A plane wave method based on approximate wave directions for two dimensional Helmholtz equations with large wave numbers

In this paper we present and analyse a high accuracy method for computing wave directions defined in the geometrical optics ansatz of Helmholtz equation with variable wave number. Then we define an “adaptive” plane wave space with small dimensions, in which each plane wave basis function is determined by such an approximate wave direction. We establish a best $L^2$ approximation of the plane wave space for the analytic solutions of homogeneous Helmholtz equations with large wave numbers and report some numerical results to illustrate the efficiency of the proposed method.