A Wavenumber Independent Boundary Element Method for an Acoustic Scattering Problem

Abstract

In this paper we consider the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data, a problem which models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve good approximation at high frequencies with a relatively low number of degrees of freedom, we propose a novel Galerkin boundary element method, using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that, on each element, the approximation space contains polynomials (of degree $\nu$) multiplied by traces of plane waves on the boundary. We prove stability and convergence and show that the error in computing the total acoustic field is ${\cal O}(N^{-(\nu+1)} \log^{1/2}N)$, where the number of degrees of freedom is proportional to $N\log N$. This error estimate is independent of the wavenumber, and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity.