Domain Decomposition Algorithms with Small Overlap

Numerical experiments have shown that two-level Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear system of equations that remains after that the variables interior to the subregions have been eliminated. In this paper, a supporting theory is developed. AMS(MOS) subject classiications. 65F10, 65N30, 65N55 1. Introduction. Over the last decade, a considerable interest has developed in Schwarz methods and other domain decomposition methods for partial diierential equations; cf. e.g. the proceedings of ve international symposia 34,17,18,35,19]. A general theory has evolved and a substantial number of new algorithms has been designed , analyzed and tested numerically. Among them are two-level, additive Schwarz methods rst introduced in 1987; cf. number of other domain decomposition methods, in particular those of Bramble, Pas-ciak, and Schatz 5,6], can also be derived and analyzed using the same framework. Recent eeorts by Bramble, Pasciak, Wang, and Xu 7], and Xu 56] 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 46]. For other current projects, which also use the Schwarz framework, see Dryja, Smith, and Widlund 27] and Dryja and Widlund 32,33]. When a two-level method is used, the restrictions of the discrete elliptic problem to overlapping subregions, into which the given region has been decomposed, are solved exactly or approximately. These local solvers form an important part of a preconditioner for a conjugate gradient method. In addition, in order to enhance the convergence rate, the preconditioner includes a global problem of relatively modest dimension.