Optimized Schwarz methods for Maxwell's equations

Victorita Dolean Maini, M.J. Gander, L. Gerardo-Giorda

Research output: Contribution to journalArticle

99 Citations (Scopus)

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.



LanguageEnglish
Pages2193-2213
Number of pages21
JournalSIAM Journal on Scientific Computing
Volume31
Issue number3
Early online date7 May 2009
DOIs
Publication statusPublished - 2009

Fingerprint

Schwarz Methods
Maxwell equations
Maxwell's equations
Partial differential equations
Transmission Conditions
Overlap
Elliptic Problems
Hyperbolic Partial Differential Equations
Dirichlet conditions
Experiments
Elliptic Partial Differential Equations
Variable Coefficients
Asymptotic Formula
Elliptic Equations
Overlapping
Harmonic
Numerical Experiment
Scalar
Entire
Converge

Keywords

  • Schwarz algorithms
  • optimized transmission conditions
  • Maxwell's equations

Cite this

Dolean Maini, Victorita ; Gander, M.J. ; Gerardo-Giorda, L. / Optimized Schwarz methods for Maxwell's equations. In: SIAM Journal on Scientific Computing. 2009 ; Vol. 31, No. 3. pp. 2193-2213.
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Optimized Schwarz methods for Maxwell's equations. / Dolean Maini, Victorita; Gander, M.J.; Gerardo-Giorda, L.

In: SIAM Journal on Scientific Computing, Vol. 31, No. 3, 2009, p. 2193-2213.

Research output: Contribution to journalArticle

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