Diplomanden- und Doktorandenseminar
Numerische Mathematik WS 2008/09

Interessenten sind herzlich eingeladen!

Weitere Vorträge siehe auch:

• Kolloquium der Arbeitsgruppe Numerik, Modellierung, Differentialgleichungen
• Seminar des Fachgebiets Optimierung bei partiellen Differentialgleichungen (Gruppe Tröltzsch)

• Diplomanden- und Doktorandenseminar SS 2008
• Diplomanden- und Doktorandenseminar WS 2007/08
• Diplomanden- und Doktorandenseminar SS 2007
• Diplomanden- und Doktorandenseminar WS 2006/07
• Diplomanden- und Doktorandenseminar SS 2006
• Diplomanden- und Doktorandenseminar WS 2005/06
• Diplomanden- und Doktorandenseminar SS 2005
• Diplomanden- und Doktorandenseminar WS 2004/05
• Diplomanden- und Doktorandenseminar SS 2004
• Diplomanden- und Doktorandenseminar WS 2003/04
• Diplomanden- und Doktorandenseminar SS 2003
• Diplomanden- und Doktorandenseminar WS 2002/03

Abstracts zu den Vorträgen:

Abstract:
Let A be a dense square complex matrix. A palindromic eigenvalue problem is of the form A*x=lambda*A'*x, where A' is the transpose of A. One possibility to solve this structured eigenvalue problem is to find a unitary square matrix Q such that Q'*A*Q is in anti triangular form. In this talk it is assumed that A is already close to antitriangular form. A method is presented to iteratively drive the elements above the anti diagonal to zero. Applications are mixed precision arithmetic, corrections of unstable algorithms and subspace tracking for palindromic eigenvalue problems.

Abstract:
In continuous-time control theory one frequently has to deal with inputs that are step functions like, for example, the Heaviside function. If, however, the system description is not an ordinary differential equation but a differential-algebraic equation of higher index the concept of solution for such systems can not be generalized in a trivial way. To handle such systems, typically, distributions are introduced in the sense of "Schwartz, L. (1950−1951), Théorie des distributions" where a distribution is defined to be a continuous and linear mapping from some test function space into the real numbers. In this talk a different, more algebraic approach will be presented.

Abstract:
We consider a new adaptive finite element algorithm for elliptic eigenvalue problems. The accuracy of the method is guaranteed by using a posteriori estimators based on the backward error analysis. In this talk we will show error estimates and numerical examples for model problem together with theoretical analysis. Additionally some further ideas of improving our method will be discussed.

Abstract:
The higher-ranking goal of the diploma thesis is to capture the input/output (i/o) behaviour of a physical system governed by the linear Oseen equations in a closed mathematical formulation to enable an effective application of distributed control. A practical approach is to discretize the spaces of the input and output functions to obtain an approximating matrix representation of the i/o operator. For an adaptive procedure error estimates by means of an analytical form of the i/o map are of maximum interest. Considering the semidiscretized Oseen equation as a linear differential algebraic equation (DAE) with constant coefficients, one has an explicit solution formula. If applied to a system with a control term, it can function as the base for a closed-form i/o map. The approach admits a wide range of spatial discretization methods, a special focus lies on mixed q1p0 finite elements. The investigation is accompanied and backed by numerical tests. In my talk I will give an insight into the problem definition, present current results, and address upcoming issues.

Abstract:
A posteriori error estimators for non symmetric eigenvalue problems of PDE's have been first studied by [Heuveline and Rannacher, A posteriori error control for finite element approximations of elliptic eigenvalue problems, 2001]. In this talk error estimates for the eigenvalue error of a simple convection-diffusion problem will be shortly derived using the variational formulation rather than the non linear ansatz of Becker and Rannacher. In addition error estimators for the eigenvalue error based on averaging techniques are developed. In the case of linear P₁ finite elements and piecewise constant coefficients, reduced error estimates of the residual and averaging error estimators are presented. Moreover several DWR techniques are compared numerically and two new weighted error estimators are proposed. The first new estimator utilises an auxiliary Raviard-Thomas solution and the second uses the averaging techniques in combination with the ideas of DWR.