ar X iv : m at h / 06 10 53 1 v 3 [ m at h . N A ] 2 4 O ct 2 00 6 OPTIMIZED SCHWARZ METHODS FOR MAXWELL EQUATIONS

Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, and it was 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. Over the last decade, optimized Schwarz methods have been developed for elliptic partial differential equations. These methods use more effective transmission conditions between subdomains, and are also convergent without overlap for elliptic problems. We show here why the classical Schwarz method applied to the hyperbolic problem converges without overlap for the Maxwell's equations. The reason is that the method is equivalent to a simple optimized Schwarz method for an equivalent elliptic problem. Using this link, we show how to develop more efficient Schwarz methods than the classical ones for the Maxwell's equations. We illustrate our findings with numerical results. 1. Introduction. Schwarz algorithms experienced a second youth over the last decades, when distributed computers became more and more performant 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, see [2, 38, 37], and books, see [32, 30, 36]. The Schwarz methods were also extended to systems of partial differential equations, such as the time harmonic Maxwell equations, see [6, 10, 1], or to linear elasticity [16, 17], but much less is known about the behavior of the Schwarz methods applied to systems of equations. This is true in particular for the Euler equations, to which the Schwarz algorithm was first applied in [28, 29], where classical (characteristic) transmission conditions are used at the interfaces, or with more general interface conditions in [5]. The analysis of such algorithms applied to systems proved to be very different from the scalar case, see [13, 14]. Over the last decade, a new class of Schwarz methods was developed for scalar partial differential equations, namely the optimized Schwarz methods. These methods 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 overlap, see [25] for Robin conditions. The second motivation for changing the transmission conditions was to obtain a convergent Schwarz method for the Helmholtz …