduce a family of two point flux expressions for charge carrier transport described by drift-diffusion problems in degenerate semiconductors with non-Boltzmann statistics which can be used in Voronoï finite volume discretizations. In the case of Boltzmann statistics, Scharfetter and Gummel derived such fluxes by solving a linear two point boundary value problem yielding a closed form expression for the flux. Instead, a generalization of this approach to the nonlinear case yields a flux value given implicitly as the solution of a nonlinear integral equation. We examine the solution of this integral equation numerically via quadrature rules to approximate the integral as well as Newton's method to solve the resulting approximate integral equation. This approach results into a family of quadrature-based Scharfetter-Gummel flux approximations. We focus on four quadrature rules and compare the resulting schemes with respect to execution time and accuracy. A convergence study reveals that the solution of the approximate integral equation converges exponentially in terms of the number of quadrature points. With very few integration nodes they are already more accurate than a state-of-the-art reference flux, especially in the challenging physical scenario of high nonlinear diffusion. Finally, we show that thermodynamic consistency is practically guaranteed.

  • H. Stephan, A. Stephan, Memory equations as reduced Markov processes, Preprint no. 2496, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2496 .
    Abstract, PDF (315 kByte)
    A large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with such a memory equation, we give an explicit construction of the corresponding Markov process. From a physical point of view the Markov process can be understood as the change of the type of some quasiparticles along one-way loops. Typically, the arising Markov process does not have the detailed balance property. The method leads to a more realisitc modeling of memory equations. Moreover, it carries over the large number of investigation tools for Markov processes to memory equations, like the calculation of the equilibrium state, the asymptotic behavior and so on. The method can be used for an approximative solution of some degenerate memory equations like delay differential equations.

  • 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, Preprint no. 2489, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2489 .
    PDF (857 kByte)

  • P. Farrell, D. Peschka, Challenges for drift-diffusion simulations of semiconductors: A comparative study of different discretization philosophies, Preprint no. 2486, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2486 .
    Abstract, PDF (2457 kByte)
    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.

  • L.O. Müller, A. Caiazzo, P.J. Blanco, Reduced-order unscented Kalman filter in the frequency domain: Application to computational hemodynamics, Preprint no. 2484, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2484 .
    Abstract, PDF (1912 kByte)
    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.

  • C. Bartsch, V. John, R.I.A. Patterson, Simulations of an ASA flow crystallizer with a coupled stochastic-deterministic approach, Preprint no. 2483, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2483 .
    Abstract, PDF (378 kByte)
    A coupled solver for population balance systems is presented, where the flow, temperature, and concentration equations are solved with finite element methods, and the particle size distribution is simulated with a stochastic simulation algorithm, a so-called kinetic Monte-Carlo method. This novel approach is applied for the simulation of an axisymmetric model of a tubular flow crystallizer. The numerical results are compared with experimental data.

  • G.R. Barrenechea, V. John, P. Knobloch, R. Rankin, A unified analysis of Algebraic Flux Correction schemes for convection-diffusion equations, Preprint no. 2475, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2475 .
    Abstract, PDF (1693 kByte)
    Recent results on the numerical analysis of Algebraic Flux Correction (AFC) finite element schemes for scalar convection-diffusion equations are reviewed and presented in a unified way. A general form of the method is presented using a link between AFC schemes and nonlinear edge-based diffusion scheme. Then, specific versions of the method, this is, different definitions for the flux limiters, are reviewed and their main results stated. Numerical studies compare the different versions of the scheme.

  • P.L. Lederer, Ch. Merdon, J. Schöberl, Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods, Preprint no. 2462, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2462 .
    Abstract, PDF (854 kByte)
    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.

  • R. Schlundt, A multilevel Schur complement preconditioner for complex symmetric matrices, Preprint no. 2452, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2452 .
    Abstract, PDF (112 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. Using the example of Maxwell's equations the generality of the approach is demonstrated.

  • M. Akbas, A. Linke, L.G. Rebholz, P.W. Schroeder, An analogue of grad-div stabilization in nonconforming methods for incompressible flows, Preprint no. 2448, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2448 .
    Abstract, PDF (4822 kByte)
    Grad-div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems, for mitigating velocity errors that are sometimes called poor mass conservation. Such errors arise due to the relaxation of the divergence constraint in classical mixed methods, and are excited whenever the spacial discretization has to deal with comparably large and complicated pressures. In this contribution, an analogue of grad-div stabilization is presented for nonconforming flow discretizations of Discontinuous Galerkin or nonconforming finite element type. Here the key is the penalization of the jumps of the normal velocities over facets of the triangulation, which controls the measure-valued part of the distributional divergence of the discrete velocity solution. Furthermore, we characterize the limit for arbitrarily large penalization parameters, which shows that the proposed nonconforming Discontinuous Galerkin methods remain robust and accurate in this limit. Several numerical examples illustrate the theory and show their relevance for the simulation of practical, nontrivial flows.

  • 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, Preprint no. 2436, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2436 .
    Abstract, PDF (798 kByte)
    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.

  • S. Mohammadi, Ch. D'alonzo, L. Ruthotto, J. Polzehl, I. Ellerbrock, M.F. Callaghan, N. Weiskopf, K. Tabelow, Simultaneous adaptive smoothing of relaxometry and quantitative magnetization transfer mapping, Preprint no. 2432, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2432 .
    Abstract, PDF (3888 kByte)
    Attempts for in-vivo histology require a high spatial resolution that comes with the price of a decreased signal-to-noise ratio. We present a novel iterative and multi-scale smoothing method for quantitative Magnetic Resonance Imaging (MRI) data that yield proton density, apparent transverse and longitudinal relaxation, and magnetization transfer maps. The method is based on the propagation-separation approach. The adaptivity of the procedure avoids the inherent bias from blurring subtle features in the calculated maps that is common for non-adaptive smoothing approaches. The characteristics of the methods were evaluated on a high-resolution data set (500 μ isotropic) from a single subject and quantified on data from a multi-subject study. The results show that the adaptive method is able to increase the signal-to-noise ratio in the calculated quantitative maps while largely avoiding the bias that is otherwise introduced by spatially blurring values across tissue borders. As a consequence, it preserves the intensity contrast between white and gray matter and the thin cortical ribbon.

  • V. John, P. Knobloch, J. Novo, Finite elements for scalar convection-dominated equations and incompressible flow problems -- A never ending story?, Preprint no. 2410, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2410 .
    Abstract, PDF (283 kByte)
    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.

  • V. John, S. Kaya, J. Novo, Finite element error analysis of a mantle convection model, Preprint no. 2403, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2403 .
    Abstract, PDF (539 kByte)
    A mantle convection model consisting of the stationary Stokes equations and a time-dependent convection-diffusion equation for the temperature is studied. The Stokes problem is discretized with a conforming inf-sup stable pair of finite element spaces and the temperature equation is stabilized with the SUPG method. Finite element error estimates are derived which show the dependency of the error of the solution of one problem on the error of the solution of the other equation. The dependency of the error bounds on the coefficients of the problem is monitored.

  • W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes. Part III: Compactness and convergence, Preprint no. 2397, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2397 .
    Abstract, PDF (327 kByte)
    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, which continues the investigations of [DDGG17a, DDGG17b], we prove the compactness of the solution vector, and existence and convergence for the approximation schemes. We point at simple structural PDE arguments as an adequate substitute to the Aubin--Lions compactness Lemma and its generalisations: These familiar techniques attain their limit in the context of our model in which the relationship between time derivatives (transport) and diffusion gradients is highly non linear.

  • W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes. Part II: Approximation and a priori estimates, Preprint no. 2396, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2396 .
    Abstract, PDF (355 kByte)
    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, which continues the investigation of [DDGG17a], we derive for thermodynamically consistent approximation schemes the natural uniform estimates associated with the dissipations. Our results essentially improve our former study [DDGG16], in particular the a priori estimates concerning the relative chemical potentials.

  • W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes. Part I: Derivation of the model and survey of the results, Preprint no. 2395, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2395 .
    Abstract, PDF (343 kByte)
    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.

  • A. Linke, Ch. Merdon, M. Neilan, F. Neumann, Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes problem, Preprint no. 2374, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2374 .
    Abstract, PDF (334 kByte)
    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.

  • 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, Preprint no. 2372, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2372 .
    Abstract, PDF (3076 kByte)
    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.

Talks, Poster

  • R. Ahrens, F. Anker, C. Bartsch, A. Voigt, V. Wiedmeyer, K. Sundmacher, V. John, S. Le Borne, Advanced numerical methods for the simulation of population balance systems, 6th International Conference on Population Balance Modelling (PBM2018), Belgium, May 7 - 9, 2018.

  • C. Bartsch, V. John, R.I.A. Patterson, A new mixed stochastic-deterministic simulation approach to particle populations in fluid flows, 6th International Conference on Population Balance Modelling (PBM2018), Belgium, May 7 - 9, 2018.

  • A. Caiazzo, A benchmark study for CFD solvers: Simulation of air flow in livestock husbandry, 3rd Leibniz MMS Days 2018, February 28 - March 2, 2018, Wissenschaftszentrum Leipzig, February 28, 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.

  • W. Dreyer, Non-Newtonian fluids and the 2nd law of thermodynamics, Technische Universität Darmstadt, Mathematische Modellierung und Analysis, April 11, 2018.

  • J. Fuhrmann, Computational assessment of the derivation of the Butler--Volmer kinetics as a limit case of the Nernst--Planck equations with surface reactions, Workshop ``Numerical Optimization of the PEM Fuel Cell Bipolar Plate'', Universität Ulm, Zentrum für Solarenergie- und Wasserstoff-Forschung, March 20, 2018.

  • J. Fuhrmann, Handling research software: Recommendations to users, developers and research managers, 3rd Leibniz MMS Days 2018, February 28 - March 2, 2018, Wissenschaftszentrum Leipzig, March 1, 2018.

  • J. Fuhrmann, Robust quality preserving numerical methods for electroosmotic flows, 15th Symposium on Modeling and Validation of Electrochemical Energy Devices, April 12 - 13, 2018, Paul Scherrer Institut, Aarau, Switzerland, April 12, 2018.

  • J. Fuhrmann, Thermodynamically consistent models and numerical methods for electroosmotic flow, Tschechische Technische Universität Prag, Czech Republic, February 19, 2018.

  • J. Fuhrmann, Thermodynamically consistent finite volumes and pressure robust finite elements for electroosmotic flow, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2018), March 21 - 23, 2018, Technische Universität München, March 22, 2018.

  • V. John, A new mixed stochastic-deterministic simulation approach for particle populations in fluid flows, 6th European Seminar on Computing, June 3 - 8, 2018, Pilsen, Czech Republic, June 6, 2018.

  • A. Linke, On the role of the Helmholtz--Leray projector for a novel pressure-robust discretization theory for the incompressible Navier--Stokes equations, Georg-August-Universität Göttingen, Institut für Numerische und Angewandte Mathematik, January 23, 2018.

  • A. Linke, On the role of the Helmholtz--Leray projector in the space discretization of the Navier--Stokes equations, 3rd Leibniz MMS Days 2018, February 28 - March 2, 2018, Wissenschaftszentrum Leipzig, March 1, 2018.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2018), March 19 - 23, 2018, Technische Universität München, March 22, 2018.

  • A. Linke, n.n., April 3 - 6, 2018, Aix Marseille Université, Institut de Mathématiques, France.

  • H. Si, Challenges in 3D unstructured mesh generation and adaptation, Challenges in the Computational Modeling of Beijing's Air Pollution and Traffic, March 19 - 23, 2018, Beijing University of Technology, China, March 22, 2018.

  • W. Dreyer, J. Fuhrmann, P. Gajewski, C. Guhlke, M. Landstorfer, M. Maurelli, R. Müller, Stochastic model for LiFePO4-electrodes, ModVal14 -- 14th Symposium on Fuel Cell and Battery Modeling and Experimental Validation, Karlsruhe, March 2 - 3, 2017.

  • P. Farrell, Numerical solution of PDEs via RBFs and FVM with focus on semiconductor problems, Technische Universität Hamburg, Institut für Mathematik, Harburg, January 6, 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.

  • N. Kumar, J.H.M. Ten Thije Boonkkamp, B. Koren, A. Linke, A nonlinear flux approximation scheme for the viscous Burgers equation, 8th International Symposium on Finite Volumes for Complex Applications (FVCA 8), Université Lille 1, Villeneuve d'Ascq, France, June 12 - 16, 2017.

  • N. Ahmed, A numerical study of residual based variational multiscale methods for turbulent incompressible flow problems, American University of the Middle East, Dasman, Kuwait, November 2, 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, A mixed stochastic -- Numeric algorithm for transported interacting particles, 38th Northern German Colloquium on Applied Analysis and Numerical Mathematics (NoKo 2017), May 4 - 5, 2017, Technische Universität Hamburg, Institut für Mathematik, May 5, 2017.

  • C. Bartsch, A mixed stochastic-deterministic approach to particles interacting in a flow, SIAM Conference on Mathematical and Computational Issues in the Geosciences, September 11 - 14, 2017, Friedrich-Alexander-Universität Erlangen-Nürnberg, September 14, 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.

  • W. Dreyer, Space-time transformations and the principle of material objectivity, May 3 - 5, 2017, Technische Universität Darmstadt, Fachbereich Mathematik.

  • W. Dreyer, Thermodynamically consistent modeling of fluids, 2nd Leibniz MMS Days 2017, February 22 - 24, 2017, Technische Informationsbibliothek, Hannover, February 23, 2017.

  • P. Farrell, How do electrons move in space? Flux discretizations for non-Boltzmann statistics, SIAM Conference on Computational Science and Engineering (CSE17), February 27 - March 3, 2017, Hilton Atlanta, Georgia, USA, March 1, 2017.

  • J. Fuhrmann, A. Glitzky, M. Liero, Hybrid finite-volume/finite-element schemes for p(x)-Laplace thermistor models, 8th International Symposium on Finite Volumes for Complex Applications (FVCA 8), Université Lille 1, Villeneuve d'Ascq, France, June 15, 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.

  • J. Fuhrmann, A coupled FEM-FVM method for electroosmotic flow, 30th Chemnitz FEM Symposium, September 25 - 27, 2017, Bundesinstitut für Erwachsenenbildung, St. Wolfgang / Strobl, Austria, September 27, 2017.

  • J. Fuhrmann, A finite volume scheme for Nernst-Planck-Poisson systems with ion size and solvation effects, 8th International Symposium on Finite Volumes for Complex Applications (FVCA 8), June 12 - 16, 2017, Université Lille 1, Villeneuve d'Ascq, France, June 14, 2017.

  • J. Fuhrmann, A policy level helping hand to deal with research software, FORCE2017 Research Communication and e-¬Scholarship Conference, October 25 - 27, 2017, Berlin, August 27, 2018.

  • J. Fuhrmann, Ionic mixtures with volume constraints: Models and numerical approaches, International workshop on liquid metal battery fluid dynamics, May 15 - 17, 2017, Dresden, May 16, 2017.

  • J. Fuhrmann, Robust quality preserving numerical methods for electroosmotic flow, CIM-WIAS Workshop ``Topics in Applied Analysis and Optimisation'', December 6 - 8, 2017, International Center for Mathematics, University of Lisbon, Portugal, December 7, 2017.

  • J. Fuhrmann, Strategien für die Verwertung wissenschaftlicher Software: praktische Erfahrungen und aktuelle Entwicklungen, Frühjahrstreffen des Arbeitskreises Wissenstransfer der Leibniz-Gemeinschaft, Leibniz-Zentrum für Marine Tropenforschung, Bremen, May 10, 2017.

  • V. John, Analytical and numerical results for algebraic flux correction schemes, 12th International Workshop on Variational Multiscale and Stabilization Methods (VMS-2017), April 26 - 28, 2017, Edificio Celestino Mutis, Campus Reina Mercedes, Sevilla, Spain, April 26, 2017.

  • V. John, Finite element methods for incompressible flow problems, May 14 - 18, 2017, Beijing Computational Science Research Center, Applied and Computational Mathematics, China.

  • V. John, Finite elements for scalar convection-dominated equations and incompressible flow problems --- A never ending story?, 30th Chemnitz FEM Symposium, September 25 - 27, 2017, Bundesinstitut für Erwachsenenbildung, St. Wolfgang / Strobl, Austria, September 27, 2017.

  • V. John, Variational multiscale (VMS) methods for the simulation of turbulent incompressible flows, CDS: Computational Science Symposium, March 16 - 18, 2017, Indian Institute of Science, Department of Computer and Data Sciences, Bangalore, India, March 16, 2017.

  • V. John, Variational multiscale (VMS) methods for the simulation of turbulent incompressible flows, Mahindra École Centrale, School of Natural Sciences, Hyderabad, India, March 9, 2017.

  • V. John, Variational multiscale (VMS) methods for the simulation of turbulent incompressible flows, Chinese Academy of Sciences, Academy of Mathematics and Systems Science, Beijing, May 10, 2017.

  • V. John, Variational multiscale (VMS) methods for the simulation of turbulent incompressible flows, Peking University, School of Mathematical Sciences, Beijing, China, May 11, 2017.

  • M. Liero, A. Glitzky, Th. Koprucki, J. Fuhrmann, 3D electrothermal simulations of organic LEDs showing negative differential resistance, Multiscale Modelling of Organic Semiconductors: From Elementary Processes to Devices, Grenoble, France, September 12 - 15, 2017.

  • A. Linke, On new developments in the discretization theory for PDEs, possibly relevant for CFD & GFD, 2nd Leibniz MMS Days 2017, February 22 - 24, 2017, Technische Informationsbibliothek, Hannover, February 23, 2017.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, 12th International Workshop on Variational Multiscale and Stabilization Methods (VMS-2017), April 26 - 28, 2017, Edificio Celestino Mutis, Campus Reina Mercedes, Sevilla, Spain, April 27, 2017.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, 8th International Symposium on Finite Volumes for Complex Applications (FVCA 8), June 12 - 16, 2017, Université Lille 1, Villeneuve d'Ascq, France, June 13, 2017.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--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 23, 2017.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, 30th Chemnitz FEM Symposium, September 25 - 27, 2017, Bundesinstitut für Erwachsenenbildung, St. Wolfgang / Strobl, Austria, September 27, 2017.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, Universität der Bundeswehr München, Institut für Mathematik und Bauinformatik, Neubiberg, January 18, 2017.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, Technische Universität Dortmund, Institut für Angewandte Mathematik, March 23, 2017.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, Freie Universität Berlin, Institut für Mathematik, May 3, 2017.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, Technische Universität Darmstadt, Fachbereich Mathematik, July 20, 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.

  • H. Si, An introduction to Delaunay-based mesh generation and adaptation, 10th National Symposium on Geometric Design and Computing (GDC 2017), August 12 - 14, 2017, Shandong Business School, Yantai, China, August 12, 2017.

  • H. Si, Challenges in tetrahedral mesh generation, PaMPA: Parallel Mesh Partitioning and Adaptation, 1st PaMPA Day Workshop, October 18, 2017, INRIA Bordeaux -- Sud-Ouest, France, October 18, 2017.

  • H. Si, On tetrahedralisations containing knotted and linked line segments, 26th International Meshing Roundtable and User Forum ``Mesh Modeling for Simulations and Visualization'', Session 4A ``Tet Meshing'', September 18 - 22, 2017, Barcelona, Spain, September 19, 2017.

  • H. Si, On tetrahedralisations containing knotted and linked line segments, Dalian University, School of Software and Technology, China, August 10, 2017.

  • H. Si, Tetrahedral mesh improvement using moving mesh smoothing and lazy searching flips, University Beijing, School of Mathematics and Systems Science, China, December 1, 2017.

  • K. Tabelow, Ch. D'alonzo, L. Ruthotto, M.F. Callaghan, N. Weiskopf, J. Polzehl, S. Mohammadi, Removing the estimation bias due to the noise floor in multi-parameter maps, The International Society for Magnetic Resonance in Medicine (ISMRM) 25th Annual Meeting & Exhibition, Honolulu, USA, April 22 - 27, 2017.

  • K. Tabelow, Ch. D'alonzo, J. Polzehl, Toward in-vivo histology of the brain, 2nd Leibniz MMs Days 2017, Technische Informationsbibliothek, Hannover, February 22 - 24, 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.

External Preprints

  • J. Fuhrmann, K.S. Scheliga, H. Pampel, H. Bernstein, B. Fritzsch, ET AL., Helmholtz Open Science Workshop ``Zugang zu und Nachnutzung von wissenschaftlicher Software'', Report, Deutsches GeoForschungsZentrum GFZ, 2017, DOI 10.2312/lis.17.01 .