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We present a novel hybrid numerical-asymptotic boundary element method forhigh frequency acoustic and electromagnetic scattering by penetrable(dielectric) convex polygons. Our method is based on a standard reformulationof the associated transmission boundary value problem as a direct boundaryintegral equation for the unknown Cauchy data, but with a nonstandard numericaldiscretization which efficiently captures the high frequency oscillatorybehaviour. The Cauchy data is represented as a sum of the classical geometricaloptics approximation, computed by a beam tracing algorithm, plus a contributiondue to diffraction, computed by a Galerkin boundary element method usingoscillatory basis functions chosen according to the principles of theGeometrical Theory of Diffraction. We demonstrate with a range of numericalexperiments that our boundary element method can achieve a fixed accuracy ofapproximation using only a relatively small, frequency-independent number ofdegrees of freedom. Moreover, for the scattering scenarios we consider, theinclusion of the diffraction term provides an order of magnitude improvement inaccuracy over the geometrical optics approximation alone.

A hybrid numerical-asymptotic boundary element method for high frequency scattering by penetrable convex polygons