Seminar Numerische Mathematik / Numerical mathematics seminars aktuelles Programm / current program

Donnerstag, 09. 12. 2004, 14:00 Uhr

Prof. Dr. A. Reusken   (Rheinisch-Westfälische Technische Hochschule Aachen, Lehrstuhl für Numerische Mathematik)
Schnelle iterative Löser für diskrete Stokes-Gleichungen

Abstract:
3 moderne block-iterative Verfahren werden miteinander verglichen. Dazu werden theoretische Konvergenzresultate und Ergebnisse von numerischen Experimente behandelt.

Dienstag, 23. 11. 2004, 10:00 Uhr

Ch. Ziegler   (Fraunhofer-Institut für Solare Energiesysteme)
Investigation of the dynamics of the proton exchange membrane fuel cell

Abstract: A theoretical study on the dynamics of the proton exchange membrane fuel cell (PEMFC) is presented. This contribution contains the development of a mathematical model for the PEMFC, the validation of the model and a study on the dynamic behavior of the PEMFC. The mathematical model presented is time-dependent, one-dimensional and isothermal. The following transport processes are described: migration of electrons and protons, multicomponent diffusion of the gaseous species, diffusion and electro-osmotic drag of water across the membrane. The phase transition of water is considered. The Butler-Volmer equation is used for the description of the catalyst layers. The validation of the model is performed by comparison of measured data taken at a reference cell. The model is applied to study the dynamic behavior of the PEMFC. Different physical processes influence the dynamic behavior of the PEMFC, for example, the double layer capacitance, convection and diffusion within the electrodes, heat conduction, convective heat transfer and flooding and drying-out of the electrodes and the membrane. A model-based analysis of the impact of these processes on the dynamic behavior of the PEMFC is performed.

Dienstag, 23. 11. 2004, 10:30 Uhr

D. Gerteisen   (Fraunhofer-Institut für Solare Energiesysteme)
Modelistic ac impedance study on porous electrodes of proton exchange membrane fuel cells using an agglomerate model

Abstract:
A one-dimensional model of the PEM fuel cell cathode is developed to analyse AC impedance spectra and polarisation curves. The porous gas diffusion electrode is assumed to consist of a network of dispersed catalyst (Pt/C) forming spherically shaped agglomerated zones that are filled with electrolyte. The coupled differential equation system describes: ternary gas diffusion in the backing (, water vapour), Fickian diffusion and Tafel kinetics for the oxygen reduction reaction (ORR) inside the agglomerates, proton migration with ohmic losses and double-layer charging in the electrode. Measurements are made of a temperature-controlled fuel cell with a geometric area of 1.4 × 1.4. Lateral homogeneity is ensured by using a high air stoichiometry of 12. The experimentally observed doubling of the Tafel slope is reproduced by the model due to diffusion limitation inside the agglomerates. The model also predicts the behaviour of measured impedance spectra. Model parameters e.g. Tafel slope, ionic resistance and agglomerate radius are varied. It is found that the density of the active agglomerates changes due to the humidification of the electrode and that the Tafel slope changes according to a change in the ORR mechanism. A sensitivity analysis of the model parameters is conducted.

Donnerstag, 08. 07. 2004, 14:00 Uhr

A. Linke   (FU Berlin, DFG-FZT 86)
Algebraische Mehrgitter-Methoden und (symmetrische) Sattelpunktprobleme

Abstract:
Im Vortrag werden einige grundlegende Ideen von Algebraischen Mehrgitter-Methoden (AMG) vorgestellt. Zur Sprache kommen klassische Verfahren wie das Ruge-Stüben-AMG f¨r symmetrische M-Matrizen und neuere Methoden für symmetrische positiv-definite Matrizen nach Falgout und Vassilevski. Am Ende des Vortrags wird auf die Schwierigkeiten eingegangen, wenn man AMG-Methoden für symmetrische Sattelpunktprobleme konstruieren möchte.

Donnerstag, 08. 07. 2004, 15:00 Uhr

Dr. J. Bloch   (FU Berlin, DFG-FZT 86)
Branch continuation and bifurcations of coupled reaction-diffusion equations

Abstract:
An algorithm for branch continuation and detection of bifurcations for large systems of diffusion-reaction equations will be presented. A test-function was specially designed to make the algorithm efficient for systems with large sparse Jacobians, and to avoid the omission or spurious attribution of bifurcation points. Its application is illustrated with the well-known Brusselator.

Donnerstag, 01. 07. 2004, 14:00 Uhr

Prof. M. Grote   Prof. M. Grote (University of Basel)
Nonreflecting boundary conditions for multiple scattering problems

Donnerstag, 24. 06. 2004, 14:00 Uhr

M. Lipinski   (Ruhr-Universität Bochum)
Two a posteriori error estimators for a saddle-point formulation of the Poisson equation with Dirichlet boundary conditions

Abstract:
By introducing a Lagrangian multiplier, which can be interpreted as the normal derivative on the boundary, we will create a weak formulation of the Poisson equation with Dirichlet boundary conditions on a polyhedral domain and a corresponding discretization. We will show that the variational and the discrete problems yield unique solutions. We will further consider a Scott-Zhang type interpolator for trace spaces, in order to approximate the given boundary data. We will then establish inverse estimates for these approximations. By doing so, we will overcome the "non-localness" of trace spaces. Finally, we will formulate two a posteriori error estimators, one based on estimating the residual, the other one based on solving local auxiliary problems. We will show that both estimators are efficient and reliable.

Donnerstag, 17. 06. 2004, 14:00 Uhr

G. Enchery   (Institut Francais du Petrole, France)
Numerical approximation of a two-phase flow problem in a porous medium with space discontinuous capillary forces

Abstract:
We consider a simplified model of a two-phase flow through a heterogeneous porous medium. Focusing on the capillary forces motion, a nonlinear degenerate parabolic problem is approximated in a domain shared in two homogeneous parts. Each subdomain is characterized by its relative permeability and capillary curves which are functions of the phase saturation. We first give a weak form of the conservation equations on the whole domain. This weak form includes a new general expression of the conditions that the traces of the phase saturations must satisfy at the interface between the two domains. We then propose a finite volume scheme for the approximation of the solution. This scheme is shown to converge to a weak solution in 1D, 2D or 3D domains. We conclude this talk by presenting some numerical tests.

Donnerstag, 17. 06. 2004, 15:00 Uhr

Prof. R. Eymard   (Université de Marne-la-Vallée, France)
Convergence of a finite volume scheme for the transient Navier-Stokes equations

Donnerstag, 27. 05. 2004, 14:00 Uhr

H. Wilke, K.A. Cliffe   (MBI Berlin)
A modular software concept for the Institute of Crystal Growth

Abstract:
The majority of physical problems to be solved numerically in the departments of IKZ are governed by nonlinear PDE's. The variety and complexity of crystal growth facilities lead to different types of PDE's and bc's usually not implemented in commercially available software products. Moreover, a realistic simulation of growth processes mostly needs a 3D and/or transient approach. In order to meet these requirements the software must be both flexible and highly efficient. Our concept consists of a number of modules around a central software core. The latter does the work needed in all circumstances and links the modules. Unlike this fixed core, the modules can be exchanged according to the requirements of the problem to be solved. At the moment we propose four main modules, i.e. for grid generation, incorporation of equations, solvers and postprocessing. Using this strategy the benefits are twofold. On one hand there are several state of the art free software products supplied in the web and on the other hand a wrapper around the core serves as an adjustable interface configured by the user. It seems to be clear that the core software has an outstanding position in this software scheme. The ENTWIFE cfl package has been chosen for this purpose because it is flexible, relatively cheep and well tested. It runs on almost all hardware platforms and can be used already by adding the default modules supplied with the core package. However, these modules should be replaced by better ones step by step as we would like to explain during the talk.

Donnerstag, 06. 05. 2004, 14:00 Uhr

Prof. W. Govaerts   (Rijksuniversiteit Gent, Dept. of Applied Mathematics and Computer Science)
Numerical methods for periodic orbits and their bifurcations

Abstract:
The Matlab program MATCONT allows to compute families of limit cycles, flip, fold and Neimark Sacker bifurcations of limit cycles, and of branch points of cycles. It can perform branch switching at branch points and period-doubling points. MATCONT discretizes the BVP for limit cycles as in AUTO [1] and CONTENT [3], i.e. by orthogonal collocation. The systems that arise in this way are typically sparse and their sparsity increases with the number of test intervals used in the discretization. The sparsity of the linearized systems is exploited by using the Matlab sparse matrix routines. We discuss in some detail the implementation of the algorithms in [2] for the continuation of flip, fold and Neimark-Sacker bifurcations as well as the more recent implementation of branch points of limit cycles. These use minimally extended systems, i.e. we append one scalar equation to the definition of limit cycles in the case of flip and fold; we introduce an additional variable and append two equations in the case of Neimark-Sacker. Branch points also require two additional equations. A related issue is the computation of normal form coefficients for bifurcations of limit cycles. We concentrate on some technical-numerical issues, in particular the relation between the undiscretized systems (functions) and the discretized systems (long vectors) present in the computer. The duality between functions represented by their values in the meshpoints of a mesh and their evaluation values in the collocation points of the mesh intervals requires a careful handling in the algorithms that we discuss. It appears to be related to the duality between left and right null spaces of singular matrices and operators and is helpful in the implementation of efficient algorithms.
References:
[1] E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Yu. A. Kuznetsov, B. Sandstede and X. J. Wang, AUTO97-AUTO2000 : Continuation and Bifurcation Software for Ordinary Differential Equations (with Hom-Cont), User's Guide, Concordia University, Montreal, Canada (1997-2000). (http://indy.cs.concordia.ca).
[2] E. J. Doedel, W. Govaerts and Yu. A. Kuznetsov, Computation of Periodic Solution Bifurcations in ODEs using Bordered Systems, SIAM Journal on Numerical Analysis 41,2 (2003) 401-435.
[3] Yu. A. Kuznetsov and V. V. Levitin,CONTENT: Integrated Environment for analysis of dynamical systems. CWI, Amsterdam (1997): ftp://ftp.cwi.nl/pub/CONTENT

Donnerstag, 22. 04. 2004, 14:00 Uhr

Ch. Hilgers   (RWTH Aachen, Geologie-Endogene Dynamik)
The role of fluids during deformation - inferences from veins

Abstract:
Fluids play a major role in the earth' s crust. They trigger brittle deformation due to reduction of the total stress, forming fluid conduits in the subsurface, and may cause precipitation of economically important minerals. Precipitates seal the rock's porosity and fractures, often displayed in cores and rock exposures. In northern Germany and in the North Sea, salt domes may induce growth of anhydrite and halite on discrete fluid pathways, which blocks off further fluid ingress. Precipitates on discrete fluid paths such as fractures are called veins. In this study, we aim at a better understanding of the regional fluid flow regime around salt structures. Investigating the natural vein microstructure, we will derive the boundary conditions during vein formation at app. 3-4 km depth. Using see-through analogue experiments, we demonstrate how the vein microstructure varies at different supersaturation and flow rates. In these experiments, we are able to observe the growing microstructure at a grain scale. Coupled with numerical simulations, a model of precipitation patterns and sealing predictions may be established, applicable to anhydrite and halite sealing in the Central European Basin system.

Donnerstag, 11. 03. 2004, 14:00 Uhr

Prof. Dr. M.A. Efendiev   (Universität Stuttgart, Mathematisches Institut A)
Longtime behaviour of solutions of a nonlinear reaction-diffusion systems arising in modeling of biofilms

Abstract:
We consider a class of highly degenerate reaction-diffusion models which describe the spatial spreading of biomass during the development of microbial films.It comprises two non-standard diffusion effects,degeneracy as in the porous medium equations and a fast diffusion. The existence of a unique bounded solution and a global atractor is proved in dependence of the bondary conditions.This is achieved by studying an auxiliary approximating sequence of systems of nondegenerate evolution equations and the construction of a Lipschitz continous semigroup.

Donnerstag, 19. 02. 2004, 14:00 Uhr

Erik Burman   (Department of Mathematics, Ecole Polytechnique Federale, Lausanne, Switzerland)
Edge stabilization: an interior penalty method for the incompressible Navier-Stokes equation

Abstract:
Recently a new method was introduced for the stabilization of H^1 conforming Galerkin approximations. The idea is to add an interior penalty term giving L^2-control of the jump of the gradient over element edges to the standard Galerkin formulation. The method has been successfully applied and analysed in the case of convection-diffusion equations and the generalized Stokes' equations in previous work. In this talk we will discuss the extension of the analysis to the Oseen's equations, i.e. Stokes' equations supplemented with a convective term. For the of interior penalty formulation applied to this problem we have proven optimal and quasi optimal a priori error estimates that are independent of the local Reynolds number. Moreover we will discuss the possibility of using this stabilization technique for the velocities in combination with inf-sup satisfying mixed finite element formulations. Some numerical examples will be presented.