Solving Helmholtz equation at high wave numbers in exterior domains

This paper presents effective difference schemes for solving the Helmholtz equation at high wave numbers. Considering the problems in the polar and spherical coordinates, we show that pollution-free difference schemes can be constructed for annulus and hollow sphere domains, but the "pollution effect" cannot be avoided in the neighborhood of the origin of the coordinate. Using the fact that the "pollution effect" appears in a small neighborhood of the origin of the coordinate, we propose a local mesh refinement approach so that a fine grid is applied only for a small region near the origin of the coordinate and a much larger grid size is employed in the remaining domain. The most attractive feature of this approach is that the computational storage can be significantly reduced while keeping almost the same numerical accuracy. Moreover, by applying a two-level local refinement technique, we can further enhance the performance of the proposed scheme. Numerical simulations are reported to verify the effectiveness of the proposed schemes for the Helmholtz problem in exterior domains.