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

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

Research output: Contribution to journalArticle

Abstract

This paper rigorously analyses preconditioning the time-harmonic Maxwell equations with absorption, where the preconditioner is constructed using two-level overlapping Additive Schwarz Domain Decomposition, and the PDE is discretised using finite-element methods of fixed, arbitrary order. The theory in this paper shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters are chosen appropriately, then classical two-level overlapping Additive Schwarz Domain Decomposition preconditioning (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.

Numerical experiments illustrate this theoretical result and also (i) explore replacing the PEC boundary conditions on the subdomains by impedance boundary conditions, and (ii) show that the preconditioner for the problem with absorption is also an effective preconditioner for the problem with no absorption. The numerical results include two substantial examples arising from applications; the first (a problem with absorption arising from medical imaging) shows the 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.
LanguageEnglish
JournalMathematics of Computation (MCOM)
Early online date30 May 2019
DOIs
StatusE-pub ahead of print - 30 May 2019

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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

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), 30.05.2019.

Research output: Contribution to journalArticle

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abstract = "This paper rigorously analyses preconditioning the time-harmonic Maxwell equations with absorption, where the preconditioner is constructed using two-level overlapping Additive Schwarz Domain Decomposition, and the PDE is discretised using finite-element methods of fixed, arbitrary order. The theory in this paper shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters are chosen appropriately, then classical two-level overlapping Additive Schwarz Domain Decomposition preconditioning (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.Numerical experiments illustrate this theoretical result and also (i) explore replacing the PEC boundary conditions on the subdomains by impedance boundary conditions, and (ii) show that the preconditioner for the problem with absorption is also an effective preconditioner for the problem with no absorption. The numerical results include two substantial examples arising from applications; the first (a problem with absorption arising from medical imaging) shows the 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.",
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AU - Tournier , Pierre-Henri

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AB - This paper rigorously analyses preconditioning the time-harmonic Maxwell equations with absorption, where the preconditioner is constructed using two-level overlapping Additive Schwarz Domain Decomposition, and the PDE is discretised using finite-element methods of fixed, arbitrary order. The theory in this paper shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters are chosen appropriately, then classical two-level overlapping Additive Schwarz Domain Decomposition preconditioning (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.Numerical experiments illustrate this theoretical result and also (i) explore replacing the PEC boundary conditions on the subdomains by impedance boundary conditions, and (ii) show that the preconditioner for the problem with absorption is also an effective preconditioner for the problem with no absorption. The numerical results include two substantial examples arising from applications; the first (a problem with absorption arising from medical imaging) shows the 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.

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