Spectral Analysis of the Discrete Helmholtz Operator Preconditioned with a Shifted Laplacian

Shifted Laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the Helmholtz equation. In this paper we present a comprehensive spectral analysis of the Helmholtz operator preconditioned with a shifted Laplacian. Our analysis is valid under general conditions. The propagating medium can be heterogeneous, and the analysis also holds for different types of damping, including a radiation condition for the boundary of the computational domain. By combining the results of the spectral analysis of the preconditioned Helmholtz operator with an upper bound on the GMRES-residual norm, we are able to provide an optimal value for the shift and to explain the mesh-dependency of the convergence of GMRES preconditioned with a shifted Laplacian. We illustrate our results with a seismic test problem.