A Multigrid Method for the Helmholtz Equation with Optimized Coarse Grid Corrections

Abstract

We study the convergence of multigrid schemes for the Helmholtz equation, focusing in particular on the choice of the coarse scale operators. Let $G_{\rm c}$ denote the number of points per wavelength at the coarse level. If the coarse scale solutions are to approximate the true solutions, then the oscillatory nature of the solutions implies the requirement $G_{\rm c} > 2$. However, in examples the requirement is more like $G_{\rm c} \gtrsim 10$, in a trade-off involving also the amount of damping present and the number of multigrid iterations. We conjecture that this is caused by the difference in phase speeds between the coarse and fine scale operators. Standard 5-point finite differences in two dimensions are our first example. A new coarse scale 9-point operator is constructed to match the fine scale phase speeds. We then compare phase speeds and multigrid performance of standard schemes with a scheme using the new operator. The required $G_{\rm c}$ is reduced from about 10 to about 3.5, with less damping present so that waves propagate over $>$ 100 wavelengths in the new scheme. Next, we consider extensions of the method to more general cases. In three dimensions, comparable results are obtained with standard 7-point differences and optimized 27-point coarse grid operators, leading to an order of magnitude reduction in the number of unknowns for the coarsest scale linear system. Finally, we show how to include perfectly matched layers at the boundary, using a regular grid finite element method. Matching coarse scale operators can easily be constructed for other discretizations. The method is therefore potentially useful for a large class of discretized high-frequency Helmholtz equations.