A Fully Discrete Collocation Method for High Frequency Scattering by Convex Polygons

We consider the numerical solution of the problem of time-harmonic acoustic scattering in two dimensions by a sound soft convex polygon. Standard boundary or finite element methods with piecewise polynomial approximation spaces have a computational cost that grows at least linearly with respect to the frequency of the incident wave. By including in the approximation space the products of plane wave basis functions with piecewise polynomials supported on a graded mesh, with the grading optimally adapted to the decay of the diffracted waves away from corners of the polygon, it has previously been shown that an error of best approximation that depends only logarithmically on the frequency of the incident wave can be achieved. To achieve this result via a Galerkin scheme requires the numerical evaluation of many highly oscillatory double integrals; to avoid this difficulty, a related collocation scheme has also previously been proposed. Some potential improvements to this scheme are discussed here.