SEMINAIRE D'ANALYSE NUMERIQUE
Année universitaire 2008-2009
Jeudi 2 octobre 2008 :
Ernst STEPHAN
(Hannover)
Boundary Elements - Principles and Applications.
First, we consider boundary integral operators for the Laplacian in polygonal domains and apply Mellin techniques to investigate the mapping properties of these operators in weighted Sobolev spaces. The crucial Mellin symbol calculus goes back to the early work of Costabel and Stephan. As an application we analyse the explicit behaviour of the solution of the Dirichlet problem at a corner with angle $\pi/2$, where special logarithmic terms appear in the solution. Then we consider a nonlinear transmission problem and apply Costabel's symmetric FE/BE-coupling procedure. We comment on the hp-version as optimal Galerkin method. Then we present recent research results. We apply the boundary element method to Signorini and friction contact problems and discuss the use of the hp-version BEM combined with the mortar projection method and adaptive procedures. Finally, we give for the classical perfect conductor problem (from electromagnetics) a BEM approach which uses only standard Lagrangian elements. Here the convergence analysis is based on our classical concept of strong ellipticity of the boundary integral operators involved.