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Abstract
This paper rigorously analyses preconditioners for the timeharmonic Maxwell equations with absorption, where the PDE is discretised using curlconforming finiteelement 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 twolevel overlapping additive Schwarz preconditioner (with PEC boundary conditions on the subdomains) performs optimallyin the sense that GMRES converges in a wavenumberindependent number of iterationsfor the problem with absorption. An important feature of the theory is that it allows the coarse space to be built from loworder elements even if the PDE is discretised using highorder 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 (fromto)  25592604 
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 
Keywords
 Absorption
 Domain decomposition
 GMRES
 High frequency
 Iterative solvers
 Maxwell equations
 Preconditioning
ASJC Scopus subject areas
 Algebra and Number Theory
 Computational Mathematics
 Applied Mathematics
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Ivan Graham
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching