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On a class of preconditioners for solving the Helmholtz equation

Authors
Journal
Applied Numerical Mathematics
0168-9274
Publisher
Elsevier
Publication Date
Volume
50
Identifiers
DOI: 10.1016/j.apnum.2004.01.009
Keywords
  • Helmholtz Equation
  • Krylov Subspace
  • Preconditioner

Abstract

Abstract 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 Δ φ− αk 2 φ 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.

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