### Abstract

Language | English |
---|---|

Pages | 2559-2604 |

Number of pages | 46 |

Journal | Mathematics of Computation |

Volume | 88 |

Issue number | 320 |

Early online date | 30 May 2019 |

DOIs | |

Publication status | E-pub ahead of print - 30 May 2019 |

### Fingerprint

### Keywords

- Maxwell equations
- high frequency
- absorption
- iterative solvers
- preconditioning
- domain decompositions
- GMRES

### Cite this

*Mathematics of Computation*,

*88*(320), 2559-2604. https://doi.org/10.1090/mcom/3447

}

*Mathematics of Computation*, vol. 88, no. 320, pp. 2559-2604. https://doi.org/10.1090/mcom/3447

**Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption.** / Bonazzoli, M.; Dolean, V.; Graham, I. G.; Spence, E. A.; Tournier, P.-H.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption

AU - Bonazzoli, M.

AU - Dolean, V.

AU - Graham, I. G.

AU - Spence, E. A.

AU - Tournier, P.-H.

PY - 2019/5/30

Y1 - 2019/5/30

N2 - This paper rigorously analyses preconditioners for the time-harmonic Maxwell equations with absorption, where the PDE is discretised using curl-conforming finite-element methods of fixed, arbitrary order and the preconditioner is constructed using additive Schwarz domain decomposition methods. The theory developed here shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters and overlap are chosen appropriately, then the classical two-level overlapping additive Schwarz preconditioner (with PEC boundary conditions on the subdomains) performs optimally--in the sense that GMRES converges in a wavenumber-independent number of iterations--for the problem with absorption. An important feature of the theory is that it allows the coarse space to be built from low-order elements even if the PDE is discretised using high-order elements. It also shows that additive methods with minimal overlap can be robust. Numerical experiments are given that illustrate the theory and its dependence on various parameters. These experiments motivate some extensions of the preconditioners which have better robustness for problems with less absorption, including the propagative case. At the end of the paper we illustrate the performance of these on two substantial applications; the first (a problem with absorption arising from medical imaging) shows the empirical robustness of the preconditioner against heterogeneity, and the second (scattering by a COBRA cavity) shows good scalability of the preconditioner with up to 3,000 processors.

AB - This paper rigorously analyses preconditioners for the time-harmonic Maxwell equations with absorption, where the PDE is discretised using curl-conforming finite-element methods of fixed, arbitrary order and the preconditioner is constructed using additive Schwarz domain decomposition methods. The theory developed here shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters and overlap are chosen appropriately, then the classical two-level overlapping additive Schwarz preconditioner (with PEC boundary conditions on the subdomains) performs optimally--in the sense that GMRES converges in a wavenumber-independent number of iterations--for the problem with absorption. An important feature of the theory is that it allows the coarse space to be built from low-order elements even if the PDE is discretised using high-order elements. It also shows that additive methods with minimal overlap can be robust. Numerical experiments are given that illustrate the theory and its dependence on various parameters. These experiments motivate some extensions of the preconditioners which have better robustness for problems with less absorption, including the propagative case. At the end of the paper we illustrate the performance of these on two substantial applications; the first (a problem with absorption arising from medical imaging) shows the empirical robustness of the preconditioner against heterogeneity, and the second (scattering by a COBRA cavity) shows good scalability of the preconditioner with up to 3,000 processors.

KW - Maxwell equations

KW - high frequency

KW - absorption

KW - iterative solvers

KW - preconditioning

KW - domain decompositions

KW - GMRES

U2 - 10.1090/mcom/3447

DO - 10.1090/mcom/3447

M3 - Article

VL - 88

SP - 2559

EP - 2604

JO - Mathematics of Computation

T2 - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 320

ER -