On the convergence of the hp-BEM with quasi-uniform meshes for the electric field integral equation on polyhedral surfaces

TL;DR

This paper proves the quasi-optimal convergence of the hp boundary element method for the electric field integral equation on polyhedral surfaces using quasi-uniform meshes and a new H^{-1/2}(div)-conforming p-interpolation operator.

Contribution

It introduces a novel H^{-1/2}(div)-conforming p-interpolation operator and establishes convergence results for hp-BEM on polyhedral surfaces.

Findings

Established quasi-optimal convergence of hp-BEM

Developed a new H^{-1/2}(div)-conforming p-interpolation operator

Proved quasi-stability with respect to polynomial degrees

Abstract

In this paper the hp-version of the boundary element method is applied to the electric field integral equation on a piecewise plane (open or closed) Lipschitz surface. The underlying meshes are supposed to be quasi-uniform. We use H(div)-conforming discretisations with quadrilateral elements of Raviart-Thomas type and establish quasi-optimal convergence of hp-approximations. Main ingredient of our analysis is a new H^{-1/2}(div)-conforming p-interpolation operator that assumes only H^r \cap H^{-1/2}(div)-regularity (r > 0) and for which we show quasi-stability with respect to polynomial degrees.