This paper begins with an introduction to additive and multiplicative Schwarz methods. A two-level method is then reviewed and a new result on its rate of convergence is established for the case when the overlap is small. Recent results by Xuejun Zhang, on multi-level Schwarz methods, are formulated and discussed. The paper is concluded with a discussion of recent joint results with Xiao-Chuan ...

Optimized Schwarz Methods (OSM) use Robin transmission conditions across the subdomain interfaces. The Robin parameter can then be optimized to obtain the fastest convergence. A new formulation is presented with a coarse grid correction. The optimal parameter is computed for a model problem on a cylinder, together with the corresponding convergence factor which is smaller than that of classical...

The use of Dirichlet-to-Neumann operators as transmission conditions is known to yield optimal Schwarz methods that converge in a finite number of iterations when the subdomain decomposition has tree-like connectivity. However, it remains an open problem whether it is possible to construct a finitely terminating algorithm for arbitrary decompositions. In this article, we construct a Schwarz met...

We consider additive Schwarz methods for the biharmonic Dirichlet problem and show that the algorithms have optimal convergence properties for some conforming nite elements. Some multilevel methods are also discussed.

In the note, we prove that additive Schwarz type preconditioner is applicable to the nested finite dimensional subspaces generated by Hermite cubic splines for the second order elliptic problem. It is shown that the preconditioned system is well conditioned.

Two-level overlapping Schwarz methods are considered for nite element problems of 3D Maxwell's equations. N ed elec elements built on tetrahedra and hexahedra are considered. Once the relative overlap is xed, the condition number of the additive Schwarz method is bounded, independently of the diameter of the triangulation and the number of subregions. A similar result is obtained for a multipli...

In this paper an analogue of the Schwarz alternating method is considered to nd a minimal eigenvalue and its corresponding eigenvector of generalized symmetric eigenvalue problem. The technique suggested is based on decomposition of the original domain into overlapping subdomains and on consideration of local eigenvalue problems in subdomains. Both multiplicative and additive variants of the me...

The convergence of the multiplicative multisplitting-type method for solving the linear complementarity problem with an H-matrix is discussed using classical and new results from the theory of splitting. This directly results in a sufficient condition for guaranteeing the convergence of the multiplicative multisplitting method. Moreover, the multiplicative multisplitting method is applied to th...