Convergence of the Natural hp-BEM for the Electric Field Integral Equation on Polyhedral Surfaces

TL;DR

This paper proves that the natural hp-BEM method for solving the electric field integral equation on polyhedral surfaces converges optimally as the mesh is refined or polynomial degree increases.

Contribution

It establishes the asymptotic quasi-optimality of the hp-BEM for EFIE on polyhedral surfaces, a significant theoretical advancement.

Findings

Galerkins solutions are asymptotically quasi-optimal

Convergence holds for sufficiently fine meshes or high polynomial degrees

Method applies to both open and closed polyhedral surfaces

Abstract

We consider the variational formulation of the electric field integral equation (EFIE) on bounded polyhedral open or closed surfaces. We employ a conforming Galerkin discretization based on div-conforming Raviart-Thomas boundary elements (BEM) of locally variable polynomial degree on shape-regular surface meshes. We establish asymptotic quasi-optimality of Galerkin solutions on sufficiently fine meshes or for sufficiently high polynomial degree.