### 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-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.

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 |

### Fingerprint

### Keywords

- Helmholtz equation
- Krylov subspace
- Preconditioner

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Modelling and Simulation

### Cite this

*Applied Numerical Mathematics*,

*50*(3-4), 409-425. https://doi.org/10.1016/j.apnum.2004.01.009

**On a class of preconditioners for solving the Helmholtz equation.** / Erlangga, Y. A.; Vuik, C.; Oosterlee, C. W.

Research output: Contribution to journal › Article

*Applied Numerical Mathematics*, vol. 50, no. 3-4, pp. 409-425. https://doi.org/10.1016/j.apnum.2004.01.009

}

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

VL - 50

SP - 409

EP - 425

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

IS - 3-4

ER -