Publications

Monographs

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

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

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

Articles in Refereed Journals

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

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

  • D. Chaudhuri, M. O'Donovan, T. Streckenbach, O. Marquart, P. Farrell, S.K. Patra, Th. Koprucki, S. Schulz, Multiscale simulations of the electronic structure of III-nitride quantum wells with varied Indium content: Connecting atomistic and continuum-based models, Journal of Applied Physics, 129 (2021), published online on 18.02.2021, DOI 10.1063/5.0031514 .

  • D. Frerichs, V. John, On reducing spurious oscillations in discontinuous Galerkin (DG) methods for steady-state convection-diffusion equations, Journal of Computational and Applied Mathematics, 393 (2021), pp. 113487/1--113487/20 (published online on 17.02.2021), DOI 10.1016/j.cam.2021.113487 .

  • U. Wilbrandt, N. Alia, V. John, Optimal control of a buoyancy-driven liquid steel stirring modeled with single-phase Navier--Stokes equations, Journal of Mathematics in Industry, 11 (2021), pp. 00106-7/1--00106-7/22, DOI 10.1186/s13362-021-00106-7 .
    Abstract
    Gas stirring is an important process used in secondary metallurgy. It allows to homogenize the temperature and the chemical composition of the liquid steel and to remove inclusions which can be detrimental for the end-product quality. In this process, argon gas is injected from two nozzles at the bottom of the vessel and rises by buoyancy through the liquid steel thereby causing stirring, i.e., a mixing of the bath. The gas flow rates and the positions of the nozzles are two important control parameters in practice. A continuous optimization approach is pursued to find optimal values for these control variables. The effect of the gas appears as a volume force in the single-phase incompressible NavierStokes equations. Turbulence is modeled with the Smagorinsky Large Eddy Simulation (LES) model. An objective functional based on the vorticity is used to describe the mixing in the liquid bath. Optimized configurations are compared with a default one whose design is based on a setup from industrial practice.

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

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

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

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

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

  • C. Cancès, C. Chainais-Hillairet, J. Fuhrmann, B. Gaudeul, A numerical analysis focused comparison of several finite volume schemes for an unipolar degenerated drift-diffusion model, IMA Journal of Numerical Analysis, 41 (2021), pp. 271--314 (published online on 17.07.2020), DOI 10.1093/imanum/draa002 .
    Abstract
    In this paper, we consider an unipolar degenerated drift-diffusion system where the relation between the concentration of the charged species c and the chemical potential h is h(c) = log c/1-c. We design four different finite volume schemes based on four different formulations of the fluxes. We provide a stability analysis and existence results for the four schemes. The convergence proof with respect to the discretization parameters is established for two of them. Numerical experiments illustrate the behaviour of the different schemes.

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

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

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

  • A. Kirch, A. Fischer, M. Liero, J. Fuhrmann, A. Glitzky, S. Reineke, Experimental proof of Joule heating-induced switched-back regions in OLEDs, Light: Science and Applications, 9 (2020), pp. 5/1--5/10, DOI 10.1038/s41377-019-0236-9 .

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

  • L.G. Ramos, R. Kehl, R. Nabben, Projections, deflation and multigrid for non-symmetric matrices, SIAM Journal on Matrix Analysis and Applications, 41 (2020), pp. 83--105, DOI 10.1016/j.jocs.2019.101067 .

  • J. Tang, P. Cui, B. Li, Z. Yaobin, H. Si, Parallel hybrid mesh adaptation by refinement and coarsening, Graphical Models, 111 (2020), pp. 101084/1--101084/10, DOI 10.1016/j.gmod.2020.101084 .

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

  • L. Blank, E. Meneses Rioseco, U. Wilbrandt, A. Caiazzo, Modeling, simulation, and optimization of geothermal energy production from hot sedimentary aquifers, Computer & Geosciences, 25 (2021), pp. 67--104 (published online on 02.09.2020), DOI 10.1007/s10596-020-09989-8 .
    Abstract
    Geothermal district heating development has been gaining momentum in Europe with numerous deep geothermal installations and projects currently under development. With the increasing density of geothermal wells, questions related to the optimal and sustainable reservoir exploitation become more and more important. A quantitative understanding of the complex thermo-hydraulic interaction between tightly deployed geothermal wells in heterogeneous temperature and permeability fields is key for a maximum sustainable use of geothermal resources. Motivated by the geological settings of the Upper Jurassic aquifer in the Greater Munich region, we develop a computational model based on finite element analysis and gradient-free optimization to simulate groundwater flow and heat transport in hot sedimentary aquifers, and investigate numerically the optimal positioning and spacing of multi-well systems. Based on our numerical simulations, net energy production from deep geothermal reservoirs in sedimentary basins by smart geothermal multi-well arrangements provides significant amounts of energy to meet heat demand in highly urbanized regions. Our results show that taking into account heterogeneous permeability structures and variable reservoir temperature may drastically affect the results in the optimal configuration. We demonstrate that the proposed numerical framework is able to efficiently handle generic geometrical and geologocal configurations, and can be thus flexibly used in the context of multi-variable optimization problems. Hence, this numerical framework can be used to assess the extractable geothermal energy from heterogeneous deep geothermal reservoirs by the optimized deployment of smart multi-well systems.

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

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

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

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

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

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

Contributions to Collected Editions

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

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

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

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

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

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

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

Preprints, Reports, Technical Reports

  • L. Heltai, A. Caiazzo, L.O. Müller, Multiscale coupling of one-dimensional vascular models and elastic tissues, Preprint no. 2830, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2830 .
    Abstract, PDF (4777 kByte)
    We present a computational multiscale model for the efficient simulation of vascularized tissues, composed of an elastic three-dimensional matrix and a vascular network. The effect of blood vessel pressure on the elastic tissue is surrogated via hyper-singular forcing terms in the elasticity equations, which depend on the fluid pressure. In turn, the blood flow in vessels is treated as a one-dimensional network. The pressure and velocity of the blood in the vessels are simulated using a high-order finite volume scheme, while the elasticity equations for the tissue are solved using a finite element method. This work addresses the feasibility and the potential of the proposed coupled multiscale model. In particular, we assess whether the multiscale model is able to reproduce the tissue response at the effective scale (of the order of millimeters) while modeling the vasculature at the microscale. We validate the multiscale method against a full scale (three-dimensional) model, where the fluid/tissue interface is fully discretized and treated as a Neumann boundary for the elasticity equation. Next, we present simulation results obtained with the proposed approach in a realistic scenario, demonstrating that the method can robustly and efficiently handle the one-way coupling between complex fluid microstructures and the elastic matrix.

  • D. Bothe, W. Dreyer, P.-É. Druet, Multicomponent incompressible fluids -- An asymptotic study, Preprint no. 2825, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2825 .
    Abstract, PDF (519 kByte)
    This paper investigates the asymptotic behavior of the Helmholtz free energy of mixtures at small compressibility. We start from a general representation for the local free energy that is valid in stable subregions of the phase diagram. On the basis of this representation we classify the admissible data to construct a thermodynamically consistent constitutive model. We then analyze the incompressible limit, where the molar volume becomes independent of pressure. Here we are confronted with two problems:

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

  • B. Gaudeul, J. Fuhrmann, Entropy and convergence analysis for two finite volume schemes for a Nernst--Planck--Poisson system with ion volume constraints, Preprint no. 2811, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2811 .
    Abstract, PDF (6564 kByte)
    In this paper, we consider a drift-diffusion system with cross-coupling through the chemical potentials comprising a model for the motion of finite size ions in liquid electrolytes. The drift term is due to the self-consistent electric field maintained by the ions and described by a Poisson equation. We design two finite volume schemes based on different formulations of the fluxes. We also provide a stability analysis of these schemes and an existence result for the corresponding discrete solutions. A convergence proof is proposed for non-degenerate solutions. Numerical experiments show the behavior of these schemes.

  • V. Miloš, P. Vágner, D. Budáč, M. Carda, M. Paidar, J. Fuhrmann, K. Bouzek, Generalized Poisson--Nernst--Planck-based physical model of O$_2$ I LSM I YSZ electrode, Preprint no. 2797, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2797 .
    Abstract, PDF (781 kByte)
    The paper presents a generalized Poisson-Nernst-Planck model of an yttria-stabilized zirconia electrolyte developed from first principles of nonequilibrium thermodynamics which allows for spatial resolution of the space charge layer. It takes into account limitations in oxide ion concentrations due to the limited availability of oxygen vacancies. The electrolyte model is coupled with a reaction kinetic model describing the triple phase boundary with electron conducting lanthanum strontium manganite and gaseous phase oxygen. By comparing the outcome of numerical simulations based on different formulations of the kinetic equations with results of EIS and CV measurements we attempt to discern the existence of separate surface lattice sites for oxygen adatoms and O2- from the assumption of shared ones. Furthermore, we discern mass-action kinetics models from exponential kinetics models.

  • D. Abdel, P. Farrell, J. Fuhrmann, Assessing the quality of the excess chemical potential flux scheme for degenerate semiconductor device simulation, Preprint no. 2787, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2787 .
    Abstract, PDF (682 kByte)
    The van Roosbroeck system models current flows in (non-)degenerate semiconductor devices. Focusing on the stationary model, we compare the excess chemical potential discretization scheme, a flux approximation which is based on a modification of the drift term in the current densities, with another state-of-the-art Scharfetter-Gummel scheme, namely the diffusion-enhanced scheme. Physically, the diffusion-enhanced scheme can be interpreted as a flux approximation which modifies the thermal voltage. As a reference solution we consider an implicitly defined integral flux, using Blakemore statistics. The integral flux refers to the exact solution of a local two point boundary value problem for the continuous current density and can be interpreted as a generalized Scharfetter-Gummel scheme. All numerical discretization schemes can be used within a Voronoi finite volume method to simulate charge transport in (non-)degenerate semiconductor devices. The investigation includes the analysis of Taylor expansions, a derivation of error estimates and a visualization of errors in local flux approximations to extend previous discussions. Additionally, drift-diffusion simulations of a p-i-n device are performed.

  • D. Abdel, P. Vágner, J. Fuhrmann, P. Farrell, Modelling charge transport in perovskite solar cells: Potential-based and limiting ion depletion, Preprint no. 2780, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2780 .
    Abstract, PDF (1051 kByte)
    From Maxwell--Stefan diffusion and general electrostatics, we derive a drift-diffusion model for charge transport in perovskite solar cells (PSCs) where any ion in the perovskite layer may flexibly be chosen to be mobile or immobile. Unlike other models in the literature, our model is based on quasi Fermi potentials instead of densities. This allows to easily include nonlinear diffusion (based on Fermi--Dirac, Gauss--Fermi or Blakemore statistics for example) as well as limit the ion depletion (via the Fermi--Dirac integral of order-1). The latter will be motivated by a grand-canonical formalism of ideal lattice gas. Furthermore, our model allows to use different statistics for different species. We discuss the thermodynamic equilibrium, electroneutrality as well as generation/recombination. Finally, we present numerical finite volume simulations to underline the importance of limiting ion depletion.

  • P.L. Lederer, Ch. Merdon, Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations, Preprint no. 2750, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2750 .
    Abstract, PDF (422 kByte)
    This paper improves guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the errors of divergence-free primal and H(div)-conforming dual mixed methods (for the velocity gradient) with an equilibration constraint that needs special care when discretised. To relax the constraints on the primal and dual method, a more general result is derived that enables the use of a recently developed mass conserving mixed stress discretisation to design equilibrated fluxes that yield pressure-independent guaranteed upper bounds for any pressure-robust (but not necessarily divergence-free) primal discretisation. Moreover, a provably efficient local design of the equilibrated fluxes is presented that reduces the numerical costs of the error estimator. All theoretical findings are verified by numerical examples which also show that the efficiency indices of our novel guaranteed upper bounds for the velocity error are close to 1.

  • N. Ahmed, G.R. Barrenechea, E. Burman, J. Guzmán, A. Linke, Ch. Merdon, A pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation, Preprint no. 2740, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2740 .
    Abstract, PDF (843 kByte)
    Discretization of Navier--Stokes' equations using pressure-robust finite element methods is considered for the high Reynolds number regime. To counter oscillations due to dominating convection we add a stabilization based on a bulk term in the form of a residual-based least squares stabilization of the vorticity equation supplemented by a penalty term on (certain components of) the gradient jump over the elements faces. Since the stabilization is based on the vorticity equation, it is independent of the pressure gradients, which makes it pressure-robust. Thus, we prove pressureindependent error estimates in the linearized case, known as Oseen's problem. In fact, we prove an O(hk+1/2) error estimate in the L2-norm that is known to be the best that can be expected for this type of problem. Numerical examples are provided that, in addition to confirming the theoretical results, show that the present method compares favorably to the classical residual-based SUPG stabilization.

  • G. Fu, Ch. Lehrenfeld, A. Linke, T. Streckenbach, Locking free and gradient robust H(div)-conforming HDG methods for linear elasticity, Preprint no. 2680, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2680 .
    Abstract, PDF (429 kByte)
    Robust discretization methods for (nearly-incompressible) linear elasticity are free of volume-locking and gradient-robust. While volume-locking is a well-known problem that can be dealt with in many different discretization approaches, the concept of gradient-robustness for linear elasticity is new. We discuss both aspects and propose novel Hybrid Discontinuous Galerkin (HDG) methods for linear elasticity. The starting point for these methods is a divergence-conforming discretization. As a consequence of its well-behaved Stokes limit the method is gradient-robust and free of volume-locking. To improve computational efficiency, we additionally consider discretizations with relaxed divergence-conformity and a modification which re-enables gradient-robustness, yielding a robust and quasi-optimal discretization also in the sense of HDG superconvergence.

Talks, Poster

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

  • P. Vágner, Generalized Nernst--Planck--Poisson model of solid oxide YSZ LSM O_2 electrode interface (online talk), 17th Symposium on Modeling and Experimental Validation of Fuel Cells, Electrolysers and Batteries (MODVAL 17) (Online Event), April 20 - 22, 2021, EPFL-Valais/Wallis, Sion, Switzerland, April 20, 2021.

  • P. Vágner, Generalized Nernst-Planck-Poisson model of solid oxide YSZ LSM O2 electrode interface, Joint European Thermodynamics Conference, June 14 - 18, 2021, Czech Technical University, Prague, Czech Republic, June 17, 2021.

  • P. Vágner, n.n., 29th Topical Meeting of the International Society of Electrochemistry (Online Event), April 18 - 21, 2021, University of Chemistry and Technology Prague, Mikulov, Czech Republic.

  • A. Caiazzo, Multiscale coupling of one-dimensional vascular models and elastic tissues (online talk), European Congress of Mathematics, MS-ID39: Modeling, approximation, and analysis of partial differential equations involving singular source terms (Online Event), June 20 - 26, 2021, University of Primorska, Faculty of Mathematics, Portorož, June 22, 2021.

  • V. John, Finite element methods respecting the discrete maximum principle for convection-diffusion equations, Berlin International Graduate School in Model and Simulation based Research (BIMoS), Technische Universität Berlin, June 7, 2021.

  • V. John, On the convergence order of the einite element error in the kinetic energy for high reynolds number incompressible flows (online talk), International Symposium on Recent Trends in Differential Equations: Theory, Computation & Application in Symposium on Recent Trends in Numerical Method for PDEs and Applications (Online Event), March 19 - 22, 2021, Indian Institute of Technology Kanpur, India, March 19, 2021.

  • A. Linke, On pressure-robustness,well-balanced schemes andthe spatial discretization ofhigh Reynolds number flows (online talk), GAMM 91st Annual Meetingof the International Association of Applied Mathematics and Mechanics, S18: Numerical methods of differential equations (Online Event), March 15 - 19, 2021, Universität Kassel, March 19, 2021.

  • A. Linke, On the role of the Helmholtz--Leray projector for a novel pressure-robust discretization theory for the incompressible Navier--Stokes equations, Ecole de Recherche CIMPA & Workshop, May 31 - June 7, 2021, Nador, Morocco, June 2, 2021.

  • CH. Merdon, A novel gradient-robust, well-balanced discretisation for the compressible isothermal Navier--Stokes problem (online talk), GAMM 91st Annual Meetingof the International Association of Applied Mathematics and Mechanics, S18: Numerical methods of differential equations (Online Event), March 15 - 19, 2021, Universität Kassel, March 19, 2021.

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

  • A. Caiazzo, CFD Simulation for non invasive estimation of blood pressure (online talk), Leibniz MMS Symposium on Computational Geophysics and Fluid Dynamics (Online Event), October 21, 2020, October 21, 2020.

  • A. Caiazzo, Multiscale modeling of vascularized tissues (online talk), Virtual Physiological Human (VPH2020) -- When models, methods & experiments meet the clinic (Online Event), August 24 - 28, 2020, INRIA, Paris, France, August 28, 2020.

  • D. Frerichs, A pressure-robust virtual element method for the Stokes problem on polygonal/polyhedral meshes (online talk), Leibniz MMS Symposium on Computational Geophysics and Fluid Dynamics (Online Event), October 21, 2020, October 21, 2020.

  • D. Frerichs, A really pressure-robust virtual element method for the Stokes problem (online talk), Conference on Scientific Computing (ALGORITMY 2020) (Online Event), September 10 - 15, 2020, Slovak University of Technology, Podbanské, Slovakia, September 14, 2020.

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

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

  • J. Fuhrmann, Finite volume methods for nonlinear multiphysics Problemp (online talk), VoronoiFVM.jl - finite volume methods for nonlinear multiphysics problems (Online Event), Lisboa, July 31, 2020.

  • J. Fuhrmann, Quality preserving numerical methods for electroosmotic flow (online talk), Conference on Scientific Computing (ALGORITMY 2020) (Online Event), September 10 - 15, 2020, Slovak University of Technology, Podbanské, Slovakia, September 14, 2020.

  • J. Fuhrmann, VTKView.jl -- vtk based in-situ visualization in Julia (online talk), Julia VizCon (Online Event), March 12 - 19, 2020, March 15, 2020.

  • V. John, Algebraic finite element stabilizations for convection-diffusion equations, Indian Institute of Technology Roorkee, Department of Mathematics, India, January 24, 2020.

  • V. John, Numerical methods for convection-dominated equations, January 27 - 30, 2020, Indian Institute of Technology Roorkee, Department of Mathematics, India.

  • V. John, On the provable convergence order for the kinetic energy of FEMs for the incompressible Navier--Stokes equations, 7th European Seminar on Computing (ESCO 2020) (Online Event), Pilsen, Czech Republic, June 8 - 12, 2020.

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

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

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

  • A. Linke, An overview on the Leibniz MMS network and its GFD-CFD research activities (online talk), Leibniz MMS Symposium on Computational Geophysics and Fluid Dynamics (Online Event), October 21, 2020, October 21, 2020.

  • A. Linke, Gradient-robustness: A new concept assuring accurate spatial discretizations for vector-valued PDEs, Workshop ``Structure, Regularity and Robustness in the Approximation of PDEs'', Università degli Studi di Milano Statale, Italy, February 10, 2020.

  • A. Linke, Gradient-robustness: A new concept assuring accurate spatial discretizations for vector-valued PDEs (online talk), Eindhoven University of Technology, Centre for Analysis, Scientific Computing and Applications, Netherlands, June 24, 2020.

  • A. Linke, On pressure-robust space discretizations and incompressible high Reynolds number flows (online talk), Conference on Scientific Computing (ALGORITMY 2020) (Online Event), September 10 - 15, 2020, Slovak University of Technology, Podbanské, Slovakia, September 15, 2020.

  • A. Linke, On the significance of pressure-robustness for locking-free incompressible flow solvers at high Reynolds numbers (online talk), Workshop on Modeling and Simulation of Transport Phenomena 2020 (Online Event), October 13 - 15, 2020, Technische Universität Dortmund, Schloss-Hotel Petry, Treis-Karden, October 14, 2020.

  • CH. Merdon, A gradient-robust discretisation for the compressible Stokes problem (online talk), Conference on Scientific Computing (ALGORITMY 2020) (Online Event), September 10 - 15, 2020, Slovak University of Technology, Podbanské, Slovakia, September 14, 2020.

  • CH. Merdon, Gradient-robust discretisations for compressible Navier--Stokes flows (online talk), Leibniz MMS Symposium on Computational Geophysics and Fluid Dynamics (Online Event), October 21, 2020, October 21, 2020.

  • CH. Merdon, Well-balanced discretisation for the compressible Stokes problem by gradient-robustness (online talk), Finite Volumes for Complex Applications IX (Online Event), June 15 - 19, 2020, University of Bergen, Norway, June 17, 2020.

  • O. Pártl, Mathematical modeling of non-isothermal compositional compressible fluid flow in zeolite bed and above its surface (online talk), Conference on Scientific Computing (ALGORITMY 2020) (Online Event), September 10 - 15, 2020, Slovak University of Technology, Podbanské, Slovakia, September 14, 2020.

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

  • H. Si, On decomposition of embedded prismatoids in $rm I!R^3$ without additional points (online talk), 10th International Conference ``Numerical Geometry, Grid Generation and Scientific Computing'' (NUMGRID 2020) (Online Event), November 25 - 27, 2020, Russian Academy of Sciences, Federal Research Center of Information and Control, Moscow, Russian Federation, November 27, 2020.

External Preprints

  • M. Coghi, W. Dreyer, P. Gajewski, C. Guhlke, P. Friz, M. Maurelli, A McKean--Vlasov SDE and particle system with interactionfrom reflecting boundaries, Preprint no. 2102.12315v1/1--2102.12315v1/37, Cornell University Library, arXiv.org, 2021.