A Galerkin Boundary Element Method for High Frequency Scattering by Convex Polygons

Abstract

In this paper we consider the problem of time‐harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well‐known second kind combined‐layer‐potential integral equation. We provide a proof that this equation and its adjoint are well‐posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.