Some Recent Results on Schwarz Type Domain Decomposition Algorithms

Numerical experiments have shown that two-level Schwarz methods, for the solution of discrete elliptic problems, 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. A supporting theory is outlined. 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 elliptic partial diierential equations. Among them are two-level, additive Schwarz methods rst introduced in 1987; cf. a number of other domain decomposition methods, in particular those of Bramble, Pasciak, and Schatz 3,4], can also be derived and analyzed using the same framework. Recent eeorts by Bramble, Pasciak, Wang, and Xu 5], and Xu 30] have extended the general framework making a systematic study of mul-tiplicative Schwarz methods possible. The multiplicative algorithms are direct generalizations of the original alternating method discovered more than 120 years ago by H.A. Schwarz 23]. For other current projects, which also use the Schwarz framework, see Dryja, Smith, and Widlund 15], Dryja and Widlund 19,21] and Widlund 29]. Proofs of most of the results of this paper can be found in Dryja and Widlund 20] or can be derived straightforwardly using the same technical tools. Here we will only discuss the additive algorithms. We begin our discussion by reexamining the block-Jacobi/conjugate