The BEM with graded meshes for the electric field integral equation on polyhedral surfaces

TL;DR

This paper analyzes the convergence of boundary element methods using graded meshes for solving the electric field integral equation on polyhedral surfaces, introducing new stability results for Raviart-Thomas elements on anisotropic meshes.

Contribution

It establishes quasi-optimal convergence of Galerkin solutions with anisotropic graded meshes and introduces new stability properties of Raviart-Thomas interpolants on such meshes.

Findings

Proved quasi-optimal convergence under mild grading restrictions.

Developed new componentwise stability estimates for Raviart-Thomas interpolants.

Validated effectiveness of graded meshes for boundary element discretizations.

Abstract

We consider the variational formulation of the electric field integral equation on a Lipschitz polyhedral surface $\Gamma$. We study the Galerkin boundary element discretisations based on the lowest-order Raviart-Thomas surface elements on a sequence of anisotropic meshes algebraically graded towards the edges of $\Gamma$. We establish quasi-optimal convergence of Galerkin solutions under a mild restriction on the strength of grading. The key ingredient of our convergence analysis are new componentwise stability properties of the Raviart-Thomas interpolant on anisotropic elements.