Functional Analysis and Applications Seminar 

14/12/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building
Ernst Stephan, Universität Hannover, Alemanha

Corner singularities and Mellin symbols

Firstly, we show mapping properties (within countably normed spaces) of several boundary integral operators acting on polygons (e.g. the weakly singular single layer potential operator for the Laplacian and the corresponding double layer potential operator). Our analysis of the boundary integral equations is based on Mellin techniques and uses the Mellin symbols of the integral operators. Functions (with corner singularities but) belonging to countably normed spaces can be approximated very efficiently by the $hp$-version of the boundary element method when refining the mesh size $h$ and increasing the polynomial degree $p$. Secondly, we analyze the Dirichlet problem for the Laplacian in a polygonal domain. Applying Mellin techniques to the boundary integral equation we show that the solution has a decomposition into regular and singular parts which blow up at certain exceptional angles. We derive a modified decomposition which depends continuously on the angle and can be used efficiently for boundary element computations.