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

Donnerstag, 2. 11. 2006, 14:00 Uhr (ESH)

Dr. L. Grasedyck   (Max-Plank-Institut Leipzig)
Hierarchical matrices based on domain decomposition

Abstract:
Most direct methods for sparse linear systems perform an LU factorisation of the original matrix after some reordering of the indices in order to reduce fill-ins. One such popular reordering method is the so-called nested dissection, which exploits the concept of separation. A favourable property of such an ordering is that a subsequent LU factorisation maintains a major part of this sparsity structure. In order to obtain a (nearly) optimal complexity, we propose to approximate all nonzero, off-diagonal blocks in H-matrix representation and compute them using H-matrix arithmetic. In the first part of the talk hierarchical matrices will be introduced based on a general clustering strategy. In the second part of the talk the clustering strategy will be based on domain decomposition and we present numerical examples that underline the almost linear scaling of the method in the number N of unknowns for problems that stem from elliptic partial differential equations. We compare the new and some standard clustering techniques as well as block-box variants.

Dienstag, 25. 4. 2006, 13:30 Uhr (ESH)

Dr. D. Hilhorst   (Université Paris Sud, France)
Convergence, a posteriori error estimates, and adaptivity for combined finite volume - finite element schemes on nonmatching grids

Abstract:
We propose and analyze a numerical scheme for nonlinear degenerate parabolic convection-diffusion-reaction equations in two or three space dimensions. We discretize the time evolution, convection, reaction, and sources terms on a given grid, which can be nonmatching and can contain nonconvex elements, by means of the cell-centered finite volume method. To discretize the diffusion term, we construct a conforming simplicial mesh with the vertices given by the original grid and use the finite element method. We allow for general inhomogeneous and anisotropic diffusion-dispersion tensors and use the local Peclet upstream weighting in order to only add the minimal numerical diffusion necessary to avoid spurious oscillations in the convection-dominated case. We prove the convergence of the scheme and illustrate its behavior on a numerical experiment. Finally we present some a posteriori error estimates.

Donnerstag, 23. 3. 2006, 14:00 Uhr (ESH)

A. Linke   (Freie Universität Berlin)
Computing incompressible flows by stabilized lowest-order Scott-Vogelius elements

Abstract:
n this talk results on a new stabilized mixed Finite Element Method for approximating the generalized Oseen equation are presented. In 2D $P_2-P_{-1}$ and in 3D $P_3-P_{-2}$ elements are used, which ensure exact mass conservation. These elements are the lowest-order Scott-Vogelius elements, which are LBB-stable on macro-element meshes. In 2D, the meshes are constructed by macro-triangles, where each big triangle is subdivided into three subtriangles. In 3D, there is an analogous construction. Since discrete solutions are always in $H_0\!(\mathit{div}\!)$, there is no need for the cumbersome grad-div stabilization. Hence, only convection must be stabilized. This is done by the face stabilization method, which penalizes jumps of the normal derivatives across interior faces. Since this stabilization affects only the diffusion-convection-reaction part of the discrete Oseen operator, the continuity equation is not changed and mass conservation is not violated - in contrast to SUPG stabilization. The method is rather insensitive to over-stabilization and seems to be promising for computing flows at high Reynolds numbers, due to exact mass conservation. It seems to deserve special attention that the discrete Oseen system can preserve many properties of the continuous Oseen equation. Comparative numerical studies are presented for 2D and 3D test problems.

Donnerstag, 2. 3. 2006, 14:00 Uhr (ESH)

Dr. Michael Fried   (Friedrich-Alexander-Universität Erlangen-Nürnberg)
A level set based adaptive finige element algorithm for image segmentation

Abstract:
We present an adaptive finite element algorithm for segmentation with denoising of multichannel images. It is based on a level set formulation of the Mumford--Shah approach proposed by Chan and Vese. The aim is to find homogeneous regions $\Omega^i$ and their boundaries $\Gamma$ inside a given, possibly noisy image $g: \Omega \to [0,1]^{N_c}$ and a piecewise smooth approximation $u$ to $g$ such that $u$ is smooth inside the segments $\Omega^i$ but may jump across the boundary $\Gamma$. Due to the heat equation like diffusion of the classical Mumford--Shah method, possible edges and details inside the segments will be blurred and eventually lost. To avoid this, we propose a slightly changed functional which leads to a TV like denoising inside the segments, hence is able to keep more details and edges while it is still denoising. The presented algorithm is able to switch between both methods such allowing a direct comparision of them and an easy adaption of the denoising properties to the actual given situation. Several numerical results including a test for convergence to a known solution, minimal partition of $3$D images and a comparision of the denoising qualities of different approaches are presented.

Donnerstag, 16. 2. 2006, 14:00 Uhr (ESH)

Prof. I. Rubinstein   (The Jacob Blaustein Institute for Desert Research, Israel)
From equilibrium to non-equilibrium electric double layer, electro-osmosis and electro-convective instability of electric conduction from an electrolite solution into a charge selective solid

Abstract:
Electro-convection is reviewed as a mechanism of mixing in the diffusion layer of a strong electrolyte adjacent to a charge-selective solid, such as an ion exchange (electrodialysis) membrane or an electrode. Two types of electro-convection in strong electrolytes may be distinguished: bulk electro-convection , due to the action of the electric field upon the residual space charge of a quasi-electro-neutral bulk solution, and convection induced by electro-osmotic slip, due to electric forces acting in the thin electric double layer of either quasi-equilibrium or non-equilibrium type near the solid/liquid interface. According to recent studies, the latter appears to be the likely source of mixing in the diffusion layer, leading to ‘over-limiting’ conductance in electrodialysis. Electro-convection near a uniform charge selective solid/liquid interface sets on as a result of hydrodynamic instability of one-dimensional steady state electric conduction through such an interface. We discuss instabilities of this kind appearing in the full electro-convective and limiting non-equilibrium electro-osmotic formulations. The short- and long-wave aspects of these instabilities are discussed along with the wave-number selection principles and possible sources of low frequency excess electric noise experimentally observed in these systems.

Mittwoch, 15. 2. 2006, 13:30 Uhr (ESH)

A. Bradji   (CMI, Centre de Mathematiques et Informatique, France)
Finite volume methods for elliptic problems

Abstract:
Our talk can be divided into two principal parts:
a) In the First Part, we consider a Second Order Elliptic Problem posed on an Intervall in R or on a Connected Polygonal Domain \Omega \subset R^2. We introduce an Admissible Mesh T in the sense of [1]. The Convergence Order of the Finite Volume Approximate Solution u^T is in general O(h) in both norms L^2 and H^1 norms. We suggest a technique, based on the so-called Fox's Difference Correction, which allows us to obtain a new Finite Volume Approximation u^T_1 of order O(h^\alpha), where \alpha equal to 2 or \frac{3}{2}. In addition to this, the new Finite Volume Approximation u^T_1 can be computed using the same matrix that used to compute the basic Finite Volume Solution u^T . This means that the computational cost of u^T_1 is comparable to that of u^T . This allows us, in case of \Omega is rectangle, to obtain Finite Volume Approximations of Arbitrary Order. Our technique can be extended to Improve the Convergence Order of the Finite Volume Approximate Solutions of some Parabolic, Hyperbolic and Nonlinear Equations.
b) In the Second Part, we suggest Finite Element and Finite Volume Schemes for a Coupled System of Elliptic Equations, modelling Electrical Conduction and Heat Diffusion with Ohmic losses. The Ohmic losses generate L^1 right hand side, which requires an adequate procedure, adapted from L^1 theory of Boccardo and Gallouet. Among the interesting paths to follow is to use the results of the First Part to Improve the Convergence of the Finite Volume Scheme cited in the Second Part.
References: [1] R. Eymard, T. Gallouet and R. Herbin: Finite Volume Methods. Handbook of Numerical Analysis. P. G.

Mittwoch, 18. 1. 2006, 14:00 Uhr (ESH)

Prof. G. Lube   (Universität Göttingen)
Stabilized FEM for incompressible flows. A critical review and new trends.

Abstract:
The numerical solution of the nonstationary, incompressible Navier-Stokes model can be splitted into linearized auxiliary problems of Oseen type. We present in a unique way dierent stabilization techniques of nite element schemes on isotropic meshes. First we describe the state-of-the-art for the classical residual-based streamline-upwind/pressure stabilizing (SUPG/PSPG) methods. Then we discuss recent symmetric stabilization techniques (edge-oriented stabilization, local and global projection-based stabilization) which avoid some drawbacks of the classical methods. These methods are closely related to the concept of variational multiscale methods which seems to provide a new approach to large eddy simulation. Finally, we give a critical comparison of these methods.