A fast hybrid Galerkin method for high-frequency acoustic scattering

摘要

We consider the scattering of a time-harmonic acoustic incident plane wave by a smooth convex object. We formulate this problem by the direct boundary integral method, using the classical combined potential approach. Based on the known asymptotics of the solution, we devise particular expansions, valid in various zones of the boundary. To achieve a good approximation at high frequencies with a relative low number of degrees of freedom, we propose a novel Galerkin boundary element method with a hybrid approximation space, consisting of the products of plane wave basis functions with piecewise polynomials supported on several overlapping meshes: a polynomial grading on the illuminated side and a geometric grading on the shadow side. Using the asymptotic expansions of the solution, we prove that, as k -> infinity, the number vertical bar D vertical bar of degree of freedom is able to decrease only very modestly to maintain a fixed absolute error bound (vertical bar D vertical bar similar to k(-1/12) is a typical behavior). Numerical experiments also show that the method achieves a better accuracy as k -> infinity, for a fixed number of degrees.