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


Authors: M. Bonazzoli, V. Dolean, I. G. Graham, E. A. Spence and P.-H. Tournier
Journal: Math. Comp. 88 (2019), 2559-2604
MSC (2010): Primary 35Q61, 65N55, 65F08, 65F10, 65N30, 78A45
DOI: https://doi.org/10.1090/mcom/3447
Published electronically: May 30, 2019
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Abstract: 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.


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Additional Information

M. Bonazzoli
Affiliation: Sorbonne Université, Université Paris-Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions, équipe Alpines, F-75005 Paris, France
Email: marcella.bonazzoli@inria.fr

V. Dolean
Affiliation: Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH, United Kingdom; and Université Côte d’Azur, CNRS, Laboratoire J-A Dieudonné, France
Email: victorita.dolean@strath.ac.uk

I. G. Graham
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
Email: I.G.Graham@bath.ac.uk

E. A. Spence
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
Email: E.A.Spence@bath.ac.uk

P.-H. Tournier
Affiliation: Sorbonne Université, Université Paris-Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions, équipe Alpines, F-75005 Paris, France
Email: tournier@ljll.math.upmc.fr

DOI: https://doi.org/10.1090/mcom/3447
Keywords: Maxwell equations, high frequency, absorption, iterative solvers, preconditioning, domain decomposition, GMRES
Received by editor(s): November 10, 2017
Received by editor(s) in revised form: October 1, 2018
Published electronically: May 30, 2019
Additional Notes: This work was granted access to the HPC resources of TGCC at CEA and CINES under the allocations 2016-067730 and 2017-ann7330 made by GENCI
The first, second, and fifth authors gratefully acknowledge support from the French National Research Agency (ANR), project MEDIMAX, ANR-13-MONU-0012; the fourth author gratefully acknowledges support from the EPSRC grant EP/R005591/1.
Article copyright: © Copyright 2019 American Mathematical Society