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

67 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 - 2007
Externally publishedYes

Fingerprint

GMRES
Hermann Von Helmholtz
Spectral Analysis
Spectrum analysis
Radiation Condition
Helmholtz equation
Iterative Solution
Helmholtz Equation
Operator
Laplace
Preconditioner
Test Problems
Damping
Speedup
Mesh
Valid
Upper bound
Norm
Radiation

Keywords

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

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics

Cite this

Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. / Van Gijzen, M. B.; Erlangga, Y. A.; Vuik, C.

In: SIAM Journal on Scientific Computing, Vol. 29, No. 5, 2007, p. 1942-1958.

Research output: Contribution to journalArticle

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