Shifted Laplace preconditioners have attracted considerable attention as
a technique to speed up convergence of iterative solution methods for the
Helmholtz equation. In the talk we present a comprehensive spectral
analysis of the discrete 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 derive an optimal value for the shift, and to
explain the mesh-depency of the convergence of GMRES preconditioned
with a shifted Laplacian. We will illustrate our results with a seismic test
problem.
Joint work with: Yogi Erlangga (University of British Columbia) and Kees Vuik (TU Delft)