## Doktorandenseminar des WIAS

# Numerische Mathematik und Wissenschaftliches Rechnen

Forschungsgruppe

zurück zum aktuellen Programm

## Seminar Numerische Mathematik / Numerical mathematics seminars

aktuelles Programm / current program

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Donnerstag, 17. 11. 2016, 14:00 Uhr (ESH)
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Prof. J. Novo(Universidad Autonoma de Madrid, Spain)

Non inf-sup stable mixed finite element approximations to the evolutionary Stokes equationsIn the first part of the talk we will consider the pressure stabilized Petrov Galerkin (PSPG) method in space to approximate the evolutionary Stokes equations with non inf-sup stable mixed finite elements. The analytical results are based on an appropriate approach for computing the initial velocity, which we suggest. We clarify the so called instability of the discrete pressure for small time steps reported in the literature and show that this instability does not occur for our proposed initial condition. In the second part of the talk we study the approximation of the same equations by means of a projection scheme with non inf-sup stable mixed finite elements. The projection scheme we propose has a stability parameter analogous to the stability parameter of the PSPG method. As a consequence of our analysis we prove the pressure stability of the Chorin--Temam method for non inf-sup stable elements. This property was known in the literature but to our knowledge had not yet been proved.

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Donnerstag, 17. 11. 2016, 15:15 Uhr (ESH)
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Dr. St. Weißer(Universität des Saarlandes)

Convection-adapted BEM-based finite element method on tetrahedral and polyhedral meshesA new discretization method for homogeneous convection-diffusion-reaction boundary value problems in 3D is presented that is a non-standard finite element method with PDE-harmonic shape functions on polyhedral elements, see [1]. The element stiffness matrices are constructed by means of local boundary element techniques. The method, which is referred to as a BEM-based FEM, can therefore be considered a local Trefftz method with element-wise (locally) PDE-harmonic shape functions. The current research combines the results of [2] with the hierarchical construction of shape functions presented in [3]. The Dirichlet boundary data for these shape functions is chosen according to a convection-adapted procedure which solves projections of the PDE onto the edges and faces of tetrahedral and polyhedral elements, respectively. This improves the stability of the discretization method for convection-dominated problems both when compared to a standard FEM and to previous BEM-based FEM approaches, as we demonstrated in several numerical experiments. Our experiments also show an improved resolution of the exponential layer at the outflow boundary for our proposed method when compared to the SUPG method.

References:

[1] C. Hofreither, U. Langer and S. Weißer. Convection-adapted BEM-based FEM. Z. Angew. Math. Mech. doi:10.1002/zamm.201500042 (2016).

[2] C. Hofreither, U. Langer and C. Pechstein. A non-standard finite element method for convection-diffusion-reaction problems on polyhedral meshes. AIP Conference Proceedings 1404(1):397{404 (2011).

[3] S. Rjasanow and S. Weier. FEM with Trefftz trial functions on polyhedral elements. J. Comput. Appl. Math. 263:202{217 (2014).

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Donnerstag, 22. 09. 2016, 14:00 Uhr (ESH)
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Dr. P. Farrell(WIAS Berlin)

Scharfetter-Gummel schemes for non-Boltzmann statisticsWhen modelling semiconductor devices via the van Roosbroeck system one ofen uses statistical functions to describe the correspondence between carrier densities and chemical potentials. For 3D bulk semiconductors the most general choice is given by the Fermi-Dirac integral of order 1/2. However, how to numerically solve the van Roosbrock in this general (non-Boltzmann) case is still an open problem. We will present and compare several methods to generalise the well-known Scharfetter-Gummel scheme. Our main goal is to discretely preserve important properties from the continuous system such as existence and uniqueness of the solution, consistency with the thermodynamical equilibrium and unconditional stability. We also show how these new numerical schemes can be efciently implemented for 2D and 3D applications.

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Donnerstag, 15. 09. 2016, 14:00 Uhr (Raum 406)
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Prof. J. Shewchuk(University of California at Berkeley, USA)

Restricted constrained delaunay triangulationsSurface meshes are used extensively in computer graphics, boundary element methods, and many other applications. Researchers have sought surface triangulations that have formal mathematical properties similar to those enjoyed by Delaunay triangulations in the plane. They succeeded by inventing the "restricted Delaunay triangulation," which is a subcomplex of the three-dimensional Delaunay triangulation. In other words, given a smooth surface embedded in three-dimensional space, we compute a set of points lying on the surface, we compute the 3D Delaunay triangulation of those points, and we select a subset of its triangles to serve as a triangulation of the surface. This subset of triangles, the restricted Delaunay triangulation, has proven itself as a mathematically powerful tool for surface meshing and surface reconstruction.

I address a question of Bruno Levy: can we constrain restricted Delaunay triangulations to include specified edges? That is, can we define a mathematically well-behaved, Delaunay-like, edge-constrained triangulation on a smooth surface? We do so by defining a specialized Voronoi diagram on the surface, then dualizing it to obtain a triangulation. But to force specified edges to appear in this triangulation, we must perform topological surgery on the surface...and that's where things get weird and interesting.

(This work is done jointly with Marc Khoury, Bruno Levy, and Marc van Kreveld.)

Bio:

Jonathan Shewchuk is a Professor in the Department of Electrical Engineering and Computer Sciences at UC Berkeley. He is best known for his software Triangle for high-quality triangular mesh generation, which won the 2003 James Hardy Wilkinson Prize in Numerical Software, and his "Introduction to the Conjugate Gradient Method Without the Agonizing Pain." His book "Delaunay Mesh Generation," written jointly with Siu-Wing Cheng and Tamal Dey, is available from CRC Press.

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Donnerstag, 18. 08. 2016, 14:00 Uhr (ESH)
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A. Chern(TU Berlin)

Fluid simulation using incompressible Schrödinger flowSimulation of incompressible fluids has been a challenging numerical task in computer graphics as well as in many areas of science.. We describe a new approach for the purely grid based simulation of incompressible fluids. In it, the fluid state is represented by a wave function evolving under the Schrödinger equation subject to incompressibility constraints. The underlying dynamics satisfies the Euler equation modified with a Landau-Lifshitz term. The latter ensures that dynamics due to thin vortical structures are faithfully reproduced. This enables robust simulation of intricate phenomena such as vortical wakes and interacting vortex filaments, even on modestly sized grids. The numerical algorithms for time evolution are exceedingly simple. In the talk I will also discuss the underlying theory which reveals fascinating connections between classical fluids, quantum mechanics, super fluids, Landau-Lifshitz theory of ferromagnetic material, and the geometry of the 3-hypersphere.

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Donnerstag, 04. 08. 2016, 14:00 Uhr (ESH)
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Ph. Lederer(TU Wien, Österreich)

An exact divergence-free reconstruction operator for the Taylor--Hood elementIn this talk we focus on a well-known issue of discretization techniques for the incompressible (Navier) Stokes equations. The numerical solution is only discrete divergencefree, which may have a major impact on quantitative and qualitative properties of the solution. In recent years Alexander Linke and cooperators developed a methodology to reconstruct exactly divergence-free solutions from discrete divergence-free ones, and use this operator within the Navier Stokes solver. In this work we extend this approach from discontinuous pressure elements to continuous pressure elements including the popular Taylor-Hood element. While for discontinuous pressures the reconstruction operator is given by element-wise local procedures, we have to extend the construction to vertex or element patches. The reconstruction leads to non conforming methods, where the consistency error is estimated in dual norms. Convergence of optimal order is proven (see [1]). The method is implemented in NGS-Py which is based on the nite element library Netgen/NGSolve. Several examples are presented.

References

[1] P. Lederer. Pressure Robust Discretizations for Navier Stokes Equations: Divergence-free Reconstruction for Taylor-Hood Elements and High Order Hybrid Discontinuous Galerkin Methods. Master's thesis, TU Wien, Austria, 2016.

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Donnerstag, 21. 07. 2016, 14:00 Uhr (ESH)
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Dr. A. Wahab(Korea Advanced Institute of Science \& Technology)

A non-iterative algorithm for elastography using joint sparsityWe present a novel non-iterative algorithm for reconstruction of multiple small elastic inclusions and identication of their Lame parameters from discrete boundary measurements of the scattered elastic field. The inclusions with variable parameters are assumed to be compactly embedded in a homogeneous background elastic medium. We first derive a Lippmann--Schwinger type integral representation for the scattered field inside the inclusions. Then, we formulate the detection problem as a joint sparse recovery and present an efficient non-iterative location search algorithm using tools borrowed from compressed sensing literature. Using a slightly modified algorithm, we reconstruct the material parameters of the inclusions once their spatial support is identified. The efficiency and robustness of the algorithm are substantiated through numerical illustrations. This work is motivated by pertinent applications in magnetic resonance elastography and can cater to a broad range of inverse scattering problems in elastic media.

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Dienstag, 28. 06. 2016, 13:30 Uhr (ESH)
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W. Huang(University of Kansas, USA)

A new implementation of the MMPDE moving mesh method and applicationsThe MMPDE moving mesh method is a dynamic mesh adaptation method for use in the numerical solution of partial differential equations. It employs a partial differential equation (MMPDE) to move the mesh nodes continuously in time and orderly in space while adapting to evolving features in the solution of the underlying problem. The MMPDE is formulated as the gradient flow equation of a meshing functional that is typically designed based on geometric, physical, and/or accuracy considerations. In this talk, I will describe a new discretization of the MMPDE which gives the mesh velocities explicitly, analytically, and in a compact matrix form. The discretization leads to a simple, efficient, and robust implementation of the MMPDE method. In particular, it is guaranteed to produce nonsingular meshes. Some applications of the method will be discussed, including mesh smoothing (to improve mesh quality), generation of anisotropic polygonal meshes, and the numerical solution of the porous medium equation and the regularized long-wave equation.

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Dienstag, 14. 06. 2016, 13:30 Uhr (ESH)
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Prof. A. Bradji(Universit{\'e} Badji Mokhtar-Annaba, Algerien)

On the convergence order of gradient schemes for time dependent partial differential quationsGradient schemes are numerical methods, which can be conforming and nonconforming, and references therein to approximate different types of partial differental quation. They are written in a discrete variational formulation and based on the approximation of functions and gradients. The framework of gradient schemes includes for instance the Hybrid Mimetic Mixed (HMM) family. The aim of this lecture is to give some recent results concerning the convergence order of gradient schemes for time dependent partial differential quations. In particular, we quotes some results related to the case of semilinear parabolic and hyperbolic equations. We also try to discuss the case of a time dependent Joule heating system.

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Dienstag, 14. 06. 2016, 14:30 Uhr (ESH)
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Ch. Wang(Northeastern University, China)

Product design and engineering analysis technologiesThis presentation begins with a brief introduction of manufacturing industries, and a laconic discussion of significance of product design and engineering analysis technologies. Subsequently, a multidisciplinary view is presented, which tends to address product engineering issues in an integrated way. Following up, the presentation outlines some significant product design technologies in the context of specific industrial implementations. Products or engineering systems are usually subject to intensive physical loadings in operational environments. Engineering analysis technologies play an essential role in product development decision-making by computing and emulating multiphysics processes, which occur within or around a product system. In this vein, the speaker finally talks about his pragmatic experiences of developing geometric modeling, meshing, FEM, and computational visualization systems.

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Donnerstag, 09. 06. 2016, 14:00 Uhr (ESH)
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Dr. G. Printsypar(WIAS Berlin)

Modeling of membranes for forward osmosis at different scalesIn this talk we summarize and present results for mathematical modeling of forward osmosis processes using macro-, multi-, and micro-scale approaches. First, we focus on a macro-scale model for a cell element used to perform laboratory experiments for forward osmosis. This model considers the membrane as a homogenized porous medium and uses a Darcy type model for fluid flow in porous media. Second, we propose a multi-scale simulation approach to model forward osmosis processes using membranes with layered homogeneous morphology. This approach accounts for both: forward osmosis macro-scale setup and detailed microstructure of the substrate using the digitally reconstructed morphology. Then, we introduce a micro-scale model for fully or partially resolved microstructure of the membrane with heterogeneous morphology. Here processes occur at the pore scale which yields a detailed and accurate picture of the processes. We introduce mathematical models and present numerical results for all three modeling approaches. The numerical experiment of the multi-scale approach is also validated with laboratory experiments.

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Donnerstag, 02. 06. 2016, 14:00 Uhr (ESH)
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N. Alia(WIAS Berlin)

Introduction to the project about optimal control of ladle stirringThe doctoral subject "Optimal control of ladle stirring" is part of the MIMESIS project led by D. H{\"o}mberg and funded by the EU through its European Industrial Doctorate initiative. In order to produce high performance steel, refinement metallurgical processes are used by steelmakers. One example of these processes are vacuum tank degassers, where the liquid steel is contained in a vacuum ladle and is subject to an inert gas flow in order to evacuate impurities such as gaseous Hydrogen particles. The control of this process is a main issue for steel plants which aim at a competitive production of high performance steel. One possibility to achieve this is to consider the vibration measured on the ladle wall and to link it with the velocity of the flow happening inside the ladle. This needs the mathematical modeling and computation of mainly two phenomena : the turbulent multiphase flow inside the ladle and the fluid/structure interaction. Then, experiments have to be carried out in order to derive an operational index which can predict the steel quality based on vibration measurements and simulations. A general introduction to this doctoral work is presented in this seminar.

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Dienstag, 24. 05. 2016, 13:30 Uhr (ESH)
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Prof. S. Kaya Merdan(Middle East Technical University, Ankara, Turkey)

Numerical analysis of a fully discrete decoupled penalty-projection algorithm for MHD in Elsässer variablesThis talk considers numerical analysis of MHD flow that is based on Elsässer variable formulation. In the first part of the talk, I will present a fully discrete, efficient time stepping scheme that decouples the MHD system. The algorithm still provides unconditionally stability with respect to the time step and is extended a more efficient class of timestepping algorithm (penalty-projection type). Numerical simulations will be given on some benchmark problems. In the second part of the talk, I will consider the long time stability property of the MHD system. The scheme uses linearly extrapolated BDF2 time stepping scheme, together with a finite element spatial discretization. The stability restriction of the algorithm is tested.

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Donnerstag, 12. 05. 2016, 14:00 Uhr (Raum 406/405)
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Prof. W. Dreyer(WIAS Berlin)

Subtle properties of the barycentric velocity in mixture modelsIn a single substance the barycentric velocity is a primitive concept while in a mixture of several constituents it is defined via the velocities of the constituents. In both cases the evolution of the barycentric velocity is determined by the balance equations of mass and momentum. There are many inconsistencies in constitutive models in the field of the barycentric velocity. In this lecture we illustrate these inconsistencies by means of three practical ex- amples. The most prominent example concerns an incompressible material with thermal expansion. A further example is provided by a liquid-vapor phase transition. Finally we study the plating process of two copper electrodes immersed in an electrolyte.

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Dienstag, 03. 05. 2016, 13:30 Uhr (ESH)
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L. Blank(WIAS Berlin)

CGS --- Taking advantage of the irregular convergence behavior

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Donnerstag, 28. 04. 2016, 14:15 Uhr (ESH)
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F. Neumann(WIAS Berlin)

Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes-ProblemThe classical Crouzeix-Raviart element for approximating the Stokes-Problem leads to discrete velocity solutions which are not pressure-robust, in the sense that estimates of the velocity error depend on the continuous pressure. A simple variational crime resolves this issue and introduces a new pressure-robust finite element scheme. In this lecture we show quasi-optimality of the new scheme under low regularity assumptions by means of a conforming and divergence-free interpolation operator. Finally we study the influence of the used velocity-reconstruction on the obtained estimates

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Donnerstag, 21. 04. 2016, 14:00 Uhr (ESH)
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Prof. Th. Apel(Universität der Bundeswehr München)

Anisotropic mesh refinement in polyhedral domains: error estimates with data in $L^2(\Omega)$The presentation is concerned with the finite element solution of the Poisson equation with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes are used for dealing with the singular behaviour of the solution in the vicinity of the non-smooth parts of the boundary, see [1]. The discretization error is analyzed for the piecewise linear approximation in the $H^1(\Omega)$- and $L^2(\Omega)$-norms by using a new quasi-interpolation operator. This new interpolant is introduced in order to prove the estimates for $L^2(\Omega)$-data in the differential equation which is not possible for the standard nodal interpolant. These new estimates allow for the extension of certain error estimates for optimal control problems with elliptic partial differential equation and for a simpler proof of the discrete compactness property for edge elements of any order on this kind of finite element meshes. The results were achieved in collaboration with Ariel Lombardi and Max Winkler, see [2].

[1] Th. Apel, S.Nicaise: The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Methods Appl. Sci., 21:519--549, 1998.

[2] Th. Apel, A.L. Lombardi, M. Winkler: Anisotropic mesh refinement in polyhedral domains: error estimates with data in $L^2(\Omega)$. M2AN 48:1117--1145, 2014.

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Dienstag, 19. 04. 2016, 15:00 Uhr (ESH)
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M. Wolff(WIAS Berlin)

Finite-element-method for the Stokes-Darcy equations in 3DThe Stokes-Darcy equations are used in the modeling of ground water flow or filters. In these applications the fluid viscosity and the hydraulic permeability are usually small and influence the behaviour of the classical iterative subdomain methods which are presented in this talk. The convergence behaviour will be considered for a simple filter and finally some guidelines for the Robin parameters will be presented.

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Donnerstag, 14. 04. 2016, 14:00 Uhr (ESH)
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Prof. K.-M. Weitzel(Philipps-Universität Marburg)

Ionentransport durch feste Materie --- BIIT Experimente und numerische Beschreibung mittels Nernst-Planck-Poisson TheorieDer von uns entwickelte Bombardement induzierte Ionentransport (BIIT) basiert auf der quasi-thermischen Anlagerung von positiv geladenen Ionen an eine Probe, die auf der Rückseite mit einer einzelnen Metallelektrode kontaktiert ist. Die Anlagerung der Ionen generiert Gradienten des elektrischen Potentials und der Teilchenzahldichte. Diese bewirken Ionentransport, der als Neutralisationsstrom an der Rückseite gemessen wird.

Das native Ionenbombardement, bei dem Bombarder-Ion und Leitungs-Ion der Probe identisch sind, eröffnet Zugang zu Leitfähigkeiten und Aktivierungsenergien.

Das Fremdionen-Bombardement, bei dem sich Bombarder-Ion und Leitungs-Ion unterscheiden, erzeugt Elektrodiffusionsprofile, die ex situ mittels ToF-SIMS quantitativ vermessen werden. Mittels ToF-SIMS kann auch gezeigt werden, dass sich die ursprüngliche Grenzfläche in eine Zwischenphase verwandelt.

Zentraler Bestandteil der Arbeiten ist die Beschreibung der BIIT Experimente, insbesondere der Diffusionsprofile, mittels der Nernst-Planck-Poisson Theorie. Die Analyse eröffnet unter anderem Zugang zur Konzentrationsabhängigkeit von Diffusionskoeffizienten.

Es werden Beispiele für den Transport verschiedener Alkali-Ionen in verschiedenen Materialien (harte Gläser, weiche Polymerfilme, ultra-dünne Polyelektrolytmembrane) präsentiert.

Falls es die Zeit gestattet werden kurz Studien zum thermischen Elektropoling und deren theoretische Beschreibung mittels eines Monte-Carlo Ansatzes vorgestellt.

Literatur: PCCP, 13, 20112-20122, (2011)

PCCP, 13, 20123-20128, (2011)

PCCP, 15, 1481-1487, (2013)

PCCP, 18, 4345-4351, (2016)

Zeitschrift für Physikalische Chemie, 226, 341-353 (2012)

Electrochimica Acta, 170, 122?130 (2015)

Electrochimica Acta, 191, 616?623 (2016)

Solid State Ionics, 282, 70?75, (2015)

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Donnerstag, 10. 03. 2016, 14:00 Uhr (ESH)
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U. Wilbrandt(WIAS Berlin)

Iterative subdomain methods for the Stokes-Darcy problem

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Donnerstag, 03. 03. 2016, 14:00 Uhr (ESH)
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M. Hoffmann(FU Berlin)

Adaptive finite element methods for Stokes--DarcyThe general form of the finite element method involves choosing a grid for the domain of the considered partial differential equation. However, if the grid is too coarse, one might not resolve the solution sufficiently well. On the other hand, it is usually a priori not known where the grid needs to be finer.

This talk deals with residual based a posteriori error estimators and the goal oriented dual weighted residual method, which make it possible to adaptively refine a given grid. A comparison of these methods applied to Stokes and convection-diffusion equations will be presented. Finally, the coupled Stokes--Darcy system will be considered.

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Dienstag, 23. 02. 2016, 13:00 Uhr (Raum 406)
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Dr. C. Bertoglio(Universidad de Chile, Chile)

Noninvasive pressure drop estimation in blood flowsPressure gradients across stenotic blood vessels is an important clinical index for diagnosis of the pathological severity of the disease. While the clinical gold standard for its measurement is the invasive catheterization, Phase-Contrast MR-imaging has emerged as a promising tool for enabling a non-invasive quantification of the pressure drop from the measured velocity field. However, current methods for recovering the pressure drop lack of robustness with respect to data perturbations. In this work we present a family of approaches that integrate velocity measures over a control volume in order to recover the stenosis pressure drop. We test the precision and robustness of different variants of the method on several numerical examples using perturbed synthetic data.

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Donnerstag, 11. 02. 2016, 14:00 Uhr (ESH)
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C. Bartsch(WIAS Berlin)

An assessment of solvers for saddle point problems emerging from the incompressible Navier-Stokes equationsIn this talk we present an assessment of the performance of different solvers for linear saddle point problems. In fluid dynamics large linear saddle point problems emerge from the linearization and discretization of the Navier?Stokes equations. Preconditioned Krylov subspace methods such as FGMRES are a popular choice to solve these large linear systems. We employ FGMRES with two different types of preconditioners: A geometric multigrid preconditioner with different implementations of Vanka type smoothers is compared with the least squares commutator preconditioner (LSC) of Elman et al.

The LSC preconditioner has been modified recently as to include boundary conditions, and comparisons of the LSC approach with geometric multigrid preconditioners are not available so far. We incorporate the original and the boundary corrected LSC preconditioner into our studies. For comparison of these preconditioned iterative methods we also consider the direct solvers UMFPACK and PARDISO.

The solvers are appplied to several variants of the commmon benchmark example of a flow around a cylinder in two dimensions, for the steady state and the time-dependent Navier--Stokes equations.

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Donnerstag, 21. 01. 2016, 14:00 Uhr (ESH)
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Dr. Ch. Merdon(WIAS Berlin)

Pressure-robust finite element methods for Navier-Stokes discretisationstandard mixed finite element methods suffer from a pressure-dependence of the a priori velocity error estimates due to a violation of the L2 orthogonality between divergence-free test functions and gradients. This instability is termed pressure-inrobustness and can cause severe velocity errors that scale with the Reynolds number.

The remedy comes in form of a variational crime that employs local reconstruction of the test functions onto Raviart-Thomas or Brezzi-Douglas-Marini functions. The talk focusses on efficient and minimal-invasive designs of these reconstruction operators for the first order Bernardi-Raugel and second order bubble enriched finite element methods and the observation that only the bubble test functions have to be reconstructed. Several numerical examples involving transient and Navier-Stokes flows compare and illustrate the robustness and potentials of the modified method.

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Donnerstag, 14.01. 2016, 14:00 Uhr (ESH)
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Dr. J. Fuhrmann(WIAS Berlin)

Generalized Nernst-Planck-Poisson systems: Modeling aspects and numerical approachesRecent developments in electrochemical modeling have lead to an increased interest in numerical simulations of electrolytic systems which are able to resolve the polarization boundary layer. Classically, the problem is formulated based on the Nernst-Planck-Poisson system for ion transport in a self-consistent electrical field. Various model improvements are currently discussed in order to take into account the volume constraint for solute concentrations.

The talk reviews a successful finite volume discretization strategy from semiconductor analysis and discusses problem reformulations which allow for its application in the context of electrolyte modeling based on improved Nernst-Planck equations. Special emphasis is made on the proper reflection of qualitative properties of the physical model at the discrete level and on synergies with numerical approaches to semiconductor modeling with generalized carrier statistic functions. Along with calculation results for benchmark examples, the influence of various model improvements is demonstrated.