Wavenumber-explicit analysis for the Helmholtz $h$-BEM: error estimates and iteration counts for the Dirichlet problem

TL;DR

This paper provides sharp bounds on GMRES iteration counts and mesh size requirements for the Helmholtz boundary element method, ensuring error control and efficiency as the wavenumber increases, especially for analytic, curved obstacles.

Contribution

It establishes the first sharp bounds on GMRES iteration growth with wavenumber and on mesh refinement for the Helmholtz BEM in scattering problems.

Findings

GMRES iteration count grows linearly with wavenumber for analytic obstacles.

Mesh size must decrease proportionally to inverse wavenumber for quasi-optimality.

New bounds are derived for nontrapping obstacles to maintain solution accuracy.

Abstract

We consider solving the exterior Dirichlet problem for the Helmholtz equation with the $h$-version of the boundary element method (BEM) using the standard second-kind combined-field integral equations. We prove a new, sharp bound on how the number of GMRES iterations must grow with the wavenumber $k$ to have the error in the iterative solution bounded independently of $k$ as $k\rightarrow \infty$ when the boundary of the obstacle is analytic and has strictly positive curvature. To our knowledge, this result is the first-ever sharp bound on how the number of GMRES iterations depends on the wavenumber for an integral equation used to solve a scattering problem. We also prove new bounds on how $h$ must decrease with $k$ to maintain $k$-independent quasi-optimality of the Galerkin solutions as $k \rightarrow \infty$ when the obstacle is nontrapping.