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

TL;DR

This paper develops an hp-version boundary element method for solving the electric field integral equation on polyhedral surfaces, explicitly accounting for solution singularities to improve convergence analysis.

Contribution

It provides an a priori error analysis for hp-BEM applied to EFIE with singular solutions, including explicit error estimates based on singularity types, mesh size, and polynomial degree.

Findings

Derived precise error estimates for hp-BEM on singular solutions.

Showed how singularity behavior affects convergence rates.

Validated the theoretical error bounds with numerical analysis.

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.