A Boundary Element Method for High Frequency Scattering by Convex Polygons

Standard boundary or finite element methods for problems of high frequency acoustic scattering by convex polygons have a computational cost that grows in direct proportion to the frequency of the incident wave. Here we present a novel Galerkin boundary element method, using plane wave basis functions on a graded mesh, for which the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency. Introduction We consider scattering of a given incident acoustic plane wave ui by a sound-soft bounded convex polygon Ω, with boundary Γ. The total acoustic field u satisfies ∆u+ ku = 0, in D := R\Ω̄, u = 0, on Γ, supplemented with an appropriate Sommerfeld radiation condition to ensure uniqueness of solution. This can be reformulated as the well known uniquely solvable second kind boundary integral equation for the unknown complementary boundary data ∂u/∂n ∈ L2(Γ);