Regularity of mixed boundary value problems in ℝ³and boundary element methods on graded meshes

Abstract

Abstract The solution of the three‐dimensional mixed boundary value problem for the Laplacian in a polyhedral domain has special singular forms at corners and edges. A ‘tensor‐product’ decomposition of those singular forms along the edges is derived. We present a strongly elliptic system of boundary integral equations which is equivalent to the mixed boundary value problem. Regularity results for the solution of this system of integral equations are given which allow us to analyse the influence of graded meshes on the rate of convergence of the corresponding boundary element Galerkin solutions. We show that it suffices to refine the mesh only towards the edges of the surfaces to regain the optimal rate of convergence.