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Optimized Schwarz methods for Maxwell’s equations. (English) Zbl 1192.78044

Summary: 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.

MSC:

78M25 Numerical methods in optics (MSC2010)
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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