zurück zum aktuellen Programm
Seminar Numerische Mathematik / Numerical mathematics seminars
aktuelles Programm / current program
Donnerstag, 8. 12. 2011, 14:00 Uhr (ESH)
H. Tran (University of Pittsburgh)
Uncoupling the incompressible fluid flows in multi-physics and multi-domain applications
Many fluid flows in applications occur in multi-physics and multi-domain. The physical processes of each subproblem may be quite different and they can require different meshes, time steps and methods. Partitioned methods, which allow us to uncouple and solve the coupled problem by calls to the corresponding subproblems' codes, are highly desirable for such flows. We consider partitioned methods in two particular applications: the Stokes-Darcy problems and the reduced magnetohydrodynamics flows. The long time stability and accuracy of our proposed methods are analyzed. Experiment results verifying the theory are also given.
Donnerstag, 6.12. 2011, 13:30 Uhr (ESH)
Prof. E. Zhang (Oregon State University/Berlin Mathematical School)
Topological analysis and visualization of 2D asymmetric tensor fields
Asymmetric tensor fields contain much richer structures than their symmetric counterparts. In this talk I will review latest development in asymmetric tensor field analysis and visualization, with an emphasis on tensor field topology. Applications of such analysis and visualization include fluid dynamics and earthquake engineering.
Donnerstag, 24. 11. 2011, 14:00 Uhr (4. Etage, Raum 406)
Priv.-Doz. Dr. G. Thäter (Karlsruher Insitut für Technologie)
Auf Eulers Spuren im Dienste des Hochleistungsrechnens
Uns heute wohlvertraute - "klassische" - partielle Differentialgleichungen sind entstanden als Versuch, Naturphänomene zu verstehen und möglichst objektiv zu beschreiben. Durch Newtons Ideen wurde ein Prozess in Gang gesetzt, in dem sich viele kluge Menschen daran abgearbeitet haben, die Quintessenz des Fließens in Gleichungen zu destillieren. Dieser Prozess ist bis heute nicht abgeschlossen. Dies hat mehrere Gründe. Zum einen gibt es neue Konfigurationen und Materialien für die die klassischen Modelle nicht entwickelt worden sind. Zum anderen können wir alle Gleichungen numerisch mit großer Genauigkeit und Geschwindigkeit lösen und uns damit Problemen in neuen Größenordnungen zuwenden, wie z.B. der Wettervorhersage. Aber gerade hier ist es nötig, alle Gleichungen immer auch als Modelle zu sehen und zu hinterfragen, um geeignete numerische Verfahren zu wählen und Ergebnisse zu bewerten. In dem Vortrag möchte ich einen kleinen Einblick in typische Fragen des Arbeitsgebietes geben. Als Beispiel werden insbesondere natürliche Konvektionsprobleme dienen.
Donnerstag, 10. 11. 2011, 14:00 Uhr (ESH)
R. Schlundt (WIAS Berlin)
Maxwellgleichungen auf Tetraedergittern
Die Maxwellgleichungen werden in ihrer integralen Darstellung behandelt. Die Anwendung der Finiten-Integrations-Theorie (FIT) benötigt primäre und duale Gitter, die orthogonal bzw. unstrukturiert sein können. Die dualen Gitter werden mit Hilfe der baryzentrischen Koordinaten erzeugt. Während die Maxwellgleichungen auf diesen beiden Gittern exakt wiedergegeben werden, werden die Materialbeziehungen numerisch approximiert. Es werden zwei verschiedenen Approximationsmöglichkeiten betrachtet. Bei der Mikrozellen-Methode werden die primären Tetraederzellen in vier verschiedene Mikrozellen unterteilt. Die Informationen benachbarter Komponenten werden somit berücksichtigt. Bei der zweiten Methode werden zur Diskretisierung der Vektorfelder Whitney-Elemente benutzt. Die Basisfunktionen sind vektorwärtig und werden so konstruiert, dass sie den speziellen Forderungen an die Stetigkeit der Tangential- oder Normalkomponente genügen.
Donnerstag, 3. 11. 2011, 14:00 Uhr (ESH)
Prof. J. Novo (Universidad Autonoma de Madrid, Instituto de Ciencias Matematicas)
Mixed finite-element approximations to the Navier-Stokes equations: two-grid schemes and a posteriori error estimations
In the first part of the talk a two-grid scheme based on mixed finite element approximations to the incompressible Navier-Stokes equations is considered. In the first level of the method the standard mixed finite-element approximation over a coarse mesh is computed. In the second level the approximation is postprocessed by solving a discrete Oseen-type problem. In the second part of the talk a posteriori error estimates for mixed finite element discretizations of the Navier-Stokes equations are derived. We show that the task of estimating the error in the evolutionary Navier-Stokes equations can be reduced to the estimation of the error in a steady Stokes problem.
Donnerstag, 29. 9. 2011, 14:00 Uhr (ESH)
Dr. H. Si (WIAS Berlin)
Short Course: Tetrahedral Mesh Generation, algorithms and implementation
This course is arranged into two parts. The first part is about the fundamental problems in tetrahedral mesh generation, which include three-dimensional boundary recovery, mesh refinement, and anisotropic meshing. Algorithms which have theoretical guarantees and practical values for tetrahedral mesh generation are introduced. The second part is about software implementation. Efficient mesh data structures and some techniques for a robust implementation are introduced.
Donnerstag, 8. 9. 2011, 14:00 Uhr (ESH)
Prof. J. Shewchuk (University of California at Berkeley, USA)
Dynamic local remeshing for elastoplastic simulation
We describe methods for simulating physical domains that are substantially reshaped by plastic flow or fracture. We combine standard Lagrangian finite element methods with algorithms for dynamic remeshing, which update a volume mesh undergoing radical deformations so that its tetrahedra do not deteriorate below a fixed quality threshold. Our dynamic mesher is conservative: it replaces as few tetrahedra as possible, and thereby limits the visual artifacts and artificial diffusion that would be introduced if we repeatedly remeshed the domain from scratch. It also locally refines and coarsens a mesh, and even creates anisotropic tetrahedra, wherever a simulation requests it. Our simulation method addresses a range of material behavior from purely elastic to highly plastic, with particular advantages for objects that span both extremes at once. We illustrate these features with animations of elastic and plastic behavior, extreme deformations, and fracture.
Our software can also improve finite element meshes so their worst tetrahedra have a level of quality substantially better than those produced by any previous method for tetrahedral mesh generation or "mesh clean-up." Our mesh improvement program, named Stellar, usually improves meshes so that all dihedral angles are between 34 and 131 degrees. Stellar is freely available to the public at http://www.cs.berkeley.edu/~jrs/stellar/ .
Dienstag, 6. 9. 2011, 14:00 Uhr (ESH)
Prof. J. Shewchuk (University of California at Berkeley, USA)
Theoretically guaranteed Delaunay Mesh Generation - In practice
This short course is an introduction to triangular and tetrahedral mesh generation algorithms, especially those based on Delaunay triangulations. Coverage is restricted to algorithms that have two desirable qualities at once: they are mathematically guaranteed to generate high-quality meshes, and they work well enough in practice to compete with traditional, heuristic algorithms in engineering applications. Topics covered include a short review of Delaunay triangulations and constrained Delaunay triangulations; extensive coverage of Delaunay refinement algorithms for triangular and tetrahedral mesh generation, including methods by Chew, Ruppert, Ungor, Boivin/Ollivier-Gooch, Miller/Walkington/Pav, and me; handling of 2D domains with curved boundaries; handling of 2D and 3D domains with small angles; sliver elimination; anisotropy; and the octree-based isosurface stuffing algorithm.
Donnerstag, 1. 9. 2011, 14:00 Uhr (ESH)
G. R. Barrenechea (University of Strathclyde, Scotland)
Computable error bounds and an adaptive selection of the stabilization parameter for the Stokes problem
In this talk I will present some recent results on a-posteriori error estimation for the Stokes problem. The estimators are fully computable and present a robust and rigorous bound for the error. First, the case of a stable pair of elements (the Fortin-Soulie element) is treated, and the estimator takes advantage of the particularities of this element. In the second part of the talk this framework is extended to stabilized finite element methods, where a general analysis is carried out and bounds covering a wide range of stabilized finite element methods for the Stokes problem are presented. Some applications to the a-posteriori selection of the stabilization parameter are also presented.
Dienstag, 30. 8. 2011, 14:00 Uhr (ESH)
Dr. P. Knobloch (Charles University, Institute of Numerical Mathematics, Czech Republic)
Improved stability and error analysis for a class of local projection stabilizations applied to the Oseen problem
We consider a class of local projection stabilizations with projection spaces defined on (possibly) overlapping sets applied to the Oseen problem. We prove that the underlying bilinear form satisfies an inf-sup condition with respect to a stronger norm than coercitivy suggests. A modification of the stabilization of the convection allows an optimal estimation of the consistency error. A priori estimates in the stronger norm and in the L2 norm for the pressure are established. Discontinuous pressure approximations are included in the analysis.
Dienstag, 21. 6. 2011, 13:30 Uhr (ESH)
Dr. D. Kourounis (Stanford University)
Adjoint formulations for gradient-based optimization of compositional flow
Freitag, 17. 6. 2011, 10:15 Uhr (ESH)
Prof. Gary L. Miller (Carnegie Mellon University, USA)
Algorithm design using spectral graph theory
Spectral Graph Theory is the interplay between linear algebra and combinatorial graph theory. One application of this interplay is a nearly linear time solvers Symmetric Diagonally Dominate system (SDD). This seemingly restrictive class of systems has received substantial interest and in the last 15 years both algorithm design theory and practical implementations have made substantial progress. There is also a growing number of problems that can be efficiently solved using SDD solvers including: image segmentation, image denoising, finding solutions to elliptic equations, computing maximum flow in a graph, graph sparsification, and graphics.
Theoretically, we have recently shown that if $A$ is an $nxn$ SDD matrix with $m$ entries the linear system $Ax =b$ can be solve to constant precision in $O(m \log n)$ time ignoring lower order factors. On the implementation side, the latest code runs in linear time experimentally for large sparse systems. In fact the problem sizes are now large enough that on modern multicore workstations the memory bandwidth is a limiting factor. Compression techniques are now used to substantially speedup the solver.
This represents joint work with: Guy Blelloch, Ioannis Koutis, Richard Pang, Kanat Tangwongsan, and David Tolliver
Donnerstag, 14. 6. 2011, 14:00 Uhr (ESH)
Dr. R. Hanke-Rauschenbach (MPI für Dynamik komplexer technischer Systeme, Magdeburg)
Oscillations and pattern formation in an electrochemical membrane reactor exposed to H_2/CO mixtures
Dienstag, 24. 5. 2011, 13:30 Uhr (ESH) sowie Donnerstag, 26. 5. 2011, 14:00 Uhr (ESH)
Dr. M. Yvinec (INRIA, France)
Titel 24. 5. 2011 CGAL mesh / Titel 26. 5. 2011 Anisotropic Delaunay meshes
Abstract für 26. 5. 2011:
Anisotropic meshes are triangulations of a given domain in the plane or in higher dimensions, with elements elongated along prescribed directions. Anisotropic triangulations have been shown to be well suited for interpolation of functions or solving PDEs. Assuming that the anisotropic shape requirements for mesh elements are given through a spatially variable metric field, the talk will present a new approach to anisotropic mesh generation. The approach relies on the well established notions of restricted Delaunay triangulation and Delaunay refinement and on a new notion of locally uniform anisotropic meshes, in which the star of each vertex is Delaunay with respect to the metric at the location of the vertex. The notion works in any dimension and leads to a mesh generation algorithm that is easy to implement and provides anisotropic meshes for 3D domains or anisotropic surface meshes. In both cases, mesh elements are shaped according to the metric field.
Donnerstag, 12. 5. 2011, 14:00 Uhr (ESH)
Dr. S. Matera (Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin)
A first-principles based multiscale modeling approach from the electronic to the continuum regime for model catalyst studies
I present an electronic structure - or first principles - based multiscale modeling approach to heterogeneous catalysis. The focus is on how to efficiently integrate first-principles kinetic Monte Carlo (1p-kMC) simulations for the surface chemistry into a fluid dynamical treatment of mass and heat transport in the reactor. Using CO oxidation at RuO2 (110) as a prototypical example I will first introduce the 1p-kMC methodology and compare this with phenomenological kinetics commonly used in chemical engineering. I will then present how we couple 1p-kMC with fluid dynamical simulations and demonstrate the results for two simple flow geometries.
Donnerstag, 5. 5. 2011, 14:00 Uhr (ESH)
Prof. G. Matthies (Universität Kassel,
MooNMD - a program package based on mapped finite element methods
Donnerstag, 7. 4. 2011, 14:00 Uhr (ESH)
Prof. R. Kornhuber (FU Berlin)
On Ritz-Galerkin methods for stochastic partial differential equations
Since the pioneering work of Ghanem and Spanos twenty years ago, Hilbert-space approaches to stochastic partial differential equations and associated Ritz-Galerkin or collocation methods based on polynomial ansatz spaces (polynomial chaos) have attracted more and more attention. After a short introduction of the basic concepts with the help of a Poisson equation with stochastic coefficients, we concentrate on extensions to variational inequalities and applications to Richards equation for saturated/unsaturated groundwater flow.
Donnerstag, 17. 2. 2011, 14:00 Uhr (ESH)
Prof. U. Krewer (MPI Magdeburg)
Influence of reactant transport on electrochemical energy systems
Electrochemical energy systems convert the chemical energy of reactants into electrical energy. Prerequisite for any such reaction is sufficient transport of reactants to and removal from the electrodes. This talk will discuss the influence of reactants on operation of three electrochemical systems: the direct methanol fuel cell and its system, the alkaline direct methanol fuel cell, and the zinc air secondary battery. Based on simulation results, requirements for cell material, system design and system operation are developed. It is furthermore shown that analysis of the dynamic cell voltage response yields information on reactant concentration. Finally, the concept of micro separation is shown to be applicable to recycle reactants at any system orientation.
Donnerstag, 20. 1. 2011, 14:00 Uhr (ESH)
N. Ahmed (Universität Magdeburg)
Operator splitting and alternating direction Galerkin methods applied to population balance equation
In population balance equations, the distribution of entities depends not only on space and time but also on their own properties referred as internal coordinates. The operator splitting method is used to transform the whole problem into two or more unsteady subproblems of smaller complexity. In our case, the first subproblem is a time-dependent convection-diusion problem while the second one is a one-dimensional transport problem with pure advection. We use backward Euler method to discretize the subproblems in time. Since the first problem is convection dominated, the local projection (LPS) method is used as stabilization in space. To discretize the pure transport problem a discontinuous Galerkin (dG) method is used. We show the unconditional stability and optimal error estimates. The aim in alternating direction scheme is the same as in the operator splitting method, i.e., reducing the high dimensional problem into a set of lower ones. First we discretize the whole problem in space and internal coordinate using LPS and dG method, respectively. Then the backward Euler time stepping method is used to obtain a fully discrete scheme. The matrices in the fully discrete scheme are tensor products of the space and internal coordinate direction. Therefore we are able to derive two steps alternating direction method. Based on an equivalent one step formulation we discuss stability and convergence of the method. Finally, we present some computational results supporting our theoretical predictions.
Donnerstag, 13. 01. 2011, 14:00 Uhr (ESH)
Prof. R. Eymard (Universite Paris Est, Departement de mathematiques)
Low degree nonconforming methods for the biharmonic problem
Some schemes for the Stokes problem in 2D involve the resolution of the biharmonic problem. We consider new methods, from finite volumes with orthogonality condition to general mesh method, including P1 finite elements, for the approximation of the biharmonic problem. These methods are shown to be convergent in any space dimension. This is confirmed by numerical examples.