Multigrid Methods for Helmholtz Problems: A Convergent Scheme in 1D Using Standard Components

We analyze in detail two-grid methods for solving the 1D Helmholtz equation discretized by a standard finite-difference scheme. We explain why both basic components, smoothing and coarse-grid correction, fail for high wave numbers, and show how these components can be modified to obtain a convergent iteration. We show how the parameters of a two-step Jacobi method can be chosen to yield a stable and convergent smoother for the Helmholtz equation. We also stabilize the coarse-grid correction by using a modified wave number determined by dispersion analysis on the coarse grid. Using these modified components we obtain a convergent multigrid iteration for a large range of wave numbers. We also present a complexity analysis which shows that the work scales favorably with the wave number.