On a class of preconditioners for solving the Helmholtz equation

Y. A. Erlangga, C. Vuik, C. W. Oosterlee

Research output: Contribution to journalArticlepeer-review

167 Citations (Scopus)


In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year's Report, St. Hugh's College, Oxford, 2001] proposed a preconditioner where an extra term is added to the Laplace operator. This term is similar to the zeroth order term in the Helmholtz equation but with reversed sign. In this paper, both approaches are further generalized to a new class of preconditioners, the so-called "shifted Laplace" preconditioners of the form Δφ-αk2φ with α∈ℂ. Numerical experiments for various wavenumbers indicate the effectiveness of the preconditioner. The preconditioner is evaluated in combination with GMRES, Bi-CGSTAB, and CGNR.

Original languageEnglish
Pages (from-to)409-425
Number of pages17
JournalApplied Numerical Mathematics
Issue number3-4
Publication statusPublished - Sep 2004


  • Helmholtz equation
  • Krylov subspace
  • Preconditioner

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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