Abstract. Optimized Schwarz methods are a new class of Schwarz methods with greatly enhanced convergence properties. They converge uniformly faster than classical Schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical Schwarz methods if the overlap is of the order of the mesh parameter, which is often the case in practical applications. They achieve this performance by using new transmission conditions between subdomains which greatly enhance the information exchange between subdomains and are motivated by the physics of the underlying problem. We analyze in this paper these new methods for symmetric positive definite problems and show their relation to other modern domain decomposition methods like the new Finite Element Tearing and Interconnect (FETI) variants.

Abstract. 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 Cauchy-Riemann equations and 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 Cauchy-Riemann and Maxwell’s equations. We illustrate our findings with numerical results. Key words. Schwarz algorithms, optimized interface conditions, Maxwell equations AMS subject classifications. 1. Introduction. Schwarz

this report is the design and experimental study of optimizing boundary parameter coupled with particular choices of inner and outer splittings. We are interested here in extending some work of San and Tang [HT96] and Tang [Tan92] to parabolic problems. There is a parameter # that acts like a feedback gain across the artificial interfaces. The primary aspect of this article is to construct and demonstrate e#ective multi-splitting methods as depending on the interface boundary condition. Consider the numerical solution of parabolic problems of form: #u #t

the uneven convergence properties of the classical Schwarz method. In the classical Schwarz method high frequency components converge very fast, whereas low frequency components are only converging very slowly and hence slow down the performance of the overall method. This can be corrected by replacing the Dirichlet transmission conditions in the classical Schwarz

Abstract. 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. Key words. Schwarz algorithms, optimized interface conditions, Maxwell equations AMS subject classifications. 1. Introduction. Schwarz