The hp-Version of the BEM with Geometric Meshes in 3D

Abstract

The solutions of three-dimensional elliptic boundary value problems in polyhedral domains or on screens have special singular forms at corners and edges. If those problems are converted via the direct method into boundary integral equations, the solutions of the latter inherit those edge and corner singularities. The singularities disturb the convergence of numerical schemes, e g finite element or boundary element methods. Numerical schemes converge as fast as the solution is approximated by the chosen trial functions due to the quasioptimality of the Galerkin method. Thus the rate of convergence is determined by the regularity of the solution. Even for smooth data, the regularity is reduced by edge and corner singularities, and therefore the convergence of these procedures is much slower than in case of smooth domains.