Optimized Schwarz methods for Maxwell's equations
Dolean Maini, Victorita and Gander, M.J. and Gerardo-Giorda, L. (2009) Optimized Schwarz methods for Maxwell's equations. SIAM Journal on Scientific Computing, 31 (3). pp. 2193-2213. ISSN 1064-8275 (https://doi.org/10.1137/080728536)
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Abstract
Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, using characteristic transmission conditions, and it has been observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for which overlap is essential for convergence. More recently, optimized Schwarz methods have been developed for elliptic partial differential equations. These methods use more effective transmission conditions between subdomains than the classical Dirichlet conditions, and optimized Schwarz methods can be used both with and without overlap for elliptic problems. We show here why the classical Schwarz method applied to both the time harmonic and time discretized Maxwell's equations converges without overlap: the method has the same convergence factor as a simple optimized Schwarz method for a scalar elliptic equation. Based on this insight, we develop an entire new hierarchy of optimized overlapping and nonoverlapping Schwarz methods for Maxwell's equations with greatly enhanced performance compared to the classical Schwarz method. We also derive for each algorithm asymptotic formulas for the optimized transmission conditions, which can easily be used in implementations of the algorithms for problems with variable coefficients. We illustrate our findings with numerical experiments.
ORCID iDs
Dolean Maini, Victorita ORCID: https://orcid.org/0000-0002-5885-1903, Gander, M.J. and Gerardo-Giorda, L.;-
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Item type: Article ID code: 44790 Dates: DateEvent2009Published7 May 2009Published OnlineSubjects: Science > Mathematics Department: Faculty of Science > Mathematics and Statistics Depositing user: Pure Administrator Date deposited: 11 Sep 2013 14:34 Last modified: 16 Nov 2024 01:07 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/44790