Abstract The classical Schwarz alternating method has recently been generalized in several directions This e ort has resulted in a number of new powerful domain decomposition methods for elliptic problems in new insight into multigrid methods and in the development of a very useful framework for the analysis of a variety of iterative methods Most of this work has focused on positive de nite sym...

In this paper, we partially answer open questions about the convergence of overlapping Schwarz methods. We prove that overlapping Schwarz methods with Dirichlet transmission conditions for semilinear elliptic and parabolic equations always converge, while overlapping Schwarz methods with Robin transmission conditions only converge for semilinear parabolic equations, but the convergence is not g...

We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are considered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the origin...

The additive Schwarz method with harmonic extension (ASH) was introduced by Cai and Sarkis (1999) as an efficient variant of the additive Schwarz method that converges faster and requires less communication. We show that ASH can also be used with optimized transmission conditions to obtain faster convergence. We show that when the decomposition into subdomains contains no cross points, optimize...

We describe and compare some recent domain decomposition algorithms of Schwarz type with respect to parallel performance. A new, robust domain decomposition algorithm { Additive Average Schwarz is compared with a classical overlapping Schwarz code. Complexity estimates are given in both two and three dimensions and actual implementations are compared on a Paragon machine as well as on a cluster...

We develop a new class of overlapping Schwarz type algorithms for solving scalar convection-diiusion equations discretized by nite element or nite diierence methods. The precon-ditioners consist of two components, namely, the usual two-level additive Schwarz precon-ditioner and the sum of some quadratic terms constructed by using products of ordered neighboring subdomain preconditioners. The or...

Schwarz methods are nowadays known as parallel solvers, and there are many variants: alternating and parallel Schwarz methods at the continuous level, additive and multiplicative Schwarz methods at the discrete level, also with restricted variants, which in the additive case build the important bridge between discrete and continuous Schwarz methods, see [4]. But where did these methods come fro...

One can refer to the paper by Keyes [1992] and the book by Smith et al. [1996] for in-depth reviews of domain decomposition (DD) methods. Of particular interest here are non-overlapping schemes such as iterative substructuring (Bjørstad et al. [2001]) and FETI methods (Farhat et al. [2001]). When solving non-linear boundary value problems (BVPs) via domain decomposition it is common to use Newt...

In implicit solvation models, the electrostatic contribution to the solvation energy can be estimated by solving a system of elliptic partial differential equations modeling the reaction potential. The domain of definition of such elliptic equations is the union of the van der Waals cavities corresponding to the atoms of the solute molecule. Therefore, the computations can naturally be performe...