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.

Summary. The performance of multigrid methods for the standard Poisson problem and for the consistent Poisson problem arising in spectral element discretizations of the Navier-Stokes equations is investigated. It is demonstrated that overlapping additive Schwarz methods are effective smoothers, provided that the solution in the overlap region is weighted by the inverse counting matrix. It is also shown that spectral element based smoothers are superior to those based upon finite element discretizations. Results for several large 3D Navier-Stokes applications are presented. 1

Abstract. We analyze algebraic multilevel methods applied to non-symmetric M-matrices. Two types of multilevel approximate block factorizations are considered. The first one is related to the AMLI method. The second method is the multiplicative counterpart of the AMLI approach which we call the multiplicative algebraic multilevel (MAMLI) method. The MAMLI method is closely related to certain geometric and algebraic multigrid methods, such as the AMGr method. Although these multilevel methods work very well in practice for many problems, not much is known about theoretical convergence properties for non-symmetric problems. Here, we establish convergence results and comparison results between AMLI and MAMLI multilevel methods applied to non-symmetric M-matrices.

Abstract. We develop a class of V-cycle type multilevel restricted additive Schwarz (RAS) methods and study the numerical and parallel performance of the new fully coupled methods for solving large sparse Jacobian systems arising from the discretization of some optimization problems constrained by nonlinear partial differential equations. Straightforward extensions of the one-level RAS to multilevel do not work due to the pollution effects of the coarse interpolation. We then introduce, in this paper, a pollution removing coarse-to-fine interpolation scheme for one of the components of the multi-component linear system, and show numerically that the combination of the new interpolation scheme with the RAS smoothed multigrid method provides an effective family of techniques for solving rather difficult PDE-constrained optimization problems. Numerical examples involving the boundary control of incompressible Navier-Stokes flows are presented in detail. Key words. Schwarz preconditioners, domain decomposition, multilevel methods, parallel computing, partial differential equations constrained optimization, inexact Newton, flow control. 1. Introduction. There are two major families of Newton techniques for solving nonlinear optimization problems: reduced space methods, characterized by the partition of the problem into smaller ones at each Newton step, and full space ones. As

Domain decomposition methods are widely used for solving large-scale linear systems of equations arising from the discretization of partial differential equations (PDEs) on

this paper, we provide an extension of RAS for symmetric positive definite problems using the so-called harmonic overlaps (RASHO). Both RAS and RASHO outperform their counterparts of the classical additive Schwarz variants. Roughly speaking, the design of RASHO is based on a much deeper understanding of the behavior of Schwarz type methods in the overlapping regions, and in the construction of the overlap. Under RASHO, the overlap is obtained by extending the nonoverlapping subdomains only in the directions that do not cut the boundaries of other subdomains, and all functions are made harmonic in the overlapping regions. As a result, the subdomain problems in RASHO are smaller than those of AS, and the communication cost is also smaller when implemented on distributed memory computers, since the right-hand sides of discrete harmonic systems are always zero that do not need to be communicated. We will show numerically that RASHO preconditioned CG takes less number of iterations than the corresponding AS preconditioned CG. An almost optimal convergence theory will be # Department of Computer Science, University of Colorado, Boulder, CO 80309, (cai@cs.colorado.edu). The work was supported in part by the NSF grants ASC-9457534, ECS-9725504, and ACI-0072089. + Faculty of Mathematics, Informatics and Mechanics, Warsaw University, Warsaw, (dryja@mimuw.edu.pl). This work was supported in part by the NSF grant CCR-9732208 and in part by the Polish Science Foundation grant 2 P03A 021 16. # Mathematical Sciences Department, Worcester Polytechnic Institute, Worcester, MA 01609, (msarkis@wpi.edu). The work was supported in part by the NSF grant CCR-9984404. 1 presented for the RASHO for elliptic problems discretized with finite element methods. Recall that the basic building bl...

Newton additive and multiplicative Schwarz iterative methods

Summary. During the last few years, an algebraic formulation of Schwarz methods was developed. In this paper this algebraic formulation is used to prove new convergence results for multiplicative Schwarz methods when applied to consistent singular systems of linear equations. Coarse grid corrections are also studied. In particular, these results are applied to the numerical solutions of Markov chains. 1

Abstract. 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 Schwarz methods. A new coarse space is presented, suitable for OSM. Numerical experiments illustrating the effectiveness of OSM with a coarse grid correction, both as an iteration and as a preconditioner, are reported. 1. Introduction. A