A collocation method for high frequency scattering by convex polygons

Arden, S., Chandler-Wilde, S. N. ORCID: https://orcid.org/0000-0003-0578-1283 and Langdon, S. (2007) A collocation method for high frequency scattering by convex polygons. Journal of Computational and Applied Mathematics, 204 (2). pp. 334-343. ISSN 0377-0427 doi: 10.1016/j.cam.2006.03.028

Abstract/Summary

We consider the problem of scattering of a time-harmonic acoustic incident plane wave by a sound soft convex polygon. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave. Recently Chandler–Wilde and Langdon proposed a novel Galerkin boundary element method for this problem for which, by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space, they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency. Here we propose a related collocation method, using the same approximation space, for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the Galerkin scheme, but with a substantially reduced computational cost.