SCHWARZ ANALYSIS OF ITERATIVE SUBSTRUCTURING ALGORITHMS FOR ELLIPTIC PROBLEMS IN 3 DIMENSIONS

Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that ar ise in finite element or finite difference approximations of partial d ifferential equations. The preconditioners are constructed from exact or approximate solvers for the same partial differential equation rest ricted to a set of subregions into which the given region has been div ided. In addition, the preconditioner is often augmented by a coarse, second-level approximation that provides additional, global exchange o f information that can enhance the rate of convergence considerably. T he iterative substructuring methods, based on decompositions of the re gion into nonoverlapping subregions, form one of the main families of such algorithms. Many domain decomposition algorithms can conveniently be described and analyzed as Schwarz methods. These algorithms are fu lly defined in terms of a set of subspaces and auxiliary bilinear form s. A general theoretical framework has previously been developed. In t his paper, these techniques are used in an analysis of iterative subst ructuring methods for elliptic problems in three dimensions. A special emphasis is placed on the difficult problem of designing good coarse models and obtaining robust methods for which the rate of convergence is insensitive to large variations in the coefficients of the differen tial equation. Domain decomposition algorithms can conveniently be bui lt from modules that represent local and global components of the prec onditioner. In this paper, a number of such possibilities are explored , and it is demonstrated how a great variety of fast algorithms can be designed and analyzed.