Recent Developments on Optimized Schwarz Methods

Abstract

The classical Schwarz method [31] is based on Dirichlet boundary conditions. Overlapping subdomains are necessary to ensure convergence. As a result, when overlap is small, typically one mesh size, convergence of the algorithm is slow. A first possible remedy is the introduction of Neumann boundary conditions in the coupling between the local solutions. This idea has led to the development of the Dirichlet-Neuman algorithm [10], Neumann-Neumann method [3] and FETI methods [8]. These methods are widely used and have been the subject of many studies, improvements and extensions to various scalar or systems of partial differential equations, see for instance the following books [32], [27], [37] and [35] and references therein. A second cure to the slowness of the original Schwarz method is to use more general interface conditions, Robin conditions were proposed in [19] and pseudo-differential ones in [17]. These methods are well-suited for indefinite problems [5] and as we shall see to heterogeneous problems.