The hp-BEM with quasi-uniform meshes for the electric field integral equation on polyhedral surfaces: a priori error analysis

TL;DR

This paper provides an a priori error analysis for the hp boundary element method applied to the electric field integral equation on polyhedral surfaces, demonstrating optimal convergence rates under certain regularity assumptions.

Contribution

It introduces a rigorous a priori error estimate for the hp-BEM on quasi-uniform meshes for the electric field integral equation, extending theoretical understanding of the method's convergence.

Findings

Proves convergence rates of the hp-BEM in the energy norm.

Establishes error bounds based on mesh size h and polynomial degree p.

Validates the expected rate of convergence under regularity assumptions.

Abstract

This paper presents an a priori error analysis of the hp-version of the boundary element method for the electric field integral equation on a piecewise plane (open or closed) Lipschitz surface. We use H(div)-conforming discretisations with Raviart-Thomas elements on a sequence of quasi-uniform meshes of triangles and/or parallelograms. Assuming the regularity of the solution to the electric field integral equation in terms of Sobolev spaces of tangential vector fields, we prove an a priori error estimate of the method in the energy norm. This estimate proves the expected rate of convergence with respect to the mesh parameter h and the polynomial degree p.