Time harmonic accoustic scattering by a convex polygon is considered. S. Chandler-Wilde and S. Langdon have recently proposed an integral equation based method for the high frequency scattering problem. Using a detailed regularity analysis of the solution, they were able to design an h-version trial space that has approximation properties that depend only logarithmically on the wave number. The key features are a) the ability to identify the leading order (in the wave number) behavior of the solution and b) a precise characterization of the solution behavior near the vertices of the polygon. Since the approximation order is fixed, the achievable convergence rate is algebraic. In this talk, we extend their work to the hp-version of the BEM. It is shown that the solution can be approximated at an exponential rate from the trial space; the problem size required to achieve a given accuracy grows only logarithmically with the wave number. In this talk, we also address the question of how to set up the stiffness matrix with work independent of the wave number.
This is joint work with S. Langdon, University of Reading.