TY - JOUR
T1 - On a class of preconditioners for solving the Helmholtz equation
AU - Erlangga, Y. A.
AU - Vuik, C.
AU - Oosterlee, C. W.
N1 - Funding Information:
✩ This research is financially supported by the Dutch Ministry of Economic Affairs under the project BTS01044 “Rigorous modelling of 3D wave propagation in inhomogeneous media for geophysical and optical problems”. * Corresponding author. E-mail address: y.a.erlangga@math.tudelft.nl (Y.A. Erlangga).
PY - 2004/9
Y1 - 2004/9
N2 - 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.
AB - 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.
KW - Helmholtz equation
KW - Krylov subspace
KW - Preconditioner
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U2 - 10.1016/j.apnum.2004.01.009
DO - 10.1016/j.apnum.2004.01.009
M3 - Article
AN - SCOPUS:3142611583
VL - 50
SP - 409
EP - 425
JO - Applied Numerical Mathematics
JF - Applied Numerical Mathematics
SN - 0168-9274
IS - 3-4
ER -