## Doktorandenseminar des WIAS

# Numerische Mathematik und Wissenschaftliches Rechnen

Forschungsgruppe

zurück zum aktuellen Programm

## Seminar Numerische Mathematik / Numerical mathematics seminars

aktuelles Programm / current program

_{FG3 intern}

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Dienstag, 11. 12. 2018, 13:30 Uhr (ESH)
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Dr. W. Dreyer(WIAS Berlin)

Kinetic theory of incompressible non-newtonian fluidsWe consider a non-newtonian fluid consisting of a newtonian solvent and polymer molecules as solute. The kinetic theory models the polymer molecules as dumbbells and compiles dynamical equations that embody internal dumbbell interactions and interactions of the dumbbell with the surrounding fluid via its velocity gradient. The dynamical equations imply a generalized Fokker-Planck equation for a distribution function of the system of polymer molecules. Then the Fokker--Planck equation is used to derive macroscopic evolution equations for certain mean values. For given velocity gradient of the fluid, three evolution equations are needed to end up with a closed equation system for the polymer molecules. The corresponding three thermodynamic variables are the number density, the mean squared end-to end distance of a polymer molecule and the polymer stress. Finally we prove an H-theorem and solve the subtle problem of thermodynamic consistency.

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Donnerstag, 29. 11. 2018, 14:00 Uhr (ESH)
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Prof. J.H.M. ten Thije Boonkkamp(Eindhoven University of Technology, Niederlande)

Flux vector approximation schemes for systems of conservation lawsConservation laws in continuum physics are often coupled, for example the continuity equations for a reacting gas mixture or a plasma are coupled through multi-species diffusion and a complicated reaction mechanism. For space discretisation of these equations we employ the finite volume method. The purpose of this talk is to present novel flux vector approximation schemes that incorporate this coupling in the discretisation. More specifically, we consider as model problems linear advection-diffusion systems with a nonlinear source and linear diffusion-reaction systems, also with a nonlinear source. The new flux approximation schemes are inspired by the complete flux scheme for scalar equations, see [1]. An extension to systems of equations is presented in [2]. The basic idea is to compute the numerical flux vector at a cell interface from a local inhomogeneous ODE-system, thus including the nonlinear source. As a consequence, the numerical flux vector is the superposition of a homogeneous flux, corresponding to the homogeneous ODE-system, and an inhomogeneous flux, taking into account the effect of the nonlinear source. The homogeneous ODE-system is either an advection-diffusion system or a diffusion-reaction system. In the first case, the homogeneous flux contains only real-valued exponentials, on the other hand, in the second case, also complex-valued components are possible, generating oscillatory solutions. The inclusion of the inhomogeneous flux makes that all schemes display second order convergence, uniformly in all parameters (Peclet and Damköhler numbers). The performance of the novel schemes is demonstrated for several test cases, moreover, we investigate several limiting cases.

References

[1] J.H.M. ten Thije Boonkkamp and M.J.H. Anthonissen, ``The finite volume-complete flux scheme for advection-diffusion-reaction equations'', J. Sci. Comput., 46, 47--70, (2011).

[2] J.H.M. ten Thije Boonkkamp, J. van Dijk, L. Liu and K.S.C. Peerenboom, ``Extension of the complete flux scheme to systems of comservation laws'', J. Sci. Comput., 53, 552--568, (2012).

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Donnerstag, 15. 11. 2018, 14:00 Uhr (ESH)
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Prof. J. Shewchuk(University of California at Berkeley, USA)

Higher-quality tetrahedral mesh generation for domains with small angles by constrained Delaunay refinementMost algorithms for guaranteed-quality tetrahedral mesh generation create Delaunay meshes. Delaunay triangulations have many good properties, but the requirement that all tetrahedra be Delaunay often forces mesh generators to overrefine where boundary polygons meet at small angles---that is, they produce too many tetrahedra, making them too small. Relaxing the Delaunay property makes it possible both to reduce overrefinement and to obtain higher-quality tetrahedra.

We describe a provably good algorithm that generates high-quality meshes that are *constrained* Delaunay triangulations, rather than purely Delaunay. This change has two big advantages: it allows us to generate higher-quality tetrahedra than purely Delaunay algorithms do, and it allows us to cope much more successfully with domains that have small angles. Both theory and an implementation show that our algorithm does not overrefine near small domain angles.

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Donnerstag, 13. 09. 2018, 14:00 Uhr (ESH)
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V. Milos(Charles University Prague)

Thermodynamics and electrochemical impedance spectroscopy - 1D model of YSZ electrolyte and triple phase boundary interfaceElectrochemical impedance spectroscopy (EIS) together with reliable models provides a powerful tool for analysing processes taking place in solid oxide cells (SOC). The presented model is derived within the framework of non-equilibrium thermodynamics [1, 2]. It is a suitable tool for description of the fundamental processes in charged mixtures [3, 4, 5], especially on interfaces, such as surface adsorption, bulk and surface diffusion, electrochemical reactions and formation of charged double layers. Special attention is dedicated to the thermodynamically consistent derivation of the boundary conditions and to handling of the electrical double layer. The contribution contains a derivation of 1D model for YSZ electrolyte and tripple boundary interface and also includes numerical solution of model partial differential equations. The implementation of finite element method (FEM) is done using open source library FEniCS. In order to explore the parametric space, EIS was simulated for a wide range of kinetic parameters of the model.

References:

1. De Groot, S. R., Mazur, P. Non-equilibrium thermodynamics., 1984.

2. Guhlke, Clemens. ``Theorie der elektrochemischen Grenzfläche'', TU Berlin (2015).

3. Dreyer, W., Guhlke, C., Landstorfer, M. Electrochem. Commun. 43 (2014): 75-78.

4. Dreyer, W., Clemens, Guhlke, C., Müller, R. Ph. Chem. Chem. Ph. 17.40 (2015): 27176-27194.

5. Landstorfer, M., Guhlke, C., Dreyer, W. Electrochimica Acta 201 (2016): 187-219.

und 15:00 Uhr

Dr. Mine Akbas(Duzce University, Türkei)

The analogue of grad-div stabilization in DG methods for incompressible flows: Limiting behavior and extension to tensor-product meshesGrad-div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems, for mitigating velocity errors that are sometimes called poor mass conservation. Such errors arise due to the relaxation of the divergence constraint in classical mixed methods and are excited whenever the spacial discretization has to deal with comparably large and complicated pressures. In this contribution, an analogue of grad-div stabilization for Discontinuous Galerkin methods is studied. Here, the key is the penalization of the jumps of the normal velocities over facets of the triangulation, which controls the measure-valued part of the distributional divergence of the discrete velocity solution. Our contribution is twofold: first, we characterize the limit for arbitrarily large penalization parameters, which shows that the stabilized nonconforming Discontinuous Galerkin methods remain robust and accurate in this limit; second, we extend these ideas to the case of non-simplicial meshes; here, broken grad-div stabilization must be used in addition to the normal velocity jump penalization, in order to get the desired pressure robustness effect. Several numerical examples illustrate the theory and show their relevance for the simulation of practical, nontrivial flows.

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Mittwoch, 11. 07. 2018, 11:00 Uhr (ESH)
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Dr. E. Meneses Rioseco(Leibniz-Institut für Angewandte Geophysik, Hannover)

Modeling, simulation and optimization of geothermal reservoir -- THMC (thermo-hydraulical, mechanical, chemical) processes involvedGeothermal energy constitutes an important renewable energy that can significantly contribute to the energy transition in Germany. The optimization of the geothermal energy extraction in the underground is a multi-variable and multi-scale optimization problem that has not been extensively addressed in the scientific community. Understanding and minimizing the risks, in particular induced seismicity, that the exploitation of deep geothermal energy may pose is of major public and scientific concern.

Robust mathematical models and numerical methods are required to capture the complex interplay of thermal, hydraulic, mechanical and chemical (THMC) processes in heterogeneous and anisotropic medium domains. In addition, the distribution and implementation of material properties in topologically complex 3D modeling domains is an ongoing effort. Considerable effort is required to realistically implement the complex architecture that results from highly heterogeneous and anisotropic geologic media.

This work presents a general overview of the mathematical formulations of the transport phenomena of mass, energy and momentum in porous and fractured geothermal reservoirs and emphasizes the current challenges faced for the simulation of the main risks (e.g. induced seismicity) that result from the complex interaction of THMC processes during the operation of geothermal plants and the need of complex optimization methods.

This presentation aims at encouraging the collaboration between the Leibniz Institute of Applied Geophysics (LIAG) and the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) within the framework of the Leibniz MMS network.

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Dienstag, 26. 06. 2018, 13:30 Uhr (ESH)
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Prof. T. Iliescu(Virginia Tech, USA)

Physically-constrained data-driven correction for reduced order modeling of fluid flowsWe propose a data-driven correction reduced order model (DDC-ROM) framework for the numerical simulation of fluid flows, which can be formally written as DDC-ROM = Galerkin-ROM + Correction. The new DDC-ROM is constructed by using ROM spatial filtering and data-driven ROM closure modeling (for the Correction term). Furthermore, we propose a physically-constrained DDC-ROM (CDDC-ROM), which aims at improving the physical accuracy of the DDC-ROM. The new physical constraints require that the CDDC-ROM operators satisfy the same type of physical laws (i.e., the Correction term's linear component should be dissipative and the Correction term's nonlinear component should conserve energy) as those satisfied by the fluid flow equations. To implement these physical constraints, in the data-driven modeling step, we replace the unconstrained least squares problem with a constrained least squares problem. We perform a numerical investigation of the new CDDC-ROM and standard DDC-ROM for a 2D channel flow past a circular cylinder at Reynolds numbers

Re=100, Re=500, andRe=1000. To this end, we consider a reproductive regime as well as a predictive (i.e., cross-validation) regime in which we use as little as50 %of the original training data. The numerical investigation clearly shows that the new CDDC-ROM is significantly more accurate than the DDC-ROM in both regimes.

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Donnerstag, 14. 06. 2018, 14:00 Uhr (Raum 405/406)
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Dr. Myfanwy Evans(TU Berlin)

Surfaces and tanglingIn biological systems, hyperbolic surfaces are readily identifiable as critical to function. This talk will introduce a few interesting systems that motivate thinking in this direction, followed by a more formal mathematical process that aims to probe the realm of complicated three-dimensional structure, in particular highly tangled materials.

und 15:00 Uhr

Prof. L. Shen(University of the Chinese Academy of Sciences, Beijing)

Implicitizing rational tensor product surfaces using the resultant of three moving planesImplicitizing rational surfaces is a fundamental computational task in Computer Graphics and Computer Aided Design. Ray tracing, collision detection, and solid modeling all benefit from implicitization procedures for rational surfaces. Recently a popular method is based on analysis of moving planes (syzygies): construct the implicit matrix using moving planes or compute the resultant of μ-basis. The moving planes method relies on the construction of appropriate moving planes. The μ-bases for rational surfaces are difficult to compute. Moreover, μ-bases for a rational surface often have high degrees, so these resultants generally contain many extraneous factors.

Here we develop fast algorithms to implicitize rational tensor product surfaces by computing the resultant of three moving planes corresponding to three syzygies with low degrees. These syzygies are easy to compute, and the resultants of the corresponding moving planes generally contain fewer extraneous factors than the resultants of the moving planes corresponding to μ-bases. We predict and compute all the possible extraneous factors that may appear in these resultants.

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Dienstag, 29. 05. 2018, 13:30 Uhr (ESH)
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G. Panasenko(University of Lyon, Frankreich & University of Chile)

Asymptotic analysis of wave propagation in a laminated beam with contrasting stiffness of the layersThe wave equation in a thin laminated beam is considered in the case when the ratio of the thickness and the length is a small parameter e, while the ratio of the stiffness coefficients of the layers is a great parameter w. The question of an approximation of such a beam by the one dimensional models is discussed. It is shown that the classical homogenized approximation is valid for the case of sufficiently small value of the product of e square by w. If this product is not small then the classical model should be replaced by a multicomponent homogenization model analogous to [1, 2]. The waves in such multicomponent beam model can have multiple wave velocities. This result generalizes [3] Section 2.6.

[1] Panasenko, G.P. ``Averaging of processes in strongly in homogeneous media'', Doklady Akademii Nauk SSSR, 1988, 298, 1, 76--79 (in Russian). English transl. in Soviet Phys. Dokl., 1988, 33.

[2] Panasenko, G.P. ``Multicomponent homogenization of processes in strongly non-homogeneous structures'' Mathematics USSRSbornik, 1990, 181, 1, 134--142 (in Russian); English transl. in Math. USSR Sbornik, 1991, 69, 1, 143--153.

[3] Panasenko, G.P. ``Multi-Scale Modelling for Structures and Composites'', Springer, Dordrecht, 2005, 398 pp.

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Donnerstag, 03. 05. 2018, 14:00 Uhr (ESH)
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Dr. H. Stephan(WIAS Berlin)

One million perrin pseudo primes including a few giantsPseudoprimes are integers that are no primes but behave like them in some sense. Suppose we have a theorem like the following: If n is a prime, then statement A(n) holds. In general, the opposite is not true: It may be that A(n) holds, but n is a composite number, a so-called pseudo prime with respect to statement A. Pseudoprimes are interesting if they are very rare, as for instance Perrin's pseudoprimes, the smallest of which is 271441. The talk introduces pseudoprimes which are based on recurrent sequences. In addition, some new numerical results on Perrin's pseudoprimes and a fast algorithm for their calculation are presented.

und 15:00 Uhr

L. Feierabend(Universität Duisburg-Essen)

Model development for flowing slurry electrodes in zinc-air batteriesSlurry or suspension electrodes are gaining a renewed interest for large-scale energy storage technologies due to potentially higher energy densities compared to redox flow batteries [1]. A suspension electrode typically consists of a liquid electrolyte and a solid material, which is chemically active and electrically conducting. For the investigated zinc-air flow batteries employing a suspension electrode, microscopic zinc particles are suspended in an aqueous potassium hydroxide electrolyte [2, 3]. Sedimentation of the metallic zinc is minimized by adding a gelling agent to the electrolyte. The gelling agent and the high particle loading lead to a pseudo-plastic rheological behavior. When the flowable suspension electrode is pumped through the channels of the zinc-air fuel cell setup, the dynamic percolation network within the suspension dictates the active electrode surface area and the maximum discharge power density. Therefore, it is desirable to investigate the influence of the local flow conditions on the particle percolation and consequently the electrochemical performance of the battery cell with adequate simulation methods. In the developed three-dimensional numerical model for the flowing suspension electrode, the complex, non-Newtonian two-phase flow is approximated by a coupling of an Eulerian continuum description for the electrolyte and a discrete, Lagrangian particle description for the motion and interaction of the microscopic particles. The partial differential equations for momentum, species, charge and energy are discretized by the finite volume method and implemented in the OpenFOAM library [4]. The particle motion including multiple simultaneous particle contacts is described with the discrete element method using the LIGGGHTS library [5]. Coupling between the particle and fluid phase is realized with the CFD-DEM method using the CFDEM library [5], where an empirical description accounts for the momentum exchange between the non-Newtonian fluid and the densely distributed particle assemblies. Due to relatively high particle concentrations, the volume displacement by the particles is considered in the electrolyte continuum model. A half-cell model for the anode part of the zinc-air flow battery is implemented, which accounts for the flow characteristics via the described CFD-DEM coupling method. Simultaneously, the charge and species transport is considered according to the classic porous electrode theory by Newman [6] adapted with newer formulations for concentrated electrolytes [7]. In contrast to electrodes with a static porous matrix, the heterogeneous local porosities change temporally depending on the evolving particle distributions. Additionally, the active electrode surface area in each finite volume is dependent on the percolation network from the considered local point to the current collector surface. To estimate the basic parameters for the charge transfer and species transport, a flat-plate electrode with a flowing electrolyte without particles is investigated. In this rather simple reference case, discontinuities of the current density close to the electrode-electrolyte-interface could be observed, which should be inherently excluded due to the model formulations. These inconsistencies will be discussed in detail. Funding by the Federal Ministry for Economic Affairs and Energy (BMWi) of the project ``ZnPLUS -Wiederaufladbare Zink-Luft-Batterien zur Energiespeicherung'' and the project ``ZnMobil - Mechanisch und elektrisch wiederaufladbare Zink-Luft-Batterie für automobile Anwendungen'' is gratefully acknowledged.

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Mittwoch, 18. 04. 2018, 13:30 Uhr (ESH)
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Prof. P. Deuring(Universitè du Littoral ''Côte d'Opale'', Frankreich)

Stabilitäts- und Fehlerschranken einer FE-FV-Diskretisierung von Konvektions-Diffusionsgleichung: exponentielle Abhängigkeit von ParameternWir betrachten eine FE-FV-Methode zur Bestimmung von Näherungslösungen von Konvektions-Diffusionsgleichungen. Bei dieser Methode wird der Diffusionsterm mittels Crouzeix-Raviart-Elementen diskretisiert, während man den Konvektionsterm durch baryzentrische finite Volumen approximiert. Wir gehen der Frage nach, ob sich Stabilitäts- und Fehlerschranken für diese Methode finden lassen, die von keiner Größe exponentiell abhängen, insbesondere nicht vom Kehrwert des Diffusionskoeffizienten.

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Donnerstag, 12. 04. 2018, 14:00 Uhr (ESH)
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Prof. M. J. Neilan(University of Pittsburgh, USA)

Inf-sup stable Stokes pairs on barycentric refinements producing divergence-free approximationsWe construct several stable finite element pairs for the Stokes problem on barycentric refinements in arbitrary dimensions and for any polynomial degree. A key feature of the spaces is that the divergence maps the discrete velocity space onto the discrete pressure space; thus, when applied to models of incompressible flows, the pairs yield divergence-free velocity approximations. The key ingredients to prove these results are local inf-sup stability estimates and a modification of Bernardi-Raugel bubble functions. This is joint work with Johnny Guzman.

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Donnerstag, 01. 03. 2018, 14:00 Uhr (ESH)
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Dr. F. Dassi(Politecnico di Milano, Italien)

Recent advancements of the Virtual Element Method in 3DThe Virtual Element Method is a novel way to discretize a partial differential equation. It avoids the explicit integration of shape functions and introduces an innovative construction of the stiffness matrix so that it acquires very interesting properties and advantages. One among them is the possibility to apply the VEM to general polygonal/polyhedral domain decomposition, also characterized by non-conforming and non-convex elements.

In this talk we focus on the definition/construction of the Virtual Element functional spaces in three dimensions and how apply this new strategy to set a standard Laplacian problem in 3D.

Finally we test the method to show its robustness with respect to element distortion and the polynomial approximation degree $k$. Then, we move to more involved cases: convection-diffusion-reaction problems with variable coefficients and magnetostatic Maxwell equations.

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Donnerstag, 01. 02. 2018, 14:00 Uhr (ESH)
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Dr. E. Sinibaldi(Italian Institute of Technology)

Selected modeling approaches for biomedical applications and biorobotics toolsThe effective deployment of theranostic agents (including drugs), smart?materials?based systems and interfaces to target regions (prospectively) in the human body where to perform the sought actions also requires theoretical models and physical tools. Modeling the delivery and the (remote) actuation/stimulation of the aforementioned agents/systems/interfaces, in particular, permits to compensate for experimental conditions hard to characterize and to better interpret the experimental results, thus paving the way for quantitative therapy design and control. This is complemented by model-based design of related experimental devices, flexible tools able to safely navigate anatomical pathways, and miniaturized effectors. In this talk I will firstly present the solution to an inverse problem, namely to determine the velocity profile in a vessel cross-section starting from the flow-rate, which is relevant to pulsatile biological flows (blood and cerebro-spinal fluid flows), with application, e.g., to magnetic particle targeting. Then I will address numerical models for intrathecal drug-delivery (drug infusion in the cerebro-spinal fluid, also accounting for transport to the spinal cord) and simplified analytical models for intra-tissue drug-delivery (particle transport into porous/poroelastic media, with application to intratumoral thermotherapy). Finally, I will overview model-based approaches for: piezoelectric nanoparticle-mediated cell stimulation; real-scale physical models of the blood-brain barrier; flexible biorobotics probes; miniature (bioinspired) actuators and effectors.

und 15:00 Uhr

L. Blank(WIAS Berlin)

Towards a simple and robust finite element method for the numerical simulation of porous media flowThe topic of this talk is the numerical simulation of flow through porous media based on the Brinkman model as a unified framework that allows the transit between Darcy and Stokes problems. Therefore an unconditionally stable low order finite element approach, which is robust with respect to the physical parameters, is proposed. This approach is based on the combination of stabilized equal order finite elements with a non-symmetric (penalty-free) Nitsche method for the weak imposition of essential boundary conditions. Focusing on the two-dimensional case, optimal a priori error estimates in a mesh-dependent norm, which allows to extend the results also to the Stokes and Darcy limits, are obtained.

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Donnerstag, 18. 01. 2018, 14:00 Uhr (ESH)
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Prof. W. Dreyer(WIAS Berlin)

Modeling of non-Newtonian fluids without material frame indifferenceIn the fifties the modeling of

non-Newtonian fluidsinitiated the search for invariant time derivatives with respect to certain space time transformations. ThenEuclidean Transformationswere selected to establish thePrinciple of Material Frame Indifferenceand moreover, the introduction of N^{th}grade fluids. However, the two concepts lead to serious inconsistencies in both thermodynamic modeling and experimental observations. In 1986 the subject has been resolved in a remarkable paper by I. Müller and K. Wilmanski. In this lecture we show how an appropriate model ofnon-Newtonian fluidsmay be incorporated inContinuum Thermodynamicsas it was laid down by D. Bothe and W. Dreyer.

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Donnerstag, 11. 01. 2018, 14:00 Uhr (ESH)
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Prof. G. Lube(Georg-August-Universität Göttingen)

Why exactly divergence-free H(div)-conforming FEM for transient incompressible flows?Recent results show that one can reconstruct classical inf-sup stable H1-conforming FEM for incompressible flow problems to be pressure-robust. Exactly divergence-free FEM are pressure-robust per construction. In particular, exactly divergence-free H(div)-conforming FEM combine excellent stability and conservation properties with minimal stabilization. We show that the numerical analysis of such methods allows to separate the (static) linear Stokes regime and the nonlinear dynamic regime. Semi-robust error estimates w.r.t. the Reynolds number will be given. In a hybridized form, H(div)-conforming FEM are suitable for large scale computation. Applications of the approach to two- and three-dimensional problems of vortex dynamics will be presented for high Reynolds numbers.

und 15:00 Uhr

Dr. A. Rasheed(Lahore University of Management Sciences, Pakistan)

Influence of magnetic field on dendrites during solidification of binary mixturesA phase field model has been proposed which incorporates convection and magnetic field in an isothermal environment. A numerical scheme is proposed and numerical analysis of model in two-dimensional geometry is performed. The numerical stability and error analysis of this approximation scheme which is based on mixed finite-element method are performed. An application of a nickel-copper binary alloy is considered. Influence of various magnetic fields on the dendrites during the solidification process has been discussed.