Some Schwarz Methods for Symmetric and Nonsymmetric Elliptic Problems

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 Cai on nonsymmetric and indeenite problems.nite elliptic problems AMS(MOS) subject classiications. 65F10, 65N30 1. Introduction. Over the last few years, a general theory has been developed for the study of additive and multiplicative Schwarz methods. Many domain decomposition and certain multigrid methods can now be successfully analyzed inside this framework. Early work by P.-L. Lions 23], 24] gave an important impetus to this eeort. The additive Schwarz methods were then developed by Dryja and Widlund 15], 16], 17], Matsokin and Nepomnyaschikh 25] and Nepomnyaschikh 26] and others. Recent eeorts by Bramble, Pasciak, Wang and Xu 4] and Xu 36] have extended the general framework making a systematic study of multiplicative Schwarz methods possible. The multiplicative algorithms are direct generalizations of the original alternating method discovered more than 120 years ago by H. A. Schwarz 30]. We note that most of the work in recent years has focused on the positive deenite, symmetric case. While this theory is quite general, the applications so far have primarily been to the solution of the often large linear systems of algebraic equations, which arise in the nite element discretization of elliptic and parabolic boundary value problems. As shown in P.-L. Lions 24], the classical Schwarz algorithms can conveniently be described in terms of subspaces of the given space. The relevant error propagation operator of a particular Schwarz method can be written as a polynomial of orthogonal projections onto these subspaces. The use of these projections in computations involves the evaluation of the residual of the original nite element problem and the exact, or approximate, solution of several nite element problems on subregions. An additional coarse discrete model is also often used to enhance the rate of convergence. For a discussion of many applications, see Dryja and Widlund 17]. For other current projects, which also use the Schwarz framework, see Dryja and Widlund 18], 20] and Dryja, Smith and Widlund 14].