Implementation of a Boundary Element Method for High Frequency Scattering by Convex Polygons

We consider scattering of a time-harmonic acoustic plane wave by a sound-soft convex polygon. To achieve good approximation at high frequencies with a relatively low number of degrees of freedom, we recently proposed a novel Galerkin boundary element method, for which the approximation space consists of piecewise polynomials supported on a graded mesh, with smaller elements adjacent to the corners of the polygon, multiplied by traces of plane waves on the boundary. Using this scheme, it was shown in [2] via both a rigorous error analysis and numerical results that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency of the incident wave. Here we discuss issues relating to the implementation of this scheme.