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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. 1. Introduction. Schwarz algorithms experienced a second youth over the last decades when distributed computers became more and more powerful and available. Fundamental convergence results for the classical Schwarz methods were derived for many partial differential equations and can now be found in several authoritative reviews [3,41,42] and books [34,33,39]. The Schwarz methods were also extended to systems of partial differential equations, such as the time harmonic Maxwell's equations [12,8], the time discretized Maxwell's equations [38], or to linear elasticity [18,19], but much less is known about the behavior of the Schwarz methods applied to hyperbolic systems of equations. This is true, in particular, for the Euler equations, to which the Schwarz algorithm was first applied in [31,32], where classical (characteristic) transmission conditions are used at the interfaces, or with more general transmission conditions in [7]. The analysis of such algorithms applied to systems proved to be very different from the scalar case; see [14,15].Over the last decade, a new class of overlapping Schwarz methods was developed for scalar partial differential equations, namely, the optimized Schwarz methods. These methods are based on a classical overlapping domain decomposition, but they use more effective transmission conditions than the classical Dirichlet conditions at the interfaces between subdomains. New transmission conditions were originally proposed for three different reasons: first, to obtain Schwarz algorithms that are convergent without...

Coarse grid correction is a key ingredient in order to have scalable domain decomposition methods. For smooth problems, the theory and practice of such two-level methods is well established, but this is not the case for problems with complicated variation and high contrasts in the coefficients. In a previous study, two of the authors introduced a coarse space adapted to highly heterogeneous coefficients using the low frequency modes of the subdomain DtN maps. In this work, we present a rigorous analysis of a two-level overlapping additive Schwarz method with this coarse space, which provides an automatic criterion for the number of modes that need to be added per subdomain to obtain a convergence rate of the order of the constant coefficient case. Our method is suitable for parallel implementation and its efficiency is demonstrated by numerical examples on some challenging problems with high heterogeneities for automatic partitionings.