The h-p boundary element method for solving 2- and 3-dimensional problems

In this survey paper we report on recent developments of the hp-version of the boundary element method. We consider weakly singular and hypersingular integral equations on polygons and polyhedral surfaces. We show that the Galerkin solutions (computed with the hp-version on geometric meshes) converge exponentially fast towards the exact solutions of the integral equations. The implementation of the hp-version of the boundary element method is discussed and adaptive algorithms are given. Model problems for the coupling of finite elements and boundary elements are also considered. Numerical results are presented which underline the theoretical results.