### Abstract

This paper rigorously analyses preconditioners for the timeharmonic 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.

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

Pages (from-to) | 2559-2604 |

Number of pages | 46 |

Journal | Mathematics of Computation (MCOM) |

Volume | 88 |

Issue number | 320 |

Early online date | 30 May 2019 |

DOIs | |

Publication status | Published - 30 Nov 2019 |

### Fingerprint

### Keywords

- Absorption
- Domain decomposition
- GMRES
- High frequency
- Iterative solvers
- Maxwell equations
- Preconditioning

### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

### Cite this

*Mathematics of Computation (MCOM)*,

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

**Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption.** / Bonazzoli, Marcella ; Dolean, Victorita; Graham, Ivan; Spence, Euan; Tournier , Pierre-Henri.

Research output: Contribution to journal › Article

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

}

TY - JOUR

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

AU - Bonazzoli, Marcella

AU - Dolean, Victorita

AU - Graham, Ivan

AU - Spence, Euan

AU - Tournier , Pierre-Henri

PY - 2019/11/30

Y1 - 2019/11/30

N2 - This paper rigorously analyses preconditioners for the timeharmonic 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 timeharmonic 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 - Absorption

KW - Domain decomposition

KW - GMRES

KW - High frequency

KW - Iterative solvers

KW - Maxwell equations

KW - Preconditioning

UR - http://www.scopus.com/inward/record.url?scp=85073249432&partnerID=8YFLogxK

U2 - 10.1090/mcom/3447

DO - 10.1090/mcom/3447

M3 - Article

VL - 88

SP - 2559

EP - 2604

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 1088-6842

IS - 320

ER -