### 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}

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

*The zoom link will be send to the staff of the institute about 15 minutes before the start of the talk. People who are not members of the institute and who are interested in participating at this talk should contact Alexander Linke linke@wias-berlin.de for obtaining the zoom login details.*

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Donnerstag, 25. 06. 2020, 14:00 Uhr (Online Event)
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Holger Stephan(WIAS Berlin)

An overview of Lyapunov functions for linear continuous and discrete problems including some new resultsLyapunov functions are useful for studying the time behaviour of the solution of evolutionary equations. The classical example is the L_2 norm, which decays on the solution of a diffusion equation with Neumann boundary conditions. The corresponding operator is an operator in divergent form, which forms a symmetric positive definite Dirichlet form. This concept can be generalized to arbitrary linear evolution equations with Markov generators. These equations are exactly those which preserve the type of physical quantities describing them: Intensive quantities (Banach space of continuous functions) are averaged, extensive quantities (Banach space of Radon measures) conserve their mass and sign. It turns out that for Markov generators with real spectrum, there is always a Hilbert space whose norm is a Lyapunov function and whose corresponding Dirichlet form is symmetric and positive. This generalizes the concept of detailed balance. An analogous result applies to the normality of operators with complex spectrum. This is a joint result with Artur Stephan from RG1. Furthermore, an infinite sequence of positive Lyapunov functions -- like energy, dissipation rate and so on -- can be defined recursively. This can be understood as a generalization of the well-known Bakry-Emery theory for diffusion type equations. In the lecture both the finite dimensional discrete case and the infinite dimensional continuous case are considered. Depending on which case is better suited for understanding the nature of the problem, we switch between them in an appropriate manner.

For zoom login details please contact Alexander Linke linke@wias-berlin.de

**
Donnerstag, 04. 06. 2020, 11:00 Uhr (Online Event)
**

Prof. Volker John(WIAS Berlin)

On the provable convergence order for the kinetic energy of FEMs for the incompressible Navier--Stokes equationsThe kinetic energy of a flow is proportional to the square of the

Lnorm of the velocity. Given a sufficient regular velocity field^{2}(Ω)uand a velocity finite element space with polynomials of degreer, then the best approximation error inLis of order^{2}(Ω)r+1. This talk will discuss which order of convergence for the kinetic energy can be proved with robust error estimates, i.e., with estimates where the constant does not depend on inverse powers of the viscosity. To fix ideas, the first part of the talk is devoted to evolutionary scalar convection-diffusion equations. In the second part, results for robust discretizations of the time-dependent incompressible Navier--Stokes equations will be surveyed. This survey covers as well inf-sup stable pairs of finite element spaces as pressure-stabilized discretizations.

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Donnerstag, 28. 05. 2020, 14:00 Uhr (Online Event)
**

Dr. Hang Si(WIAS Berlin)

Detri2, a 2D constrained Delaunay-Voronoi mesh generatorDetri2 is a C++ program to generate triangular meshes from arbitrary polygonal domains.It uses constrained Delaunay algorithms to conform the input line segments and maintain the locally Delaunay property of the mesh toplogy. It uses Delaunay refinement algorithms to generate meshes with good angle bounds. It can generate adaptive meshes corresponding to input mesh sizing functions. In this talk, I will give a brief tutorial of the basic functionalites of Detri2. I will demonstrate the aglorithms and many of the features of Detri2 using a graphic user interface - Detri2Qt. The source code of Detri2 and Detri2Qt are available at http://www.wias-berlin.de/people/si/detri2.html

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Donnerstag, 14. 05. 2020, 14:00 Uhr (Online Event)
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Derk Frerichs(WIAS Berlin)

Basic concepts of virtual element methods and a really pressure-robust virtual element method for the Stokes problemVirtual element methods (VEM) can be seen as a modern approach to extend finite element methods to polygonal and polyhedral meshes. This talk has two aims. First, basic ideas, concepts and advantages of VEM are presented using the well known Poisson problem. The second part concerns a VEM for the Stokes problem. For the Stokes problem exactly divergence-free VEM fail when it comes to small viscosity parameters or when the continuous pressure is complicated. In this talk a modification of the VEM is presented based on Raviart--Thomas approximations of the testfunctions which renders the method really pressure-robust, i.e. locking free for very small viscosities. The construction is also interesting for hybrid hyigh-order methods on polygonal or polyhedral meshes. Numerical results round up the theoretical findings.

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Donnerstag, 07. 05. 2020, 14:00 Uhr (Online Event)
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Dilara Abdel(WIAS Berlin)

Comparison of modified Scharfetter-Gummel schemes for generalized drift-diffusion systemsThe standard model for the description of semi-classical transport of free electrons and holes in a self-consistent electric field in a semiconductor device is the van Roosbroeck system of equations. For the discretization of this system Voronoi finite volume schemes based on different choices of flux approximations are considered. The classical Scharfetter--Gummel scheme yields a thermodynamically consistent numerical flux, but cannot be used for general charge carrier statistics. This talk aims to give an impression of the comparison and analysis of modified Scharfetter--Gummel schemes to simulate generalized drift-diffusion systems.

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Dienstag, 05. 05. 2020, 14:00 Uhr (Online Event)
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Dr. Christian Merdon(WIAS Berlin)

A novel gradient-robust, well-balanced discretisation for the compressible Stokes problemThis talk suggests a novel well-balanced discretisation of the stationary compressible isothermal Stokes problem based on the concept of gradient-robustness. Gradient-robustness means that arbitrary gradient fields in the momentum balance are well-balanced by the discrete pressure gradient - if there is enough mass in the system to compensate the force. The scheme is provably convergent and asymptotic-preserving in the sense that it degenerates for low Mach numbers to a recent inf-sup stable and pressure-robust modified Bernardi--Raugel finite element method for the incompressible Stokes equations. All properties are demonstrated in numerical examples.

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Donnerstag, 05. 03. 2020, 14:00 Uhr (ESH)
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Dr. Alexander Linke(WIAS Berlin)

Gradient-robustness: A new concept assuring accurate spatial discretizations for vector-valued PDEsVector-valued PDEs like the incompressible Navier--Stokes equations (in primitive variables velocity and pressure) describe the time evolution of a vector-valued quantity like the momentum density. For vector-valued PDEs it is quite natural to derive formally on the continuous level a derived time evolution of the vorticity and the divergence. Accordingly, any space discretization for a vector-valued PDE (implicitly) delivers a discrete vorticity and discrete divergence equation.

While the celebrated inf-sup stability will be shown to assure an accurate discrete divergence equation, the talk will actually focus on the question, which structural properties allow for an implicitly defined accurate discrete vorticity equation. The key observation is that the L^2-orthogonality of divergence-free vector fields and gradient fields is the weak equivalent to the vector calculus identity $nabla times nabla psi = mathbf0$ for arbitrary smooth scalar fields $psi$.

In the context of the incompressible Navier-Stokes equations, the concept of pressure-robustness was introduced in 2016, in order to discriminate between space discretizations with accurate and inaccurate discrete vorticity equations. H(div)-conforming finite element spaces have been found as an important means to realize the L^2-orthogonality between discretely divergence-free vector fields and (arbitrary) gradient fields. Further, it is shown that spatial discretizations that are not pressure-robust may suffer from i) a degradation of the (preasymptotic) convergence rate, and ii) large, parameter-dependent constants in error estimates. Typical flows that benefit from pressure-robustness are quasi-hydrostatic flows, quasi-geostrophic flows and vortex-dominated high Reynolds number flows.

Last but not least, the talk shows how to extend the concept of pressure-robustness to more general vector-valued PDEs, leading to the concept of gradient-robustness. Gradient robustness assures that dominant and extreme gradient fields in a vector-valued PDE will not lead to inaccuracies in the discretization. The talk will show examples from linear elasticity and compressible (Navier--)Stokes flows at low Mach numbers and in stratified flows. Thus, connections to well-balanced schemes and WENO schemes will be drawn.

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Donnerstag, 27. 02. 2020, 14:00 Uhr (ESH)
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Dr. Philip Lederer(TU Wien, Österreich)

Divergence-free tangential finite element methods for incompressible flows on surfacesIn this work we consider the numerical solution of incompressible flows on twodimensional manifolds. Whereas the compatibility demands of the velocity and the pressure spaces are known from the flat case one further has to deal with the approximation of a velocity field that lies only in the tangential space of the given geometry. Abandoning

H^{1}-conformity allows us to construct finite elements which are -- due to an application of the Piola transformation -- exactly tangential. To reintroduce continuity (in a weak sense) we make use of (hybrid) discontinuous Galerkin techniques. To further improve this approach,H(div_{ Γ})-conforming finite elements can be used to obtain exactly divergence-free velocity solutions. We present several new finite element discretizations. On a number of numerical examples we examine and compare their qualitative properties and accuracy.

und 15:00 Uhr

Dr. Anton Stephan(Institut für Physik der Atmosphäre)

Modeling Navier--Stokes equations for high Reynolds number flows and complex geometries in aerodynamic applicationsThe flight of an aircraft through a turbulent environment presents a difficult numerical problem. Though it can be easily described by the Navier-Stokes equations, it is impossible to perform a direct numerical simulation resolving all physically relevant scales. For several decades, numerous different models have been developed which perform very well for very specific problems. However, since the aircraft flight includes complex geometries as well as large spatial dimensions these models fail to capture all important features of the problem.

The vision of establishing a simulation system for virtual flight in a realistic environment is addressed by the coupling of two separate flow solvers in a bi-directional manner. This enables for example the simulation of a flight through gusts, trailing vortices and clear air turbulence, including the effects on the aircraft and the roll-up of trailing vortices and their further development until final decay. A compressible Reynolds-Averaged Navier-Stokes (RANS) solver resolves the near-field around an aircraft model. An incompressible Large-Eddy Simulation (LES) solver is used to model the atmosphere around the aircraft with its wake footprint in the LES domain. This hybrid simulation setup is used to investigate complex physical problems occurring in different flight scenarios.

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Donnerstag, 20. 02. 2020, 14:00 Uhr (ESH)
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Prof. Wolfgang Dreyer(WIAS Berlin)

Classical tests of electro-chemistryVolume changes, heat of dilution, osmosis, saturation pressure decrease and boiling temperature increase of electrolytes are classical phenomena that can be used to test and adjust parameter of electrochemical modeling. In this lecture we apply the electrolyte model developed at WIAS to these phenomena in order to critically examine the underlying assumptions. The observations require generalizations of the notions of incompressibility and surface tension.

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Dienstag, 18. 02. 2020, 13:30 Uhr (ESH)
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Ondřej Pártl(WIAS Berlin)

Mathematical modeling of non-isothermal compositional compressible fluid flow in zeolite bed and above its surfaceI will present a mathematical model and numerical scheme for the simulation of non-isothermal compositional compressible fluid flow in a heterogeneous porous medium and in the coupled atmospheric boundary layer above the surface of this porous medium, where one of the flowing components adsorbs on the porous matrix. I will discuss the application of this model to the simulation of the hydration of a zeolite bed.

My model is based on the two-domain approach, i.e., the domain in which the flow occurs is divided into the porous medium subdomain and the free flow subdomain. In each subdomain, the flow is described by the corresponding balance equations for mass, momentum and energy. On the interface between the subdomains, coupling conditions are prescribed.

In both subdomains, the spacial discretization of the governing equations is carried out via the finite volume method. For time integration, we use a special time stepping procedure.

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Donnerstag, 13. 02. 2020, 14:00 Uhr (ESH)
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Dr. Gabriel R. Barrenechea(University of Strathclyde, Scotland)

Low-order divergence-free finite element methodsIn this talk I will review results on a divergence-free reconstruction of the lowest order pair for the Navier-Stokes equation. More precisely, from a stabilised P1xP0 scheme, a divergence-free velocity field is built as the result of a lift of the pressure jumps, and it is then plugged in the convective term of the momentum equation. This process provides a method that can be proven stable without the need to suppose the mesh refined enough. We first apply this idea to the transient Navier-Stokes equations, where estimates independent of the viscosity are derived. Then, the applicability of this idea is extended to a steady-state generalised Boussinesq system.

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Donnerstag, 06. 02. 2020, 14:00 Uhr (ESH)
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Dr. Holger Stephan(WIAS Berlin)

A general concept of majorization and a corresponding Robin Hood methodThe Robin Hood method is a numerical method for constructing a double-stochastic matrix that maps a given vector to another vector that is also given. This is possible if the preimage majorizes the image. The majorization of vectors is a special order sometimes called the Lorentz order. A double-stochastic matrix is a special case of a Markov (or stochastic) matrix. In the case of a general Markov matrix, it is not yet clear when such a construction is possible, what kind of condition similar to the majorization must be fulfilled by the given vectors, nor is a numerical method for determining the matrix known (regardless some heuristic iterative versions of the Robin-Hood method).

In the talk, we give a complete solution to all these problems, including a general direct Robin Hood method. It turns out that the construction of a Markov matrix from two given pairs of two vectors is possible if and only if the pairs satisfy a certain order condition. If one interprets the vectors as states of a physical system, then this order is exactly the order known as the natural time order or the second law of thermodynamics.

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Donnerstag, 09. 01. 2020, 14:00 Uhr (ESH)
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Lia Strenge(Technische Universität Berlin)

A multilayer, multi-timescale model approach for economic and frequency control in power grids using JuliaPower systems are subject to fundamental changes due to the increasing infeed of decentralized renewable energy sources and storage. The decentral nature of the new actors in the system requires new concepts for structuring the power grid, and achieving a wide range of control tasks ranging from seconds to days. Here we introduce a multilayer dynamical network model covering a wide range of control time scales. Crucially we combine a decentralized, self-organized low-level control and a smart grid layer of devices that can aggregate information from remote sources. The stability critical task of frequency control is performed by the former, the economic objective of demand matching dispatch by the latter. Having both aspects present in the same model allows us to study the interaction between the layers. Remarkably we find that adding communication in the form of aggregation does not improve the performance in the cases considered. Instead the self-organised state of the system already contains the information required to learn the demand structure in the entire power grid. The model introduced here is highly flexible, and can accommodate a wide range of scenarios relevant to future power grids. We expect that it will be especially useful in the context of low-energy microgrids with distributed generation. All simulations and numerical experiments for control design and analysis with sampling-based methods are performed in Julia 1.1.0. The overall model is implemented as stiff nonlinear ordinary differential equation (ODE) with periodic callbacks for the control actions. The ODE has dimension 4 N, where N is the number of edges of the graph representing the power grid (i.e., N feed-in/load connections). It is planned to use automatic differentiation to learn more about the overall nonlinear model.