Optimized Schwarz Methods in the Numerical Solution of PDEs

Schwarz algorithms have experienced a second youth over the last decades, when distributed computers became more and more powerful and available. In the classical Schwarz algorithm the computational domain is divided into subdomains and Dirichlet continuity is enforced on the interfaces between subdomains. Fundamental convergence results for the classical Schwarz methods have been derived for many partial differential equations. Within this framework the overlap between subdomains is essential for convergence. More recently, Optimized Schwarz Methods have been developed: based on more effective transmission conditions than the classical Dirichlet conditions at the interfaces between subdomains, such algorithms can be used both with and without overlap. On the other hand, such algorithms show greatly enhanced performance compared to the classical Schwarz method.
I will present a survey of Optimized Schwarz Methods for the numerical approximation of partial differential equation, focusing mainly on heterogeneous convection-diffusion and electromagnetic problems.