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 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 language | English |
|---|---|
| Pages (from-to) | 409-425 |
| Number of pages | 17 |
| Journal | Applied Numerical Mathematics |
| Volume | 50 |
| Issue number | 3-4 |
| DOIs | |
| Publication status | Published - Sep 2004 |
| Externally published | Yes |
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Keywords
- Helmholtz equation
- Krylov subspace
- Preconditioner
ASJC Scopus subject areas
- Applied Mathematics
- Computational Mathematics
- Modelling and Simulation
Cite this
On a class of preconditioners for solving the Helmholtz equation. / Erlangga, Y. A.; Vuik, C.; Oosterlee, C. W.
In: Applied Numerical Mathematics, Vol. 50, No. 3-4, 09.2004, p. 409-425.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - On a class of preconditioners for solving the Helmholtz equation
AU - Erlangga, Y. A.
AU - Vuik, C.
AU - Oosterlee, C. W.
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
UR - http://www.scopus.com/inward/record.url?scp=3142611583&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=3142611583&partnerID=8YFLogxK
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 -