Numerische Mathematik und Wissenschaftliches Rechnen

Publikationen

Monografien

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

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

Artikel in Referierten Journalen

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

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

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

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

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

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

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

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

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

• 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. 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.

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

• 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.

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

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

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

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

• F. Galarce Marín, K. Tabelow, J. Polzehl, Ch.P. Papanikas, V. Vavourakis, L. Lilaj, I. Sack, A. Caiazzo, Displacement and pressure reconstruction from magnetic resonance elastography images: Application to an in silico brain model, SIAM Journal on Imaging Sciences, 16 (2023), pp. 996--1027, DOI 10.1137/22M149363X .
  Abstract
  This paper investigates a data assimilation approach for non-invasive quantification of intracranial pressure from partial displacement data, acquired through magnetic resonance elastography. Data assimilation is based on a parametrized-background data weak methodology, in which the state of the physical system tissue displacements and pressure fields is reconstructed from partially available data assuming an underlying poroelastic biomechanics model. For this purpose, a physics-informed manifold is built by sampling the space of parameters describing the tissue model close to their physiological ranges, to simulate the corresponding poroelastic problem, and compute a reduced basis. Displacements and pressure reconstruction is sought in a reduced space after solving a minimization problem that encompasses both the structure of the reduced-order model and the available measurements. The proposed pipeline is validated using synthetic data obtained after simulating the poroelastic mechanics on a physiological brain. The numerical experiments demonstrate that the framework can exhibit accurate joint reconstructions of both displacement and pressure fields. The methodology can be formulated for an arbitrary resolution of available displacement data from pertinent images. It can also inherently handle uncertainty on the physical parameters of the mechanical model by enlarging the physics-informed manifold accordingly. Moreover, the framework can be used to characterize, in silico, biomarkers for pathological conditions, by appropriately training the reduced-order model. A first application for the estimation of ventricular pressure as an indicator of abnormal intracranial pressure is shown in this contribution.

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

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

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

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

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

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

• 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, published online on 26.08.2023, 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.

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

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

• P. Ral, A.K. Giri, V. John, Instantaneous gelation and non-existence of weak solutions for the Oort--Hulst--Safronov coagulation model, Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences, 479 (2023), pp. 20220385/1--20220385/13, DOI 10.1098/rspa.2022.0385 .
  Abstract
  The possible occurrence of instantaneous gelation to the Oort--Hulst--Safronov (OHS) coagulation equation is investigated for a certain class of unbounded coagulation kernels. The existence of instantaneous gelation is confirmed by showing the non-existence of mass-conserving weak solutions. Finally, it is shown that for such kernels, there is no weak solution to the OHS coagulation equation at any time interval.

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

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

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

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

• O. Pártl, U. Wilbrandt, J. Mura, A. Caiazzo, Reconstruction of flow domain boundaries from velocity data via multi-step optimization of distributed resistance, Computers & Mathematics with Applications. An International Journal, 129 (2023), pp. 11--33, DOI 10.1016/j.camwa.2022.11.006 .
  Abstract
  We reconstruct the unknown shape of a flow domain using partially available internal velocity measurements. This inverse problem is motivated by applications in cardiovascular imaging where motion-sensitive protocols, such as phase-contrast MRI, can be used to recover three-dimensional velocity fields inside blood vessels. In this context, the information about the domain shape serves to quantify the severity of pathological conditions, such as vessel obstructions. We consider a flow modeled by a linear Brinkman problem with a fictitious resistance accounting for the presence of additional boundaries. To reconstruct these boundaries, we employ a multi-step gradient-based variational method to compute a resistance that minimizes the difference between the computed flow velocity and the available data. Afterward, we apply different post-processing steps to reconstruct the shape of the internal boundaries. To limit the overall computational cost, we use a stabilized equal-order finite element method. We prove the stability and the well-posedness of the considered optimization problem. We validate our method on three-dimensional examples based on synthetic velocity data and using realistic geometries obtained from cardiovascular imaging.

Beiträge zu Sammelwerken

• L. Ermoneit, B. Schmidt, J. Fuhrmann, Th. Koprucki, L.R. Schreiber, M. Kantner, Simulation of single-electron shuttling for spin-qubit transport in a SiGe quantum bus, in: Book of Abstracts of the International Workshop on Computational Nanotechnology 2023 (IWCN 2023), X. Orios Plaedvall, G. Abadal Berini, X. Cartoixà Soler, A. Cummings, C.F. Destefani, D. Jiménez Jiménez, J. Mart'in Mart'inez, R. Rodr'iguez Mart'inez, A. Benali, eds., pp. 88-89.

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

• R. Finn, M. O'Donovan, P. Farrell, T. Streckenbach, J. Moatti, Th. Koprucki, S. Schulz, Theoretical investigation of carrier transport and recombination processes for deep UV (Al,Ga)N light emitters, in: 23nd International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2023), P. Bardella, A. Tibaldi, eds., IEEE, 2023, pp. 83--84, DOI 10.1109/NUSOD59562.2023.10273485 .
  Abstract
  We present a theoretical study on the impact of alloy disorder on carrier transport and recombination rates in an (Al,Ga)N single quantum well based LED operating in the deep UV spectral range. Our calculations indicate that alloy fluctuations enable percolative pathways which can result in improved carrier injection into the well, but may also increase carrier leakage from the well. Additionally, we find that alloy disorder induces carrier localization effects, a feature particularly noticeable for the holes. These localization effects can lead to locally increased carrier densities when compared to a virtual crystal approximation which neglects alloy disorder. We observe that both radiative and non-radiative recombination rates are increased. Our calculations also indicate that Auger--Meitner recombination increases faster than the radiative rate, based on a comparison with a virtual crystal approximation.

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

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

Preprints, Reports, Technical Reports

• 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.

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

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

• D. Frerichs-Mihov, M. Zainelabdeen, V. John, On collocation points for physics-informed neural networks applied to convection-dominated convection-diffusion problems, Preprint no. 3074, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3074 .
  Abstract, PDF (623 kByte)
  In recent years physics-informed neural networks (PINNs) for approximating the solution to (initial-)boundary value problems gained a lot of interest. PINNs are trained to minimize several residuals of the problem in collocation points. In this work we tackle convection-dominated convection-diffusion problems, whose solutions usually possess layers, which are small regions where the solution has a steep gradient. Inspired by classical Shishkin meshes, we compare hard- constrained PINNs trained with layer-adapted collocation points with ones trained with equispaced and uniformly randomly chosen points. We observe that layer-adapted points work the best for a problem with an interior layer and the worst for a problem with boundary layers. For both problems at most acceptable solutions can be obtained with PINNs.

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

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

• V. John, Ch. Merdon, M. Zainelabdeen, Augmenting the grad-div stabilization for Taylor--Hood finite elements with a vorticity stabilization, Preprint no. 3055, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3055 .
  Abstract, PDF (2139 kByte)
  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.

Vorträge, Poster

• 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.

• 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.

• 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, M. Kantner, Th. Koprucki, J. Fuhrmann, B. Schmidt, Optimal control of a Si/SiGe quantum bus for scalable quantum computing architectures, QUANTUM OPTIMAL CONTROL From Mathematical Foundations to Quantum Technologies, Berlin, May 21, 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, Multiscale FSI for the effective modeling of vascular tissues, VPH (Virtual Physiological Human) 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, Robert-Koch-Forum, Wilhelmstraße 67, Berlin, June 7, 2024.

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

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

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

• J. Fuhrmann, 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, High Tatra Mountains, Podbanske, Slovakia, March 19, 2024.

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

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

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

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

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

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

• 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.

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

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

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

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

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

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

• S. Katz, Impact of turbulence modeling on the full and reduced simulations of aortic blood flow, 22nd Computational Fluids Conference (CFC 2023), April 25 - 28, 2023, International Association for Computational Mechanics (IACM), Cannes, France, April 28, 2023.

• L. Ermoneit, B. Schmidt, J. Fuhrmann, Th. Koprucki, M. Kantner, Coherent spin-qubit shuttling in a SiGe quantum bus: Device-scale modeling, simulation and optimal control, Leibniz MMS Days 2023, Potsdam, April 17 - 19, 2023.

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

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

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

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

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

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

• C. Cárcamo Sanchez, F. Galarce Marín, A. Caiazzo, I. Sack, K. Tabelow, Quantitative tissue pressure imaging via PDE-informed assimilation of MR-data, MATH+ Day, Humboldt-Universität zu Berlin, October 20, 2023.

• P. Farrell, Charge transport in Perovskites devices: modeling, numerical analysis and simulations, Workshop on Applied Mathematics: Quantum and Classical Models, Università degli Studi di Firenze, Dipartimento di Matematica e Informatica 'Ulisse Dini', Italy, November 29, 2023.

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

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

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

• D. Frerichs-Mihov, On deep learning techniques for solving convection-dominated convection-diffusion equations, 10th International Congress on Industrial and Applied Mathematics (ICIAM), Minisymposium 00747 ''Analysis and Numerics on Deep Learning Based Methods for Solving PDEs'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 23, 2023.

• D. Frerichs-Mihov, Using deep learning techniques for solving convection-dominated convection-diffusion equations, 22nd Computational Fluids Conference (CFC 2023), April 25 - 28, 2023, International Association for Computational Mechanics (IACM), Cannes, France, April 28, 2023.

• 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, Humboldt-Universität zu Berlin, October 20, 2023.

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

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

• J. Fuhrmann, VORONOIFVM.JL -- A multiphysics finite volume solver for elliptic and parabolic systems, SIAM Conference on Computational Science and Engineering (CSE23), Minisymposium MS67 ``Research Software Engineering with Julia'', February 26 - March 3, 2023, Amsterdam, Netherlands, February 27, 2023.

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

• V. John, A SUPG-stabilized POD-ROM method for convection-diffusion-reaction problems (online talk), Numerical Analysis of Galerkin ROMs seminar series (Online Event), February 28, 2023.

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

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

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

• M. Kantner, L. Ermoneit, B. Schmidt, J. Fuhrmann, A. Sala, L.R. Schreiber, Th. Koprucki, Optimal control of a SiGe-quantum bus for coherent electron shuttling in scalable quantum computing architectures, Silicon Quantum Electronics Workshop 2023, Kyoto, Japan, October 31 - November 2, 2023.

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

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

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

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

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

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

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

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

• J.P. Thiele, Competencies and responsibilities of an RSE and how to acquire them (in Germany), FG RSE 2023: Fachgruppentreffen, Gesellschaft für Informatik, October 10 - 11, 2023, Leibniz Universität Hannover, October 10, 2023.

Preprints im Fremdverlag

• 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, https://www.dealii.org/, 2024.

• G. Alì, P. Farrell, N. Rotundo, Forward lateral photovoltage scanning problem: Perturbation approach and existence-uniqueness analysis, Preprint no. 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. 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.

• J.P. Thiele, ideal.II: a Galerkin space-time extension to the finite element library deal.II, Preprint no. 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.

• R. Araya, A. Caiazzo, F. Chouly, Stokes problem with slip boundary conditions using stabilized finite elements combined with Nitsche, Preprint no. hal-04077986, Hyper Articles en Ligne (HAL), 2023.
  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 are established, and illustrated with various numerical experiments.

• R. Finn, M. O'Donovan, P. Farrell, J. Moatti, T. Streckenbach, Th. Koprucki, S. Schulz, Theoretical study of the impact of alloy disorder on carrier transport and recombination processes in deep UV (Al, Ga)N light emitters, Preprint no. hal-04037215, Hyper Articles en Ligne (HAL), 2023.
  Abstract
  Aluminium gallium nitride ((Al,Ga)N) has gained significant attention in recent years due to its potential for highly efficient light emitters operating in the deep ultra-violet (UV) range (< 280 nm). However, given that current devices exhibit extremely low efficiencies, understanding the fundamental properties of (Al,Ga)N-based systems is of key importance. Here, using a multi-scale simulation framework, we study the impact of alloy disorder on carrier transport, radiative and non-radiative recombination processes in a c-plane Al0.7Ga0.3N/Al0.8Ga0.2N quantum well embedded in a p-i-n junction. Our calculations reveal that alloy fluctuations can open "percolative" pathways that promote transport for the electrons and holes into the quantum well region. Such an effect is neglected in conventional, and widely used transport simulations. Moreover, we find also that the resulting increased carrier density and alloy induced carrier localization effects significantly increase non-radiative Auger-Meitner recombination in comparison to the radiative process. Thus, to avoid such non-radiative process and potentially related material degradation, a careful design (wider well, multi quantum wells) of the active region is required to improve the efficiency of deep UV light emitters.

• F. Galarce, J. Mura, A. Caiazzo, Bias and multiscale correction methods for variational state estimation algorithms, Preprint no. arXiv:2311.14031, Cornell University, 2023, DOI 10.48550/arXiv.2311.14031 .
  Abstract
  The integration of experimental data into mathematical and computational models is crucial for enhancing their predictive power in real-world scenarios. However, the performance of data assimilation algorithms can be significantly degraded when measurements are corrupted by biased noise, altering the signal magnitude, or when the system dynamics lack smoothness, such as in the presence of fast oscillations or discontinuities. This paper focuses on variational state estimation using the so-called 'Parameterized Background Data Weak' method, which relies on a parameterized background by a set of constraints, enabling state estimation by solving a minimization problem on a reduced-order background model, subject to constraints imposed by the input measurements. To address biased noise in observations, a modified formulation is proposed, incorporating a correction mechanism to handle rapid oscillations by treating them as slow-decaying modes based on a two-scale splitting of the classical reconstruction algorithm. The effectiveness of the proposed algorithms is demonstrated through various examples, including discontinuous signals and simulated Doppler ultrasound data.

• 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, Preprint no. arXiv:2304.08313, Cornell University, 2023, DOI 10.48550/arXiv.2304.08313 .
  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.

• F. Goth, R. Alves, M. Braun, L.J. Castro, G. Chourdakis, S. Christ, J. Cohen, F. Erxleben, J.-N. Grad, M. Hagdorn, T. Hodges, G. Juckeland, D. Kempf, A.-L. Lamprecht, J. Linxweiler, M. Schwarzmeier, H. Seibold, J.P. Thiele, H. VON Waldow, S. Wittke, Foundational competencies and responsibilities of a research software engineer, Preprint no. arXiv:2311.11457, Cornell University, 2023, DOI 10.48550/arXiv.2311.11457 .
  Abstract
  The term Research Software Engineer, or RSE, emerged a little over 10 years ago as a way to represent individuals working in the research community but focusing on software development. The term has been widely adopted and there are a number of high-level definitions of what an RSE is. However, the roles of RSEs vary depending on the institutional context they work in. At one end of the spectrum, RSE roles may look similar to a traditional research role. At the other extreme, they resemble that of a software engineer in industry. Most RSE roles inhabit the space between these two extremes. Therefore, providing a straightforward, comprehensive definition of what an RSE does and what experience, skills and competencies are required to become one is challenging. In this community paper we define the broad notion of what an RSE is, explore the different types of work they undertake, and define a list of fundamental competencies as well as values that define the general profile of an RSE. On this basis, we elaborate on the progression of these skills along different dimensions, looking at specific types of RSE roles, proposing recommendations for organisations, and giving examples of future specialisations. An appendix details how existing curricula fit into this framework.

• P.L. Lederer, Ch. Merdon, Gradient-robust hybrid DG discretizations for the compressible Stokes equations, Preprint no. arXiv.2311.06098, Cornell University, 2023, DOI 10.48550/arXiv.2311.06098 .
  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.

• B. Spetzler, D. Abdel, F. Schwierz, M. Ziegler, P. Farrell, The role of mobile point defects in two-dimensional memristive devices, Preprint no. arXiv:2304.06527, Cornell University, 2023, DOI 10.48550/arXiv.2304.06527 .
  Abstract
  Two-dimensional (2D) layered transition metal dichalcogenides (TMDCs) are promising memristive materials for neuromorphic computing systems as they could solve the problem of the excessively high energy consumption of conventional von Neumann computer architectures. Despite extensive experimental work, the underlying switching mechanisms are still not understood, impeding progress in material and device functionality. This study reveals the dominant role of mobile defects in the switching dynamics of 2D TMDC materials. The switching process is governed by the formation and annihilation dynamics of a local vacancy depletion zone. Moreover, minor changes in the interface potential barriers cause fundamentally different device behavior previously thought to originate from multiple mechanisms. The key mechanisms are identified with a charge transport model for electrons, holes, and ionic point defects, including image-charge-induced Schottky barrier lowering (SBL). The model is validated by comparing simulations to measurements for various 2D MoS2-based devices, strongly corroborating the relevance of vacancies in TMDC devices and offering a new perspective on the switching mechanisms. The insights gained from this study can be used to extend the functional behavior of 2D TMDC memristive devices in future neuromorphic computing applications.

• P. Farrell, J. Moatti, M. O'Donovan, S. Schulz, Th. Koprucki, Importance of satisfying thermodynamic consistency in light emitting diode simulations, Preprint no. hal-04012467, Hyper Articles en Ligne (HAL), 2023.
  Abstract
  We show the importance of using a thermodynamically consistent flux discretization when describing drift-diffusion processes within light emitting diode simulations. Using the classical Scharfetter-Gummel scheme with Fermi-Dirac statistics is an example of such an inconsistent scheme. In this case, for an (In,Ga)N multi quantum well device, the Fermi levels show steep gradients on one side of the quantum wells which are not to be expected. This result originates from neglecting diffusion enhancement associated with Fermi-Dirac statistics in the numerical flux approximation. For a thermodynamically consistent scheme, such as the SEDAN scheme, the spikes in the Fermi levels disappear. We will show that thermodynamic inconsistency has far reaching implications on the current-voltage curves and recombination rates.