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In this paper we present and analyse a high accuracy method for computingwave directions defined in the geometrical optics ansatz of Helmholtz equationwith variable wave number. Then we define an "adaptive" plane wave space withsmall dimensions, in which each plane wave basis function is determined by suchan approximate wave direction. We establish a best $L^2$ approximation of theplane wave space for the analytic solutions of homogeneous Helmholtz equationswith large wave numbers and report some numerical results to illustrate theefficiency of the proposed method.

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