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

Marcella Bonazzoli, Victorita Dolean, Ivan Graham, Euan Spence, Pierre-Henri Tournier

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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 languageEnglish
Pages (from-to)2559-2604
Number of pages46
JournalMathematics of Computation (MCOM)
Volume88
Issue number320
DOIs
Publication statusPublished - 30 May 2019

Fingerprint

Domain Decomposition
Preconditioning
Maxwell's equations
Preconditioner
Absorption
Harmonic
Additive Schwarz
Boundary conditions
Overlapping
Higher-order Elements
Element Order
GMRES
Medical Imaging
Impedance
Scalability
Cavity
Finite Element Method
Numerical Experiment
Scattering
Mesh

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

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

In: Mathematics of Computation (MCOM), Vol. 88, No. 320, 30.05.2019, p. 2559-2604.

Research output: Contribution to journalArticle

Bonazzoli, Marcella ; Dolean, Victorita ; Graham, Ivan ; Spence, Euan ; Tournier , Pierre-Henri. / Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption. In: Mathematics of Computation (MCOM). 2019 ; Vol. 88, No. 320. pp. 2559-2604.
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