The convergence of multiplicative Schwarz-type methods for solving linear systems when the coefficient matrix is either a nonsingular M-matrix or a symmetric positive definite matrix is studied using classical and new results from the theory of splittings. The effect on convergence of algorithmic parameters such as the number of subdomains, the amount of overlap, the result of inexact local solves and of “coarse grid” corrections (global coarse solves) is analyzed in an algebraic setting. Results on algebraic additive Schwarz are also included.

Convergence results for the restrictive additive Schwarz (RAS) method of Cai and Sarkis [SIAM J. Sci. Comput., 21 (1999), pp. 792–797] for the solution of linear systems of the form Ax = b are provided using an algebraic view of additive Schwarz methods and the theory of multisplittings. The linear systems studied are usually discretizations of partial differential equations in two or three dimensions. It is shown that in the case of A symmetric positive definite, the projections defined by the methods are not orthogonal with respect to the inner product defined by A, and therefore the standard analysis cannot be used here. The convergence results presented are for the class of M-matrices (and more generally for H-matrices) using weighted max norms. Comparison between different versions of the RAS method are given in terms of these norms. A comparison theorem with respect to the classical additive Schwarz method makes it possible to indirectly get quantitative results on rates of convergence which otherwise cannot be obtained by the theory. Several RAS variants are considered, including new ones and two-level schemes.

Convergence properties of additive and multiplicative Schwarz iterations for solving linear systems of equations with a symmetric positive semidefinite matrix are analyzed. The analysis presented applies to matrices whose principal submatrices are nonsingular, i.e., positive definite. These matrices appear in discretizations of some elliptic partial differential equations, e.g., those with Neumann or periodic boundary conditions.

Convergence results for the restricted multiplicative Schwarz (RMS) method, the multiplicative version of the restricted additive Schwarz (RAS) method for the solution of linear systems of the form Ax = b, are provided. An algebraic approach is used to prove convergence results for nonsymmetric M-matrices. Several comparison theorems are also established. These theorems compare the asymptotic rate of convergence with respect to the amount of overlap, the exactness of the subdomain solver, and the number of domains. Moreover, comparison theorems are given between the RMS and RAS methods as well as between the RMS and the classical multiplicative Schwarz method.

The convergence of additive and multiplicative Schwarz methods for computing certain characteristics of Markov chains such as stationary probability vectors and mean first passage matrices is studied. Our main result is a convergence theorem for multiplicative Schwarz iterations when applied to singular systems. As a by-product we also obtain a convergence result for alternating iterations. It is also shown that, when the Markov chain is ergodic, additive and multiplicative Schwarz methods can be applied to the nonsingular systems that result from reducing the equations. The so-called coarse grid corrections are are also studied.

A convergence analysis is presented for additive Schwarz iterations when applied to consistent singular systems of equations of the form Ax = b. The theory applies to singular M-matrices with one-dimensional null space and is applicable in particular to systems representing ergodic Markov chains, and to certain discretizations of partial differential equations. Additive Schwarz can be seen as a generalization of block Jacobi, where the set of indices defining the diagonal blocks have nonempty intersection; this is called the overlap. The presence of overlap is known to accelerate the convergence of the methods in the nonsingular case. By providing convergence results, as well as some characteristics of the induced splitting, we hope to encourage the use of this additional computational tool for the solution of Markov chains and other singular systems. We present several numerical examples showing that additive Schwarz performs better than block Jacobi. For completeness, a few numerical experiments with block Gauss–Seidel and multiplicative Schwarz are also included.

Abstract. The Schwarz method can be used for the iterative solution of elliptic boundary value problems on a large domain Ω. One subdivides Ω into smaller, more manageable, subdomains and solves the differential equation in these subdomains using appropriate boundary conditions. Schwarz-Robin methods use Robin conditions on the artificial interfaces for information exchange at each iteration. Optimized Schwarz Methods (OSM) are those in which one optimizes the Robin parameters. While the convergence theory of classical Schwarz methods (with Dirichlet conditions on the artificial interface) is well understood, the overlapping Schwarz-Robin methods still lack a complete theory. In this paper, an abstract Hilbert space version of the OSM is presented, together with an analysis of conditions for its convergence. It is also shown that if the overlap is relatively uniform, these convergence conditions are met for Schwarz-Robin methods for two-dimensional elliptic problems, for any positive Robin parameter. In the discrete setting, we obtain that the convergence rate ω(h) varies like a polylogarithm of h. Numerical experiments show that the methods work well and that the convergence rate does not appear to depend on h. 1. Introduction. Schwarz

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A simple proof is presented of a quite general theorem on the convergence of stationary iterations for solving singular linear systems whose coefficient matrix is Hermitian and positive semidefinite. In this manner, elegant proofs are obtained of some known convergence results, including the necessity of the P-regular splitting result due to Keller, as well as recent results involving generalized inverses. Other generalizations are also presented. These results are then used to analyze the convergence of several versions of algebraic additive and multiplicative Schwarz methods for Hermitian positive semidefinite systems.

. In the convergence theory of multisplittings for symmetric positive definite (s.p.d.) matrices it is usually assumed that the weighting matrices are scalar matrices, i.e., multiples of the identity. In this paper, this restrictive condition is eliminated. In its place it is assumed that more than one (inner) iteration is performed in each processor (or block). The theory developed here is applied to nonstationary multisplittings for s.p.d. matrices, as well as to two-stage multisplittings for symmetric positive semidefinite matrices. Key words. iterative methods, linear systems, symmetric positive definite matrices, block methods, parallel algorithms, multisplitting, two-stage, nonstationary AMS subject classifications. 65F10, 65F15 PII. S0895479800367038 1.