A collocation method for high-frequency scattering by convex polygons

  • Authors:

  • S. Arden;S. N. Chandler-Wilde;S. Langdon

  • Affiliations:

  • Department of Mathematics, University of Reading, Whiteknights, P.O. Box 220, Berkshire RG6 6AX, UK;Department of Mathematics, University of Reading, Whiteknights, P.O. Box 220, Berkshire RG6 6AX, UK;Department of Mathematics, University of Reading, Whiteknights, P.O. Box 220, Berkshire RG6 6AX, UK

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2007

Quantified Score

Hi-index 7.29

Visualization

Abstract

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.