### Abstract

A new, coercive formulation of the Helmholtz equation was introduced in Moiola and Spence (2014). In this paper we investigate h-version Galerkin discretisations of this formulation, and the iterative solution of the resulting linear systems. We find that the coercive formulation behaves similarly to the standard formulation in terms of the pollution effect (i.e. to maintain accuracy as k→∞ h must decrease with k at the same rate as for the standard formulation). We prove k-explicit bounds on the number of GMRES iterations required to solve the linear system of the new formulation when it is preconditioned with a prescribed symmetric positive-definite matrix. Even though the number of iterations grows with k, these are the first such rigorous bounds on the number of GMRES iterations for a preconditioned formulation of the Helmholtz equation, where the preconditioner is a symmetric positive-definite matrix.

Original language | English |
---|---|

Pages (from-to) | 110-131 |

Number of pages | 22 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 352 |

Early online date | 6 Dec 2018 |

DOIs | |

Publication status | Published - 15 May 2019 |

### Keywords

- math.NA
- 35J05, 65N30, 65F10
- Finite element method
- GMRES
- Helmholtz equation
- Wavenumber-explicit analysis
- Coercive variational formulation
- Pollution effect

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

*Journal of Computational and Applied Mathematics*,

*352*, 110-131. https://doi.org/10.1016/j.cam.2018.11.035

**Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?** / Diwan, Ganesh C.; Moiola, Andrea; Spence, Euan A.

Research output: Contribution to journal › Article

*Journal of Computational and Applied Mathematics*, vol. 352, pp. 110-131. https://doi.org/10.1016/j.cam.2018.11.035

}

TY - JOUR

T1 - Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?

AU - Diwan, Ganesh C.

AU - Moiola, Andrea

AU - Spence, Euan A.

N1 - 27 pages, 7 figures

PY - 2019/5/15

Y1 - 2019/5/15

N2 - A new, coercive formulation of the Helmholtz equation was introduced in Moiola and Spence (2014). In this paper we investigate h-version Galerkin discretisations of this formulation, and the iterative solution of the resulting linear systems. We find that the coercive formulation behaves similarly to the standard formulation in terms of the pollution effect (i.e. to maintain accuracy as k→∞ h must decrease with k at the same rate as for the standard formulation). We prove k-explicit bounds on the number of GMRES iterations required to solve the linear system of the new formulation when it is preconditioned with a prescribed symmetric positive-definite matrix. Even though the number of iterations grows with k, these are the first such rigorous bounds on the number of GMRES iterations for a preconditioned formulation of the Helmholtz equation, where the preconditioner is a symmetric positive-definite matrix.

AB - A new, coercive formulation of the Helmholtz equation was introduced in Moiola and Spence (2014). In this paper we investigate h-version Galerkin discretisations of this formulation, and the iterative solution of the resulting linear systems. We find that the coercive formulation behaves similarly to the standard formulation in terms of the pollution effect (i.e. to maintain accuracy as k→∞ h must decrease with k at the same rate as for the standard formulation). We prove k-explicit bounds on the number of GMRES iterations required to solve the linear system of the new formulation when it is preconditioned with a prescribed symmetric positive-definite matrix. Even though the number of iterations grows with k, these are the first such rigorous bounds on the number of GMRES iterations for a preconditioned formulation of the Helmholtz equation, where the preconditioner is a symmetric positive-definite matrix.

KW - math.NA

KW - 35J05, 65N30, 65F10

KW - Finite element method

KW - GMRES

KW - Helmholtz equation

KW - Wavenumber-explicit analysis

KW - Coercive variational formulation

KW - Pollution effect

UR - http://www.scopus.com/inward/record.url?scp=85058552662&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2018.11.035

DO - 10.1016/j.cam.2018.11.035

M3 - Article

VL - 352

SP - 110

EP - 131

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

ER -