Dr. Stefan A. Funken Institut für Angewandte Mathematik Universität Hannover Welfengarten 1 30167 Hannover GERMANY

Abstract for the 9th GAMM–Workshop on Parallel Multigrid Methods

Adaptive Multilevel Boundary Element Methods for the Single Layer Potential in IR^{2}

Stefan A. Funken

We consider weakly singular integral equations on polygons in IR^{2}. To obtain approximate solutions we use h-versions of the Galerkin Boundary Element Method. We show that the Multilevel Additive Schwarz operator with a hierarchical basis yields a preconditioned linear system whose condition number is bounded. We use the properties of the hierchical basis to derive an a posteriori error estimate for the difference between the exact solution and the additive multilevel solution. In order to compute good approximations with the Galerkin method we may need refined, non-uniform meshes, which may be produced by adaptive processes. For these meshes we show reliability and efficiency of our estimate. Based on this estimate we introduce an Adaptive Multilevel Algorithm with easily computable local error indicators. The theoretical results are illustrated by some numerical examples.

Dr. Stefan A. Funken Institut für Angewandte Mathematik Universität Hannover, Welfengarten 1, 30167 Hannover, GERMANY Phone:[49] 511–762–2230

E-Mail: funken@ifam.uni-hannover.de