Multigrid Simulation for High-Frequency Solutions of the Helmholtz Problem in Heterogeneous Media

Abstract

The Helmholtz problem is hard to solve in heterogeneous media, in particular, when the wave number is real and large. The problem is neither coercive nor Hermitian symmetric. This article concerns the V-cycle multigrid (MG) method for high-frequency solutions of the Helmholtz problem. Since we need to choose at least 10--12 grid points per wavelength for stability, the coarse grid problem is still large. To solve the coarse grid problem efficiently, a nonoverlapping domain decomposition method is adopted without introducing another coarser subspace correction. Various numerical experiments have shown that the convergence rate of the resulting MG method is independent on the grid size and the wave number, provided that the coarse grid problem is fine enough for the solution to capture characteristics of the physical problem.