Multiplicative Schwarz Algorithms for Some Nonsymmetric and Indefinite Problems

The classical Schwarz alternating method has recently been generalized in several directions. This effort has resulted in a number of new powerful domain decomposition methods for elliptic problems, in new insight into multigrid methods, and in the development of a very useful framework for the analysis of a variety of iterative methods. Most of this work has focused on positive definite, symmetric problems. In this paper, a general framework is developed for multiplicative Schwarz algorithms for nonsymmetric and indefinite problems. Several applications are then discussed including two- and multilevel Schwarz methods and iterative substructuring algorithms. Some new results on additive Schwarz methods are also presented.