Hybrid numerical-asymptotic approximation for high frequency scattering by penetrable convex polygons

We consider time-harmonic scattering by penetrable convex polygons, a Helmholtz transmission problem. Standard numerical schemes based on piecewise polynomial approximation spaces become impractical at high frequencies due to the requirement that the number of degrees of freedom in any approximation must grow at least linearly with respect to frequency in order to represent the oscillatory solution. High frequency asymptotic methods on the other hand are non-convergent and may be insufficiently accurate at low to medium frequencies. Here, we design a hybrid numerical-asymptotic boundary element approximation space that combines the best features of both approaches. Specifically, we compute the classical geometrical optics solution using a beam tracing algorithm, and then we approximate the remaining diffracted field using an approximation space enriched with carefully chosen oscillatory basis functions. We demonstrate via numerical simulations that this approach permits the accurate and efficient representation of the boundary solution and the far field pattern.