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

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

Research output: Contribution to journalArticle

70 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1942-1958
Number of pages17
JournalSIAM Journal on Scientific Computing
Volume29
Issue number5
DOIs
Publication statusPublished - Dec 1 2007

Keywords

  • Convergence analysis
  • GMRES
  • Helmholtz equation
  • Shifted Laplace preconditioner iterative solution methods

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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