Natural hp-BEM for the electric field integral equation with singular solutions

Abstract

We apply the hp-version of the boundary element method (BEM) for the numerical solution of the electric field integral equation (EFIE) on a Lipschitz polyhedral surface G. The underlying meshes are supposed to be quasi-uniform triangulations of G, and the approximations are based on either Raviart-Thomas or Brezzi-Douglas-Marini families of surface elements. Non-smoothness of G leads to singularities in the solution of the EFIE, severely affecting convergence rates of the BEM. However, the singular behaviour of the solution can be explicitly specified using a finite set of power functions (vertex-, edge-, and vertex-edge singularities). In this paper we use this fact to perform an a priori error analysis of the hp-BEM on quasi-uniform meshes. We prove precise error estimates in terms of the polynomial degree p, the mesh size h, and the singularity exponents.