### Abstract

Standard boundary integral methods for high frequency scattering problems, with piecewise polynomial approximation spaces, suffer from the debilitating restriction that the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to the frequency. For problems of acoustic scattering by sound soft convex polygons, it has recently been demonstrated by Chandler-Wilde and Langdon that by including plane wave basis functions supported on a graded mesh in the approximation space, with smaller elements closer to the corners of the polygon, convergence rates that depend only logarithmically on the frequency can be achieved. In this talk we review this approach, and consider some improvements and extensions to more complicated scattering problems.