Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian

Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian

van Gijzen, M.B.
Erlangga, Y.A.
Vuik, C.

Electrical Engineering, Mathematics and Computer Science

2006

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 meshdependency of the convergence of GMRES preconditioned with a shifted Laplacian. We illustrate our results with a seismic test problem.

Helmholtz equation
shifted Laplace preconditioner
iterative solution methods
GMRES
convergence analysis

http://resolver.tudelft.nl/uuid:0d5a64ef-90b2-45e6-95e5-fb6c75b4a58b

Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics

1389-6520

Reports of the Department of Applied Mathematical Analysis, 06-16

Institutional Repository

report

(c) 2006 van Gijzen, M.B.; Erlangga, Y.A.; Vuik, C.