Numerische Mathematik und Wissenschaftliches Rechnen

Publikationen

Monografien

• G.R. Barrenechea, V. John, P. Knobloch, Monotone Discretizations for Elliptic Second Order Partial Differential Equations, 61 of Springer Series in Computational Mathematics, Springer, Cham, 2025, pp. 1--649, (Monograph Published), DOI 978-3-031-80684-1 .
  Abstract
  This book offers a comprehensive presentation of numerical methods for elliptic boundary value problems that satisfy discrete maximum principles (DMPs). The satisfaction of DMPs ensures that numerical solutions possess physically admissible values, which is of utmost importance in numerous applications. A general framework for the proofs of monotonicity and discrete maximum principles is developed for both linear and nonlinear discretizations. Starting with the Poisson problem, the focus is on convection-diffusion-reaction problems with dominant convection, a situation which leads to a numerical problem with multi-scale character. The emphasis of this book is on finite element methods, where classical (usually linear) and modern nonlinear discretizations are presented in a unified way. In addition, popular finite difference and finite volume methods are discussed. Besides DMPs, other important properties of the methods, like convergence, are studied. Proofs are presented step by step, allowing readers to understand the analytic techniques more easily. Numerical examples illustrate the behavior of the methods.

• A. Caiazzo, L. Heltai, I.E. Vignon-Clementel, Part 1/Chapter 2: Mathematical Modeling of Blood Flow in the Cardiovascular System, in: Quantification of Biophysical Parameters in Medical Imaging, Second Edition, I. Sack, T. Schaeffter, eds., Springer, Cham, 2024, pp. 39--61, (Chapter Published), DOI 10.1007/978-3-031-61846-8_3 .

Artikel in Referierten Journalen

• C.L. Manganelli, D. Spirito, P. Farrell, J. Frigerio, A. De Lacovo, D. Marian, M. Virgilio, Strain engineering in semiconductor materials, physica status solidi (RLL) -- Rapid Research Letters (pss RRL), 19 (2025), pp. 2400383/1--2400383/3, DOI 10.1002/pssr.202400383 .
  Abstract
  Strain engineering has become an essential strategy in the advancement of semiconductor technologies, providing a power- ful mean to modulate the electronic, optical, and mechanical properties of materials. By introducing controlled deformation into crystal lattices, this approach enables enhanced carrier mobility, tailored bandgap energies, and improved device perfor- mance across applications in photonics, optoelectronics, and quantum technologies.

• TH. Anandh, D. Ghose, H. Jain, P. Sunkad, S. Ganesan, V. John, Improving hp-variational physics-informed neural networks for steady-state convection-dominated problems, Computer Methods in Applied Mechanics and Engineering, 438 (2025), pp. 117797/1--117797/25, DOI 10.1016/j.cma.2025.117797 .
  Abstract
  This paper proposes and studies two extensions of applying hp-variational physics-informed neural networks, more precisely the FastVPINNs framework, to convection-dominated convection-diffusion-reaction problems. First, a term in the spirit of a SUPG stabilization is included in the loss functional and a network architecture is proposed that predicts spatially varying stabilization parameters. Having observed that the selection of the indicator function in hard-constrained Dirichlet boundary conditions has a big impact on the accuracy of the computed solutions, the second novelty is the proposal of a network architecture that learns good parameters for a class of indicator functions. Numerical studies show that both proposals lead to noticeably more accurate results than approaches that can be found in the literature.

• B. García-Archilla, V. John, J. Novo, POD-ROM methods: From a finite set of snapshots to continuous-in-time approximations, SIAM Journal on Numerical Analysis, 63 (2025), pp. 800-826, DOI 10.1137/24M1645681 .
  Abstract
  This paper studies discretization of time-dependent partial differential equations (PDEs) by proper orthogonal decomposition reduced order models (POD-ROMs). Most of the analysis in the literature has been performed on fully discrete methods using first order methods in time, typically the implicit Euler time integrator. Our aim is to show which kind of error bounds can be obtained using any time integrator, both in the full order model (FOM), applied to compute the snapshots, and in the POD-ROM method. To this end, we analyze in this paper the continuous-in-time case for both the FOM and POD-ROM methods, although the POD basis is obtained from snapshots taken at a discrete (i.e., not continuous) set of times. Two cases for the set of snapshots are considered: the case in which the snapshots are based on first order divided differences in time and the case in which they are based on temporal derivatives. Optimal pointwise-in-time error bounds between the FOM and the POD-ROM solutions are proved for the L high 2 (omega) norm of the error for a semilinear reaction-diffusion model problem. The dependency of the errors on the distance in time between two consecutive snapshots and on the tail of the POD eigenvalues is tracked. Our detailed analysis allows us to show that, in some situations, a small number of snapshots in a given time interval might be sufficient to accurately approximate the solution in the full interval. Numerical studies support the error analysis.

• V. John, Ch. Merdon, M. Zainelabdeen, Augmenting the grad-div stabilization for Taylor--Hood finite elements with a vorticity stabilization, Journal of Numerical Mathematics, 33 (2025), pp. 37--54, DOI 10.1515/jnma-2023-0118 .
  Abstract
  The least squares vorticity stabilization (LSVS), proposed in Ahmed et al. for the Scott--Vogelius finite element discretization of the Oseen equations, is studied as an augmentation of the popular grad-div stabilized Taylor--Hood pair of spaces. An error analysis is presented which exploits the situation that the velocity spaces of Scott--Vogelius and Taylor--Hood are identical. Convection-robust error bounds are derived under the assumption that the Scott--Vogelius discretization is well posed on the considered grid. Numerical studies support the analytic results and they show that the LSVS-grad-div method might lead to notable error reductions compared with the standard grad-div method.

• F. Romor, G. Stabile, G. Rozza, Explicable hyper-reduced order models on nonlinearly approximated solution manifolds of compressible and incompressible Navier--Stokes equations, Journal of Computational Physics, 524 (2025), pp. 113729/1--113729/8, DOI 10.1016/j.jcp.2025.113729 .
  Abstract
  A slow decaying Kolmogorov n-width of the solution manifold of a parametric partial differential equation precludes the realization of efficient linear projection-based reduced-order models. This is due to the high dimensionality of the reduced space needed to approximate with sufficient accuracy the solution manifold. To solve this problem, neural networks, in the form of different architectures, have been employed to perform accurate nonlinear regressions of the solution manifolds. However, the majority of the implementations are non-intrusive black-box surrogate models and only a part of them perform dimension reduction from the number of degrees of freedom of the discretized parametric models to a latent dimension. We present a new intrusive and explicable methodology for reduced-order modeling that employs neural networks for the solution manifold approximation but that does not discard the physical and numerical models underneath in the predictive/online stage. We will focus on autoencoders used to compress further the dimensionality of linear approximants of solution manifolds, achieving in the end a nonlinear dimension reduction. After having obtained an accurate nonlinear approximant, we seek for the solutions on the latent manifold with the residual-based nonlinear least-squares Petrov--Galerkin method, opportunely hyper-reduced in order to be independent of the number of degrees of freedom. New adaptive hyper-reduction strategies are developed along with the employment of local nonlinear approximants. We test our methodology on two nonlinear time dependent parametric benchmarks involving a supersonic flow past a NACA airfoil with changing Mach number and an incompressible turbulent flow around the Ahmed body with changing slant angle.

• F. Romor, D. Torlo, G. Rozza, Friedrichs' systems discretized with the DGM: domain decomposable model order reduction and Graph Neural Networks approximating vanishing viscosity solutions, Journal of Computational Physics, 531 (2025), pp. 113915/1--113915/40, DOI 10.1016/j.jcp.2025.113915 .
  Abstract
  Friedrichs' systems (FS) are symmetric positive linear systems of first-order partial differential equations (PDEs), which provide a unified framework for describing various elliptic, parabolic and hyperbolic semi-linear PDEs such as the linearized Euler equations of gas dynamics, the equations of compressible linear elasticity and the Dirac-Klein-Gordon system. FS were studied to approximate PDEs of mixed elliptic and hyperbolic type in the same domain. For this and other reasons, the discontinuous Galerkin method (DGM) represents the most common and versatile choice of approximation space for FS in the literature. We implement a distributed memory solver for stationary FS in deal.II. Our focus is model order reduction. Since FS model hyperbolic PDEs, they often suffer from a slow Kolmogorov n-width decay. We develop and combine two approaches to tackle this problem in the context of large-scale applications. The first is domain decomposable reduced-order models (DD-ROMs). We will show that the DGM offers a natural formulation of DD-ROMs, in particular regarding interface penalties, compared to the continuous finite element method. We also develop new repartitioning strategies to obtain more efficient local approximations of the solution manifold. The second approach involves shallow graph neural networks used to infer the limit of a succession of projection-based linear ROMs corresponding to lower viscosity constants: the heuristic behind concerns the development of a multi-fidelity super-resolution paradigm to mimic the mathematical convergence to vanishing viscosity solutions while exploiting to the most interpretable and certified projection-based DD-ROMs.

• Y. Hadjimichael, Ch. Merdon, M. Liero, P. Farrell, An energy-based finite-strain model for 3D heterostructured materials and its validation by curvature analysis, International Journal for Numerical Methods in Engineering, 125 (2024), pp. e7508/1--e7508/28, DOI 10.1002/nme.7508 .
  Abstract
  This paper presents a comprehensive study of the intrinsic strain response of 3D het- erostructures arising from lattice mismatch. Combining materials with different lattice constants induces strain, leading to the bending of these heterostructures. We propose a model for nonlinear elastic heterostructures such as bimetallic beams or nanowires that takes into account local prestrain within each distinct material region. The resulting system of partial differential equations (PDEs) in Lagrangian coordinates incorporates a nonlinear strain and a linear stress-strain relationship governed by Hooke?s law. To validate our model, we apply it to bimetallic beams and hexagonal hetero-nanowires and perform numerical simulations using finite element methods (FEM). Our simulations ex- amine how these structures undergo bending under varying material compositions and cross-sectional geometries. In order to assess the fidelity of the model and the accuracy of simulations, we compare the calculated curvature with analytically derived formula- tions. We derive these analytical expressions through an energy-based approach as well as a kinetic framework, adeptly accounting for the lattice constant mismatch present at each compound material of the heterostructures. The outcomes of our study yield valuable insights into the behavior of strained bent heterostructures. This is particularly significant as the strain has the potential to influence the electronic band structure, piezoelectricity, and the dynamics of charge carriers.

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

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

• Z. Li, Y. VAN Gennip, V. John, An MBO method for modularity optimisation based on total variation and signless total variation, European Journal of Applied Mathematics, published online on 25.11.2024, DOI 10.1017/S095679252400072X .
  Abstract
  In network science, one of the significant and challenging subjects is the detection of communities. Modularity [1] is a measure of community structure that compares connectivity in the network with the expected connectivity in a graph sampled from a random null model. Its optimisation is a common approach to tackle the community detection problem. We present a new method for modularity maximisation, which is based on the observation that modularity can be expressed in terms of total variation on the graph and signless total variation on the null model. The resulting algorithm is of Merriman--Bence--Osher (MBO) type. Different from earlier methods of this type, the new method can easily accommodate different choices of the null model. Besides theoretical investigations of the method, we include in this paper numerical comparisons with other community detection methods, among which the MBO-type methods of Hu et al. [2] and Boyd et al. [3], and the Leiden algorithm [4].

• P.C. Africa, D. Arndt, W. Bangerth, B. Blais, M. Fehling, R. Gassmöller, T. Heister, L. Heltai, S. Kinnewig, M. Kronbichler, M. Maier, P. Munch, M. Schreter-Fleischhacker, J.P. Thiele, B. Turcksin, D. Wells, V. Yushutin, The deal.II library, Version 9.6, Journal of Numerical Mathematics, 32 (2024), pp. 0137/1--0137/10, DOI 10.1515/jnma-2024-0137 .
  Abstract
  This paper provides an overview of the new features of the finite element library deal.II, version 9.6.

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

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

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

• G.R. Barrenechea, V. John, P. Knobloch, Finite element methods respecting the discrete maximum principle for convection-diffusion equations, SIAM Review, 66 (2024), pp. 3--88, DOI 10.1137/22M1488934 .
  Abstract
  Convection-diffusion-reaction equations model the conservation of scalar quantities. From the analytic point of view, solutions of these equations satisfy, under certain conditions, maximum principles, which represent physical bounds of the solution. That the same bounds are respected by numerical approximations of the solution is often of utmost importance in practice. The mathematical formulation of this property, which contributes to the physical consistency of a method, is called the discrete maximum principle (DMP). In many applications, convection dominates diffusion by several orders of magnitude. It is well known that standard discretizations typically do not satisfy the DMP in this convection-dominated regime. In fact, in this case it turns out to be a challenging problem to construct discretizations that, on the one hand, respect the DMP and, on the other hand, compute accurate solutions. This paper presents a survey on finite element methods, with the main focus on the convection-dominated regime, that satisfy a local or a global DMP. The concepts of the underlying numerical analysis are discussed. The survey reveals that for the steady-state problem there are only a few discretizations, all of them nonlinear, that at the same time both satisfy the DMP and compute reasonably accurate solutions, e.g., algebraically stabilized schemes. Moreover, most of these discretizations have been developed in recent years, showing the enormous progress that has been achieved lately. Similarly, methods based on algebraic stabilization, both nonlinear and linear, are currently the only finite element methods that combine the satisfaction of the global DMP and accurate numerical results for the evolutionary equations in the convection-dominated scenario.

• D. Budáč, V. Miloš, M. Carda, M. Paidar, K. Bouzek, J. Fuhrmann, A Monte Carlo approach for simulating electrical conductivity in highly porous ceramic composites: Impact of internal structure, ACS Applied Materials and Interfaces, 16 (2024), pp. 62292--62300, DOI 10.1021/acsami.4c08287 .
  Abstract
  Porous ceramic composites play an important role in several applications. This is due to their unique properties resulting from a combination of various materials. Determination of the composite properties and structure is crucial for their further development and optimization. However, composite analysis often requires complex, expensive, and time-demanding experimental work. Mathematical modeling represents an effective tool to substitute experimental approach. The present study employs a Monte Carlo 3D equivalent electronic circuit network model developed to analyze a highly porous composite on the basis of minimum easily obtainable input parameters. Solid oxide cell electrodes were used as a model example, and this study focuses primarily on materials with a porosity of 55% and higher, characterized by deviation of behavior from those of lower void fraction share. This task is approached by adding to the original Monte Carlo model an additional parameter defining the void phase coalescence phenomenon. The enhanced model accurately simulates electrical conductivity for experimental samples of up to 75% porosity. Using sample composition, single-phase properties, and experimentally determined conductivity, this model allows us to estimate data of the internal structure of the material. This approach offers a rapid and cost-effective method to study material microstructure, providing insights into properties, such as electrical conductivity and heat conductivity. The present research thus contributes to advancing predictive capabilities in understanding and optimizing the performance of composite materials with potential in various technological applications.

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

• P.L. Lederer, Ch. Merdon, Gradient-robust hybrid DG discretizations for the compressible Stokes equations, Journal of Scientific Computing, 100 (2024), pp. 54/1--54/26, DOI 10.1007/s10915-024-02605-2 .
  Abstract
  This paper studies two hybrid discontinuous Galerkin (HDG) discretizations for the velocity-density formulation of the compressible Stokes equations with respect to several desired structural properties, namely provable convergence, the preservation of non-negativity and mass constraints for the density, and gradient-robustness. The later property dramatically enhances the accuracy in well-balanced situations, such as the hydrostatic balance where the pressure gradient balances the gravity force. One of the studied schemes employs an H (div)-conforming velocity ansatz space which ensures all mentioned properties, while a fully discontinuous method is shown to satisfy all properties but the gradient-robustness. Also higher-order schemes for both variants are presented and compared in three numerical benchmark problems. The final example shows the importance also for non-hydrostatic well-balanced states for the compressible Navier--Stokes equations.

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

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

• G. Padula, F. Romor, G. Stabile, G. Rozza, Generative models for the deformation of industrial shapes with linear geometric constraints: Model order and parameter space reductions, Computer Methods in Applied Mechanics and Engineering, 423 (2024), pp. 116823/1--116823/36, DOI 10.1016/j.cma.2024.116823 .

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

• M.U. Qureshi, S. Matera, D. Runge, Ch. Merdon, J. Fuhrmann, J.-U. Repke, G. Brösigke, Reduced order CFD modeling approach based on the asymptotic expansion -- An application for heterogeneous catalytic systems, Chemical Engineering Journal, 504 (2025), pp. 158684/1--158684/11 (appeared online on 24.12.2024), DOI 10.1016/j.cej.2024.158684 .
  Abstract
  Recent experimental techniques allow to obtain atomic scale information of heterogeneous catalysts under operando conditions, but, typically require rather complex reactor geometries. To utilize this complementary information in e.g. kinetic model development, Computational Fluid Dynamics (CFD) is needed to address the non-trivial coupling of chemical kinetics and mass transport in such chambers. However, conventional CFD approaches for solving catalytic systems have a drawback of huge computational expense, incurred by trying to solve a stiff problem. In this study, we present a reduced order approach with a significantly lower computational footprint than conventional CFD. The idea behind the approach is to estimate the solution without having to directly couple the mass transport and surface kinetics. This is achieved by a lowest-order asymptotic expansion in the catalyst sample size or, equivalently, the lateral variation of gas phase concentrations above the catalytic surface. This reduces the overall simulation time by orders of magnitude, particularly for inverse problems. We demonstrate the approach for catalytic formation of Methanol from CO2 and H2 in a two dimensional channel flow and for different applied reaction conditions, sample sizes and catalyst loadings.

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

• C. Cárcamo, A. Caiazzo, F. Galarce, J. Mura, A stabilized total pressure-formulation of the Biot's poroelasticity equations in frequency domain: Numerical analysis and applications, Computer Methods in Applied Mechanics and Engineering, 432 (2024), pp. 117353/1--117353/31, DOI 10.1016/j.cma.2024.117353 .
  Abstract
  This work focuses on the numerical solution of the dynamics of a poroelastic material in the frequency domain. We provide a detailed stability analysis based on the application of the Fredholm alternative in the continuous case, considering a total pressure formulation of the Biot's equations. In the discrete setting, we propose a stabilized equal order finite element method complemented by an additional pressure stabilization to enhance the robustness of the numerical scheme with respect to the fluid permeability. Utilizing the Fredholm alternative, we extend the well-posedness results to the discrete setting, obtaining theoretical optimal convergence for the case of linear finite elements. We present different numerical experiments to validate the proposed method. First, we consider model problems with known analytic solutions in two and three dimensions. As next, we show that the method is robust for a wide range of permeabilities, including the case of discontinuous coefficients. Lastly, we show the application for the simulation of brain elastography on a realistic brain geometry obtained from medical imaging.

• D. Frerichs-Mihov, L. Henning, V. John, On loss functionals for physics-informed neural networks for convection-dominated convection-diffusion problems, Communications on Applied Mathematics and Computation, published online on 22.05.2024, DOI 10.1007/s42967-024-00433-7 .
  Abstract
  In the convection-dominated regime, solutions of convection-diffusion problems usually possesses layers, which are regions where the solution has a steep gradient. It is well known that many classical numerical discretization techniques face difficulties when approximating the solution to these problems. In recent years, physics-informed neural networks (PINNs) for approximating the solution to (initial-)boundary value problems received a lot of interest. In this work, we study various loss functionals for PINNs that are novel in the context of PINNs and are especially designed for convection-dominated convection-diffusion problems. They are numerically compared to the vanilla and a $hp$-variational loss functional from the literature based on two benchmark problems whose solutions possess different types of layers. We observe that the best novel loss functionals reduce the $L^2(Omega)$ error by $17.3%$ for the first and $5.5%$ for the second problem compared to the methods from the literature.

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

• V. John, X. Li, Ch. Merdon, Pressure-robust $L^2 (Omega)$ error analysis for Raviart--Thomas enriched Scott--Vogelius pairs, Applied Mathematics Letters, 156 (2024), pp. 109138/1--109138/12, DOI 10.1016/j.aml.2024.109138 .
  Abstract
  Recent work shows that it is possible to enrich the Scott--Vogelius finite element pair by cer- tain Raviart--Thomas functions to obtain an inf-sup stable and divergence-free method on general shape-regular meshes. A skew-symmetric consistency term was suggested for avoiding an ad- ditional stabilization term for higher order elements, but no L² (Ω) error estimate was shown for the Stokes equations. This note closes this gap. In addition, the optimal choice of the stabilization parameter is studied numerically.

• V. John, Ch. Merdon, M. Zainelabdeen, Augmenting the grad-div stabilization for Taylor--Hood finite elements with a vorticity stabilization, Journal of Numerical Mathematics, published online on 05.11.2024, DOI 10.1515/jnma-2023-0118 .
  Abstract
  The least squares vorticity stabilization (LSVS), proposed in Ahmed et al. for the Scott--Vogelius finite element discretization of the Oseen equations, is studied as an augmentation of the popular grad-div stabilized Taylor--Hood pair of spaces. An error analysis is presented which exploits the situation that the velocity spaces of Scott--Vogelius and Taylor--Hood are identical. Convection-robust error bounds are derived under the assumption that the Scott--Vogelius discretization is well posed on the considered grid. Numerical studies support the analytic results and they show that the LSVS-grad-div method might lead to notable error reductions compared with the standard grad-div method.

• J.P. Thiele, Th. Wick, Numerical modeling and open-source implementation of variational partition-of-unity localizations of space-time dual-weighted residual estimators for parabolic problems, Journal of Scientific Computing, 99 (2024), pp. 25/1--25/40, DOI 10.1007/s10915-024-02485-6 .
  Abstract
  In this work, we consider space-time goal-oriented a posteriori error estimation for parabolic problems. Temporal and spatial discretizations are based on Galerkin finite elements of continuous and discontinuous type. The main objectives are the development and analysis of space-time estimators, in which the localization is based on a weak form employing a partition-of-unity. The resulting error indicators are used for temporal and spatial adaptivity. Our developments are substantiated with several numerical examples.

Preprints, Reports, Technical Reports

• R. Araya, C. Cárcamo, A.H. Poza, A stabilized finite element method for the Navier--Stokes/Darcy coupled problem, Preprint no. 3183, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3183 .
  Abstract, PDF (2163 kByte)
  In this work, we propose, analyze, and numerically verify a new stabilized finite element method for the Navier--Stokes/Darcy coupled problem, which models a fluid flowing through a free medium into a porous medium. At the interface between both domains, we impose mass conservation, the balance of normal forces, and the well-known Beavers--Joseph--Saffman con- ditions [9]. The stabilization terms are defined using Galerkin's least-squares stabilization for the Navier--Stokes equation and Masud--Hughes stabilization [30] for the Darcy equation. This new discrete scheme employs equal-order elements to approximate the velocity and pressure of the fluid and generalizes the scheme recently analyzed for the linear case in [4]. The well-posedness of the discrete problem is established via fixed-point theorems under small data conditions, and we prove optimal error estimates in natural norms. Finally, we present numerical examples to confirm the expected theoretical convergence orders in cases where a manufactured solution is available, as well as to demonstrate the effectiveness of our scheme in a physical model with varying permeability fields.

• M. Eigel, Ch. Merdon, A posteriori error control for stochastic Galerkin FEM with high-dimensional random parametric PDEs, Preprint no. 3174, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3174 .
  Abstract, PDF (5084 kByte)
  PDEs with random data are investigated and simulated in the field of Uncertainty Quantification (UQ), where uncertainties or (planned) variations of coefficients, forces, domains and boundary con- ditions in differential equations formally depend on random events with respect to a pre-determined probability distribution. The discretization of these PDEs typically leads to high-dimensional (determin- istic) systems, where in addition to the physical space also the (often much larger) parameter space has to be considered. A proven technique for this task is the Stochastic Galerkin Finite Element Method (SGFEM), for which a review of the state of the art is provided. Moreover, important concepts and results are summarized. A special focus lies on the a posteriori error estimation and the derivation of an adaptive algorithm that controls all discretization parameters. In addition to an explicit residual based error estimator, also an equilibration estimator with guaranteed bounds is discussed. Under cer- tain mild assumptions it can be shown that the successive refinement produced by such an adaptive algorithm leads to a sequence of approximations with guaranteed convergence to the true solution. Nu- merical examples illustrate the practical behavior for some common benchmark problems. Additionally, an adaptive algorithm for a problem with a non-affine coefficient is shown. By transforming the original PDE a convection-diffusion problem is obtained, which can be treated similarly to the standard affine case.

• O. Pártl, E. Meneses Rioseco, Efficient numerical framework for geothermal energy production optimization in fracture-controlled reservoirs, Preprint no. 3169, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3169 .
  Abstract, PDF (3112 kByte)
  We describe an open-source numerical framework for the automated search for the placements of injection and production wells in hot fracture-controlled reservoirs that sustainably optimize geothermal energy production, where we consider deviated multiwell layouts (smart multiwell arrangement). This search is carried out via 3D simulations of groundwater flow and heat transfer. We model the reservoirs as discrete fracture networks (DFN) in which the fractures are 2D manifolds with polygonal boundaries embedded in a 3D porous medium. The wells are modeled via the immersed boundary method. The flow and heat transport in the DFN-matrix system are modeled by solving the balance equations for mass, momentum, and energy. The fully developed numerical framework combines the finite element method with semi-implicit time-stepping, algebraic flux correction, and approximation of the wells via the non-matching approach. To perform the optimization, we use various gradient- free algorithms. We present the results of verification and validation tests with DFNs of simple structure and realistic physical parameter values.

• C. Cárcamo, P. Ciarlet Jr., AT-coercivity approach to the nonlinear Stokes equations, Preprint no. 3167, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3167 .
  Abstract, PDF (234 kByte)
  We address the nonlinear Stokes problem with Dirichlet boundary conditions, introducing additional variables into the standard formulation to accommodate solutions with reduced regularity requirements. To ground this analysis, we first review relevant preliminary results, emphasizing the significance of achieving T -coercivity in the context of nonlinear Stokes flows. We then introduce a specially designed operator T , proving its bijectivity and showing that it induces coercivity when applied to the test function space. This result provides a rigorous foundation for solving the quasi- Newtonian Stokes problem with minimal regularity constraint and also sets up the T -coercivity as an alternative to the well-posedness of the nonlinear Stokes problems.

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

• D. Brust, K. Hopf, J. Fuhrmann, A. Cheilytko, M. Wullenkord, Ch. Sattler, Transport of heat and mass for reactive gas mixtures in porous media: Modeling and application, Preprint no. 3139, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3139 .
  Abstract, PDF (3958 kByte)
  We present a modeling framework for multi-component, reactive gas mixtures and heat transport in porous media based on the Maxwell--Stefan and Darcy equations for multi-component diffusion and forced, viscous flow through porous media. Analysis of the model equations reveals thermodynamic con- sistency and uniqueness of steady states, while their mathematical structure facilitates discretization via the Finite-Volume approach resulting in an open- source based implementation of the modeling framework in Julia. The model allows to impose boundary conditions that accurately reflect the conditions prevailing in a photo-thermal chemical reactor that is subsequently intro- duced as a case study for the modeling framework. Comparison of numerical with experimental results reveals good agreement. Improvement options for the physical reactor are derived from simulation results demonstrating the practical utility of the modeling framework. Additionally, the framework is used for the simulation of thermodiffusion in a ternary gas mixture and has been verified with published numerical results with very good agreement.

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

Vorträge, Poster

• Z. Elsayed Amer, Numerical methods for a coupled drift-diffusion and Helmholtz model for laser applications, MESIGA25: Numerical Methods in Applied Mathematics, March 11 - 13, 2025, Institut für Mathematik der Universität Potsdam, March 12, 2025.

• A. Erhardt, Mathematical modeling and analysis of cardiac dynamics, Kolloquium des SFB 1114, Freie Unversität Berlin, April 24, 2025.

• Y. Hadjimichael, Strain distribution in zincblende and wurtzite nanowires bent by a one-sided stressor shell, Leibniz MMS Days 2025, March 26 - 28, 2025, The Leibniz Research Network "Mathematical Modeling and Simulation", Leibniz Institute for Baltic Sea Research Warnemünde (IOW), March 27, 2025.

• M. Demir, An evolve-filter-relax regularized reduced order model for the Boussinesq approximat, SIAM Conference on Computational Science and Engineering (CSE25), March 3 - 7, 2025, SIAM, Society for Industrial and Applied Mathematics, Fort Worth, Texas, USA.

• M. Zainelabdeen, Gradient-robust finite element-finite volume scheme for the compressible Stokes equations, 21st Workshop on Numerical Methods for Problems with Layer Phenomena, April 24 - 25, 2025, University of Galway, School of Mathematical and Statistical Sciences, Ireland, April 24, 2025.

• A. Caiazzo, Application of shape registration of aortic geometries to data assimilation and graph neural networks, COOMEDIA-Seminar, Computational mathematics for bio-medical applications, February 18 - 19, 2025, Inria, Sorbonne Université and CNRS, Paris, France, February 19, 2025.

• P. Farrell, Numerische Methoden für innovative Halbleiterbauteile, Transfer Workshop: ErUM-Scientists and Industry in Dialogue, February 6 - 7, 2025, ErUM Data Hub, Aachen, February 6, 2025.

• J. Fuhrmann, Finite volume based electrolyte simulations in the Julia programming language, Seminar: Theorie und computergestützte Modellierung von Materialien in der Energietechnik (IET-3), January 15 - 17, 2025, Forschungszentrum Jülich, Institute of Energy Technologies (IET), January 16, 2025.

• J. Fuhrmann, LiquidElectrolytes.jl - A generalized Poisson-Nernst-Planck solver written in Juli, ModVal 2025 - 21st Symposium on Modeling and Experimental Validation of Electrochemical Energy Technologies, Karlsruhe, March 10 - 12, 2025.

• J. Fuhrmann, The Julia language: Reproducibility infrastructure and project workflows, Leibniz MMS Days 2025, March 26 - 28, 2025, The Leibniz Research Network "Mathematical Modeling and Simulation", Leibniz Institute for Baltic Sea Research Warnemünde (IOW), March 28, 2025.

• J. Fuhrmann, Two point flux finite volume methods for mixture flows in porous media, Mixtures: Modeling, analysis and computing, February 5 - 7, 2025, Charles University, Faculty of Mathematics and Physics, Prag, Czech Republic, February 5, 2025.

• P. Jaap, WIAS-PDELib: AJulia PDE solver ecosystem in a GitHub organization, deRSE25 - 5th conference for Research Software Engineering in Germany, February 25 - 27, 2025.

• V. John, Finite element methods respecting the discrete maximum principle for convection-diffusion equations, The International Conference on Multigrid and Multiscale Methods in Computational Science (IMG) 2025, February 3 - 5, 2025, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia, February 3, 2025.

• V. John, Some experiences in using ML techniques for the numerical solution of PDEs, Leibniz MMS Days 2025, March 26 - 27, 2025, The Leibniz Research Network "Mathematical Modeling and Simulation", Leibniz Institute for Baltic Sea Research Warnemünde (IOW), March 27, 2025.

• O. Pártl, Efficient numerical framework for fracture-controlled reservoir performance optimization, Leibniz MMS Days 2025, March 26 - 28, 2025, The Leibniz Research Network "Mathematical Modeling and Simulation", Leibniz Institute for Baltic Sea Research Warnemünde (IOW), March 27, 2025.

• F. Romor, Data assimilation performed with robust shape registration and graph neural networks: Application to aortic coarctation, MESIGA25: Numerical Methods in Applied Mathematics, March 11 - 13, 2025, Institut für Mathematik der Universität Potsdam, March 12, 2025.

• F. Romor, Registration-based data assimilation from medical images, COLIBRI Focus Workshop on Computational Medicine, January 30 - 31, 2025, Universität Graz, Austria, January 30, 2025.

• F. Romor, Registration-based data assimilation from medical images, DTE & AICOMAS 2025, 3rd IACM Digital Twins an Engineering Conference (DTE 2025) & 1st ECCOMAS Artificial Intelligence and Computational Methods in Applied Science (AICOMAS 2025), February 17 - 21, 2025, Arts et Métiers - ENSAM (Paris Campus), Paris, France, February 18, 2025.

• J.P. Thiele, Easy to use tools for software quality, Leibniz MMS Days 2025, March 26 - 28, 2025, The Leibniz Research Network "Mathematical Modeling and Simulation", Leibniz Institute for Baltic Sea Research Warnemünde (IOW), March 27, 2025.

• J.P. Thiele, Software Engineering for and with Reserchers: What is required?, deRSE25 - 5th conference for Research Software Engineering in Germany, February 25 - 27, 2025, Karlsruher Institut für Technologie, February 26, 2025.

• Z. Amer, Numerical methods for coupled drift-diffusion and Helmholtz Models for laser applications, International Conference on Simulation of Organic Electronics and Photovoltaics, SimOEP, September 2 - 4, 2024, ZHAW - Zurich University of Applied Sciences, Winterthur, Switzerland, September 4, 2024.

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

• S. Katz, Simulierter Puls: Partikel in turbulentem Blutfluss, Lange Nacht der Wissenschaften, Leibniz-Gemeinschaft, Berlin, June 22, 2024.

• S. Katz, Turbulence modeling in aortic blood flow: traditional models and perspectives on machine learning, VPH (Virtual Physiological Human) Conference 2024, September 4 - 6, 2024, Universität Stuttgart, September 4, 2024.

• L. Ermoneit, B. Schmidt, J. Fuhrmann, A. Sala, N. Ciroth, L. Schreiber, T. Breiten, Th. Koprucki, M. Kantner, Optimal control of a Si/SiGe quantum bus for scalable quantum computing architectures, Quantum Optimal Control: From Mathematical Foundations to Quantum Technologies, MATH+, Berlin, May 21, 2024.

• Y. Hadjimichael, An energy-based finite-strain model for 3D heterostructured materials, Leibniz MMS Days 2024, April 10 - 12, 2024, Leibniz Network "Mathematical Modeling and Simulation", Leibniz Institut für Verbundwerkstoffe GmbH (IVW), Kaiserslautern, April 11, 2024.

• B. Schmidt, J.-P. Thiele, Code and perish?! How about publishing your software?, Leibniz MMS Days 2024, Mini Workshop, April 10 - 12, 2024, Leibniz-Institut für Verbundwerkstoffe (IVW), Kaiserslautern, April 10, 2024.

• D. Abdel, Modeling and simulation of vacancy-assisted charge transport in innovative semiconductor devices, Applied Mathematics and Simulation for Semiconductor Devices (AMaSiS 2024), September 10 - 13, 2024, WIAS Berlin, September 11, 2024.

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

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

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

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

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

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

• A. Caiazzo, Data-driven reduced-order modeling and data assimilation for the characterization of aortic coarctation, 8th International Conference on Computational and Mathematical Biomedical Engineering (CMBE24), June 24 - 26, 2024, George Mason University, Arlington, Virginia, USA, June 24, 2024.

• A. Caiazzo, Multiscale FSI for the effective modeling of vascular tissues, Virtual Physiological Human, VPH Conference 2024, September 4 - 6, 2024, Universität Stuttgart, September 4, 2024.

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

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

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

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

• P. Farrell, Charge transport in perovskites solar cells: Modeling, analysis and simulations, Inria-ECDF Partnership Kick-Off Workshop, June 5 - 7, 2024, Inria and the Einstein Center for Digital Future, Berlin, June 7, 2024.

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

• J. Fuhrmann, S. Maass, S. Ringe, Monolithic coupling of a CatMAP based microkinetic model for heterogeneous electrocatalysis and ion transport with finite ion sizes, ModVal 2024 -- 20th Symposium on Modeling and Validation of Electrochemical Energy Technologies, Villigen, Switzerland, March 13 - 14, 2024.

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

• J. Fuhrmann, J.P. Thiele, Erfahrungen mit Softwarelizenzierung und Transfer am WIAS, Leibniz Open Transfer Workshop, Leibniz Gemeinschaft, June 5, 2024.

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

• J. Fuhrmann, Finite volume simulations of drift-diffusion problems in semiconductors and electrolytes, Seminar, Laboratoire de Génie des Procédés et Matériaux, Chaire de Biotechnologie de CentraleSupélec, Pomacle, France, September 19, 2024.

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

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

• Y. Hadjimichael, Strain distribution in zincblende and wurtzite GaAs nanowires bent by a one-sided (In,Al)As shell, Applied Mathematics and Simulation for Semiconductor Devices (AMaSiS 2024), Berlin, September 10 - 13, 2024.

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

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

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

• V. John, On two modeling issues in aortic blood flow simulations, Seminar of Dr. Nagaiah Chamakuri, Scientific Computing Group (SCG), School of Mathematics, Indian Institute of Science Education and Research, Thiruvananthapuram, Kerala, India, February 20, 2024.

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

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

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

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

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

• CH. Merdon, Pressure-robustness in Navier--Stokes finite elements simulations, Wissenschaftlicher Beirat, WIAS Berlin, September 27, 2024.

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

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

• O. Pártl, Optimization of geothermal energy production from fracture-controlled reservoirs via 3D numerical modeling and simulation, General Assembly 2024 of the European Geosciences Union (EGU), April 14 - 19, 2024, European Geosciences Union (EGU), Wien, Austria, April 15, 2024, DOI 10.5194/egusphere-egu24-4164 .

• F. Romor, Efficient numerical resolution of parametric partial differential equations on solution manifolds parametrized by neural networks, 9th European Congress on Computational Methods in Applied Sciences and Engineering, June 3 - 7, 2024, ECCOMAS, scientific organization, Lissabon, Portugal, June 4, 2024.

• F. Romor, Registration-based data assimilation of aortic blood flow, Leibniz MMS Days 2024, Kaiserslautern, April 10 - 12, 2024.

• F. Romor, Registration-based data assimilation of aortic blood flow, Leibniz MMS Days 2024, April 10 - 12, 2024, Leibniz Network "Mathematical Modeling and Simulation", Leibniz Institut für Verbundwerkstoffe GmbH (IVW), Kaiserslautern.

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

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

• H. Stephan, Mathematik und Erkenntnistheorie I, Mathematik und Erkenntnistheorie II, Lange Nacht der Wissenschaften, Leibniz-Gemeinschaft, Berlin, June 22, 2024.

• H. Stephan, Zahlentheorie, Geometrie und Physik, Tag der Mathematik, Technische Universität Berlin, May 4, 2024.

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

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

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

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

Preprints im Fremdverlag

• D. Abdel, M. Herda, M. Ziegler, C. Chainais-Hillairet, B. Spetzler, P. Farrell, Numerical analysis and simulation of lateral memristive devices: Schottky, ohmic, and multi-dimensional electrode models, Preprint no. 15065, Cornell University, 2024, DOI 10.48550/arXiv.2412.15065 .
  Abstract
  In this paper, we present the numerical analysis and simulations of a multi-dimensional memristive device model. Memristive devices and memtransistors based on two-dimensional (2D) materials have demonstrated promising potential as components for next-generation artificial intelligence (AI) hardware and information technology. Our charge transport model describes the drift-diffusion of electrons, holes, and ionic defects self-consistently in an electric field. We incorporate two types of boundary models: ohmic and Schottky contacts. The coupled drift-diffusion partial differential equations are discretized using a physics-preserving Voronoi finite volume method. It relies on an implicit time-stepping scheme and the excess chemical potential flux approximation. We demonstrate that the fully discrete nonlinear scheme is unconditionally stable, preserving the free-energy structure of the continuous system and ensuring the non-negativity of carrier densities. Novel discrete entropy-dissipation inequalities for both boundary condition types in multiple dimensions allow us to prove the existence of discrete solutions. We perform multi-dimensional simulations to understand the impact of electrode configurations and device geometries, focusing on the hysteresis behavior in lateral 2D memristive devices. Three electrode configurations - side, top, and mixed contacts - are compared numerically for different geometries and boundary conditions. These simulations reveal the conditions under which a simplified one-dimensional electrode geometry can well represent the three electrode configurations. This work lays the foundations for developing accurate, efficient simulation tools for 2D memristive devices and memtransistors, offering tools and guidelines for their design and optimization in future applications.

• P. Jaap, O. Sander, How to project onto SL(n), Preprint no. 19310, Cornell University, 2025, DOI 10.48550/arXiv.2501.19310 .
  Abstract
  We consider the closest-point projection with respect to the Frobe- nius norm of a general real square matrix to the set SL (n) of matrices with unit determinant. As it turns out, it is sufficient to consider diagonal matrices only. We investigate the structure of the problem both in Euclidean coordinates and in an n-dimensional generalization of the classical hyperbolic coordinates of the positive quadrant. Using symmetry arguments we show that the global minimizer is contained in a particular cone. Based on different views of the problem, we propose four different iterative algorithms, and we give convergence results for all of them. Numerical tests show that computing the projection costs essentially as much as a singular value decomposition. Finally, we give an explicit formula for the first derivative of the projection.

• F. Romor, F. Galarce, J. Brüning, L. Goubergrits, A. Caiazzo, Data assimilation performed with robust shape registration and graph neural networks: Application to aortic coarctation, Preprint no. 2502.12097, Cornell University, 2025, DOI 10.48550/arXiv.2502.12097 .
  Abstract
  Image-based, patient-specific modelling of hemodynamics can improve diagnostic capabilities and pro- vide complementary insights to better understand the hemodynamic treatment outcomes. However, com- putational fluid dynamics simulations remain relatively costly in a clinical context. Moreover, projection- based reduced-order models and purely data-driven surrogate models struggle due to the high variability of anatomical shapes in a population. A possible solution is shape registration: a reference template geometry is designed from a cohort of available geometries, which can then be diffeomorphically mapped onto it. This provides a natural encoding that can be exploited by machine learning architectures and, at the same time, a reference computational domain in which efficient dimension-reduction strategies can be performed. We compare state-of-the-art graph neural network models with recent data assimilation strategies for the prediction of physical quantities and clinically relevant biomarkers in the context of aortic coarctation.

• C.K. Macnamara, I. Ramis-Conde, T. Lorenzi, A. Caiazzo, An agent-based modelling framework for tumour growth incorporating mechanical and evolutionary aspects of cell dynamics, Preprint no. bioRxiv:596685, Cold Spring Harbor Laboratory, 2024, DOI 10.1101/2024.05.30.596685 .
  Abstract
  We develop an agent-based modelling framework for tumour growth that in-corporates both mechanical and evolutionary aspects of the spatio-temporal dynamics of cancer cells. In this framework, cells are regarded as viscoelastic spheres that interact with other neighbouring cells through mechanical forces. The phenotypic state of each cell is described by the level of expression of an hypoxia-inducible factor that regulates the cellular response to available oxygen. The rules that govern proliferation and death of cells in different phenotypic states are then defined by integrating mechanical constraints and evolutionary principles. Computational simulations of the model are carried out under a variety of scenarios corresponding to different intra-tumoural distributions of oxygen. The results obtained, which indicate excellent agreement between simulation outputs and the results of formal analysis of phenotypic selection, recapitulate the emergence of stable phenotypic heterogeneity among cancer cells driven by inhomogeneities in the intra-tumoural distribution of oxygen. This article is intended to present a proof of concept for the ideas underlying the proposed modelling framework, with the aim to apply the related modelling methods to elucidate specific aspects of cancer progression in the future.

• P.C. Africa, D. Arndt, W. Bangerth, B. Blais, M. Fehling, R. Gassmöller, T. Heister, L. Heltai, S. Kinnewig, M. Kronbichler, M. Maier, P. Munch, M. Schreter-Fleischhacker, J.P. Thiele, B. Turcksin, D. Wells, V. Yushutin, The deal.II Library, Version 9.6, Report, hrefhttps://www.dealii.org/https://www.dealii.org/, 2024.

• G. Alì, P. Farrell, N. Rotundo, Forward lateral photovoltage scanning problem: Perturbation approach and existence-uniqueness analysis, Preprint no. arXiv:2404.10466, Cornell University, 2024, DOI 10.48550/arXiv.2404.10466 .
  Abstract
  In this paper, we present analytical results for the so-called forward lateral photovoltage scanning (LPS) problem. The (inverse) LPS model predicts doping variations in crystal by measuring the current leaving the crystal generated by a laser at various positions. The forward model consists of a set of nonlinear elliptic equations coupled with a measuring device modeled by a resistance. Standard methods to ensure the existence and uniqueness of the forward model cannot be used in a straightforward manner due to the presence of an additional generation term modeling the effect of the laser on the crystal. Hence, we scale the original forward LPS problem and employ a perturbation approach to derive the leading order system and the correction up to the second order in an appropriate small parameter. While these simplifications pose no issues from a physical standpoint, they enable us to demonstrate the analytic existence and uniqueness of solutions for the simplified system using standard arguments from elliptic theory adapted to the coupling with the measuring device.

• TH. Anandh, D. Ghose, H. Jain, P. Sunkad, S. Ganesan, V. John, Improving hp-variational physics-informed neural networks for steady-state convection-dominated problems, Preprint no. arXiv:2411.09329, Cornell University, 2024, DOI 10.48550/arXiv.2411.09329 .
  Abstract
  This paper proposes and studies two extensions of applying hp-variational physics-informed neural networks, more precisely the FastVPINNs framework, to convection-dominated convection-diffusion-reaction problems. First, a term in the spirit of a SUPG stabilization is included in the loss functional and a network architecture is proposed that predicts spatially varying stabilization parameters. Having observed that the selection of the indicator function in hard-constrained Dirichlet boundary conditions has a big impact on the accuracy of the computed solutions, the second novelty is the proposal of a network architecture that learns good parameters for a class of indicator functions. Numerical studies show that both proposals lead to noticeably more accurate results than approaches that can be found in the literature.

• R. Araya, A. Caiazzo, F. Chouly, Stokes problem with slip boundary conditions using stabilized finite elements combined with Nitsche, Preprint no. arXiv:2404.08810, Cornell University, 2024, DOI 10.48550/arXiv.2404.08810 .
  Abstract
  We discuss how slip conditions for the Stokes equation can be handled using Nitsche method, for a stabilized finite element discretization. Emphasis is made on the interplay between stabilization and Nitsche terms. Well-posedness of the discrete problem and optimal convergence rates, in natural norm for the velocity and the pressure, are established, and illustrated with various numerical experiments. The proposed method fits naturally in the context of a finite element implementation while being accurate, and allows an increased flexibility in the choice of the finite element pairs.

• TH. Belin, P. Lafitte-Godillon, V. Lescarret, J. Fuhrmann, C. Mascia, hrefhttps://hal.science/hal-04647129/file/main_article_HAL.pdfEntropy solutions of a diffusion equation with discontinuous hysteresis and their finite volume approximation, Preprint no. hal-04647129, HAL (Hyper Articles en Ligne) open science, National Center for Scientific Research (CNRS), Inria, and INRAE, 2024.
  Abstract
  We provide a finite volume approximation in dimension d>1 to a quasilinear parabolic equation with discontinuous hysteresis modelling a phase change, arising as a singluar limit of a pseudo-parabolic regularisation of a foward-backward diffusion equation. The convergence of the numerical solution to a suitable weak entropy solutions is shown under a parallelism assumption between the nonlinearities driving the evolution in each phase. The main challenge lies in the treatment of the discontinuous hysteresis operator in the proof of the compactness of the sequence of approximate solutions. This is achieved by regularising the hysteresis operator with a continuous one for which Hilpert inequalities are accessible and let us obtain crucial uniform translation estimates in L1 in space. Numerical simulations, computed using a Julia-based framework for the finite volume discretisation of reaction-diffusion equations, are shown.

• J.P. Thiele, ideal.II: A Galerkin space-time extension to the finite element library deal.II, Preprint no. arXiv:2408.08840, Cornell University, 2024, DOI 10.48550/arXiv.2408.08840 .
  Abstract
  The C++ library deal.II provides classes and functions to solve stationary problems with finite elements on one- to threedimensional domains. It also supports the typical way to solve time-dependent problems using time-stepping schemes, either with an implementation by hand or through the use of external libraries like SUNDIALS. A different approach is the usage of finite elements in time as well, which results in space-time finite element schemes. The library ideal.II (short for instationary deal.II) aims to extend deal.II to simplify implementations of the second approach.