### Doktorandenseminar des WIAS

## FG3: Numerische Mathematik und Wissenschaftliches
Rechnen /

*RG3: Numerical Mathematics
and Scientific Computing*/

LG5: Numerik für innovative
Halbleiter-Bauteile

*LG5: Numerics for
innovative semiconductor devices*

## Seminar Numerische Mathematik / *Numerical mathematics seminars*

aktuelles Programm / *current program*

_{Archiv}

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

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

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Donnerstag, 19. 01. 2023, 13:30 Uhr (WIAS-ESH/Hybrid Event)
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Michele Pütz(BTU Cottbus-Senftenberg)

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

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

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Dienstag, 13. 12. 2022, 13:30 Uhr (WIAS-ESH/Hybrid Event)
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Xu Li(Shandong University)

Inf-sup stabilized Scott--Vogelius elements for incompressible flowsThis talk is concerned about the finite element methods for incompressible flows such as Stokes problem and Navier-Stokes problem. When designing the schemes for incompressible flows, one should consider the continuity requirement, inf-sup stability and divergence constraint simultaneously, which is a challenging problem. Traditionally, most of the finite element methods do not satisfy the divergence constraint exactly. One exception is the Scott-Vogelius elements, i.e., P_k-P_k-1 pairs, which are stable on some special meshes such as barycentric refined meshes. Here we introduce a new class of finite elements which stabilizes the Scott-Vogelius elements by enriching the velocity space with some Raviart-Thomas functions on general mesh, while maintaining the divergence-free property. Starting with the Stokes equations, we show the inf-sup stability, the convergence analysis and the pressure-robustness property of the newly proposed element. Also a reduced version of the method, an equivalent P_k-P_0 discretization, is discussed. Finally we extend this element to the Navier-Stokes problem, where the convection-robustness property and the discretization of the nonlinear term is also of interest and importance.

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

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Donnerstag, 08. 09. 2022, 14:00 Uhr (WIAS-ESH/Hybrid Event)
**

Dr. Yiannis Hadjimichael(WIAS Berlin)

Implicit strong-stability-preserving Runge--Kutta methods with downwind-biased operatorsStrong stability preserving (SSP) time integrators have been developed to preserve certain nonlinear stability properties (e.g., positivity, monotonicity, boundedness) of the numerical solution in arbitrary norms, when coupled with suitable spatial discretizations. The existing general linear methods (including Runge-Kutta and linear multistep methods) either attain small time steps for strong stability preservation or are only first-order accurate. One way to relax the time-step restrictions is to consider time integrators that contain both upwind- and downwind-biased operators.

In this talk, we review SSP Runge-Kutta methods that use upwind- and downwind-biased discretizations in the framework of perturbations of Runge-Kutta methods. We show how downwinding improves the SSP properties of time-stepping methods and breaks some stability barriers. In particular, we focus on implicit downwind SSP Runge-Kutta methods whose SSP coefficient can vary with respect to the method's coefficients. We present a novel one-parameter family of third-order, three-stage perturbed Runge-Kutta methods, for which the CFL-like time-step restriction can be arbitrarily large. The stability of this family of methods is analyzed, and we demonstrate that such methods are accurate enough when large CFL numbers are used. Furthermore, we discuss the complexity of solving the nonlinear problem at each step and we propose a block factorization of the Jacobian that reduces the computational cost of Newton's method.

For zoom login details please contact Ulrich Wilbrandt ulrich.wilbrandt@wias-berlin.de

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Donnerstag, 21. 07. 2022, 14:00 Uhr (Online Event)
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Prof. Gabriel Acosta(University of Buenos Aires, Argentinien)

Nonlocal models related problemsabstract

For zoom login details please contact ulrich.wilbrandt@wias-berlin.de

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Dienstag, 12. 07. 2022, 13:30 Uhr (WIAS-ESH)
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Dr. Zahra Lakdawala(Lahore University of Management Sciences, Pakistan)

On water wave dynamics using physics informed neural networksA vanilla feed forward neural network consists of neurons and layers and the mapping between input and output is approximated using a non-linear function. The network is trained using data. In this work, we set up a neural network such that the spatio-temporal solution of time-dependent wave propagation models is learnt. This is done by providing the physical model, such as the 1D wave and shallow water wave equations and its associated boundary and initial conditions, as rules for learning the network. We investigate the feasibility of data-driven and model-driven network predictions against numerical solutions. We further investigate the accuracy of the trained network through different parameter configurations and look closely into the multi-objective loss function that is constructed by including the residual error of the physical equations and the associated initial and boundary conditions. We present the results of numerical solutions against solutions obtained from data and model-driven neural networks using data and physics informed rules for learning. Lastly, we construct a hybrid data and physics driven network and show that this significantly improves the accuracy of the physics-driven network.

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Montag, 20. 06. 2022, 16:00 Uhr (WIAS-ESH)
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Prof. Maxim A. Olshanskii(University of Houston, USA)

Numerical analysis of surface fluidsIn this talk we focus on numerical analysis for systems of PDEs governing the motion of material viscous surfaces, the topic motivated by continuum-based modeling of lateral organization in plasma membranes. We shall consider several systems of fluid and phase-field equations defined on evolving surfaces and discuss some recent results about well-posedness of such problems. We further introduce a computational approach and numerical analysis for the resulting systems of PDEs. The methods are combined to deliver a computationally tractable and thermodynamically consistent model describing the dynamics of a two-phase viscous layer. The talk closes with an illustration of the model capacity to predict lateral ordering in multicomponent vesicles of different lipid compositions.

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Donnerstag, 02. 06. 2022, 14:00 Uhr (WIAS-ESH)
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Julien Moatti(Université de Lille, Frankreich)

A structure preserving hybrid finite volume scheme for semi-conductor models on general meshesIn 1996, Gajewski and Gärtner introduced a model describing semi-conductor devices in presence of an exterior magnetic field and proposed a scheme to discretise it. From a numerical point of view, the main difficulty in dealing with this model is the impact of the magnetic field over the system, which leads to anisotropic diffusion equations. In particular their scheme does not preserve the positivity of the solutions if the magnetic field is too intense.

In this talk, I will introduce a new structure preserving scheme for discretising similar systems of equations. The scheme under study is based on the Hybrid Finite Volume method, which is devised to handle anisotropic diffusion tensors alongside with very general polygonal/polyhedral meshes. It is designed to handle general statistics (including Boltzmann and Blakemore) as well as strong magnetic fields, while ensuring that the computed densities are positive.

The analysis of the scheme relies on the preservation of an entropy structure at the discrete level, which mimics the continuous behaviour of the system. I will explain how we can use this structure to show that there exist positive solutions to the scheme. As a by product, we also establish a "good discrete long-time behaviour" property: the discrete solutions converge towards a discrete thermal equilibrium as time tends to infinity.

Alongside with the theoretical results, I will present numerical results obtained with this scheme in various situations. I will especially focus on the preservation of bounds for the carrier densities and on the long-time behaviour of the solutions.

This is a joint work with Claire Chainais-Hillairet, Maxime Herda and Simon Lemaire.

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Dienstag, 07. 06. 2022, 13:30 Uhr (WIAS-ESH)
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Dr. Gabriel R. Barrenechea(University of Strathclyde, GB)

Divergence-free finite element methods for an inviscid fluid modelIn this talk I will review some recent results on the stabilisation of linearised incompressible inviscid flows (or, with a very small viscosity).

The partial differential equation is a linearised incompressible equation similar to Euler's equation, or Oseen's equation in the vanishing viscosity limit. In the first part of the talk I will present results on the well-posedeness of the partial differential equation itself. From a numerical methods' perspective, the common point of the two parts is the aim of proving the following type of estimate:

u-_h^ _L^2^ leq C h^k+frac12 u_H^k+1^

whereuis the exact velocity andu_h^ is its finite element approximation. In the estimate above, the constantCis independent of the viscosity (if the problem has a viscosity), and, more importantly, independent of the pressure. This estimate mimicks what has been achieved for stabilised methods for the convection-diffusion equation in the past. Nevertheless, up to the best of our knowledge, this is the first time this type of estimate is obtained in a pressure-robust way.

I will first present results of discretisations using H(div)-conforming spaces, such as Raviart-Thomas, or Brezzi-Douglas-Marini where an estimate of the type eqrefmain-estimate is proven (besides an optimal estimate for the pressure). In the second part of the talk I will move on to H^1-conforming divergence-free elements, with the Scott-Vogelius element as the prime example. In this case, due to the H^1-conformity, extra control on the convective term needs to be added. After a (very brief) review of the idea of vorticity stabilisation, I will present very recent results on CIP stabilisation where a discussion on the type, and number, of jump terms will be presented. The method is independent of the pressure gradients, which makes it pressure-robust and leads to pressure-independent error estimates such as eqrefmain-estimate.

Finally, some numerical results will be presented and the present approach will be compared to the classical CIP method.

This work is a collaboration with E. Burman (UCL, UK), and E. Caceres and J. Guzmán (Brown, USA).

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Donnerstag, 31. 03. 2022, 14:00 Uhr (WIAS-ESH)
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Dr. Stefano Piani(International School for advance studies, Italien)

HDG methods for the Van Roosbroeck modelIn the early fifties of the twentieth century, Van Roosbroeck formulated the so-called fundamental semiconductor device equations: a system of three nonlinear coupled partial differential equations which describes potential distribution, carrier concentrations, and current flow in arbitrary semiconductor devices. In this model, charge carriers move because of two physical effects: diffusion and convection, with the latter often dominant. Performing a numerical simulation of the Van Roosbroeck equations represent a challenging problem, mainly because of their stiffness. A classical solution to this problem has been found in 1969 by Scharfetter and Gummel, who developed a nonstandard discretization method: a finite-volume scheme with a tailored numerical flux.

At the beginning of this century, instead, it has become more and more common to address convection-dominated problems using discontinuous Galerkin methods (DG methods). One of the main advantages of these methods is their ability to handle adaptive algorithms, working with meshes with hanging nodes and approximation of varying polynomial degrees. Unfortunately, when compared with the standard continuous finite elements, DG methods were often criticized for using too many globally coupled degrees of freedom.

During my talk, I will introduce a particular class of DG methods, the Hybridizable Discontinuous Galerkin Methods which exploit the classic techniques of static condensation to reduce the total number of globally coupled degrees of freedom and I will discuss the possibility of applying these methods to the Van Roosbroeck system, exploiting their similarities with the Scharfetter-Gummel scheme.

For zoom login details please contact ulrich.wilbrandt@wias-berlin.de.

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Donnerstag, 24. 03. 2022, 14:00 Uhr (Hybrid Event/WIAS-ESH)
**

Dr. Nicola Courtier(University of Oxford, UK)

Part I: Accurate and adaptable charge transport modelling of perovskite solar cells using IonMonger

Part II: Parameter/state estimation using the measure-moment approach to polynomial optimisationThis talk will cover the two main foci of my work to-date on the simulation of perovskite solar cell and lithium-ion battery models. Firstly, the development of a fast and accurate finite element scheme for a one-dimensional drift-diffusion model of a perovskite solar cell. This scheme underpins the open-source Matlab code IonMonger released in 2019 [1]. The latest version simulates a 100-point impedance spectroscopy measurement in less than a minute on a desktop computer. The ability of the scheme to cope with nonlinear terms, while maintaining second-order local accuracy, enables it to be applied to a variety of similar models e.g. [2].

Secondly, we address the problem of global parameter estimation of the four parameters of a reduced-order model for the impedance spectra of a perovskite solar cell. The reduced model is derived via asymptotic analysis of the drift-diffusion model [3]. Parameter estimation is performed using a novel approach to polynomial optimisation based on advancements in the field of measure-moment theory, via the numerical solution of a sequence of semi-definite programs [4]. Though limited by the curse of dimensionality, the measure-moment approach may also be used for state estimation and optimal control of reduced-order models. We benchmark the method via application to battery models containing 2 or 4 states.

[1] N.E. Courtier, G. Richardson, and J.M. Foster. A fast and robust numerical scheme for solving models of charge carrier transport and ion vacancy motion in perovskite solar cells. Applied Mathematical Modelling, 63, 329348 (2018)

[2] I. Korotkin, S. Sahu, S. E. J. O'Kane, G. Richardson, and J. M. Foster. DandeLiion v1: An Extremely Fast Solver for the Newman Model of Lithium-Ion Battery (Dis)charge. Journal of The Electrochemical Society, 168, 060544 (2021)

[3] L. J. Bennett, A. J. Riquelme, N. E. Courtier, J. A. Anta, and G. Richardson. A new ideality factor for perovskite solar cells and an analytical theory for their impedance spectroscopy response. arXiv:2105.11226 (2021)

[4] D. Henrion, J.-B. Lasserre, and J. Löfberg. GloptiPoly 3: moments, optimization and semidefinite programming. Optimization Methods & Software, 24:4-5, 761-779 (2009)

For zoom login details please contact ulrich.wilbrandt@wias-berlin.de

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Donnerstag, 10. 03. 2022, 14:00 Uhr (Hybrid Event/WIAS-ESH)
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Xu Li(Shandong University, China)

Two divergence-free reconstructions for Navier--Stokes simulations regarding EMA and robust estimates of kinetic energy errorNavier--Stokes simulations with divergence-free elements have many fascinating properties such as EMA-conservation (EMA: kinetic energy, linear momentum and angular momentum), pressure-robustness and Re-semi-robustness. Due to the complexity of divergence-free elements, developing novel discretizations for the commonly used elements which possess similar properties is a very popular way in Navier-Stokes community. In this talk, we discuss two divergence-free reconstruction methods for Navier--Stokes simulations. The first method is obtained by replacing the advective velocity of the convective trilinear form with its a divergence-free approximation. The resulting method preserves kinetic energy and linear momentum under some appropriate senses. Regarding its error estimates, we prove that the Gronwall constant in the kinetic energy error bound does not depend on the inverse powers of viscosity explicitly, which is similar to the EMAC method. The second method is based on the pressure-robust reconstruction formulation in [A. Linke & C. Merdon, CMAME, 2016]. We propose a novel trilinear form which preserves (redefined) energy, momentum and angular momentum, while maintaining pressure-robustness. It can also be proven that the constants in the kinetic energy error bound do not depend on the inverse powers of viscosity explicitly, which is similar to the methods with divergence-free elements. In both methods we discuss the reconstruction operators in detail. Finally, we also show some numerical experiments with Taylor--Hood, MINI and Bernardi--Raugel elements, doing some comparisons with the EMAC method and some low order divergence-free element methods.

(This is a joint work with Hongxing Rui, Shandong University)

For zoom login details please contact ulrich.wilbrandt@wias-berlin.de