Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions

Domain decomposition methods provide powerful preconditioners for the iterative solution of large systems of algebraic equations that arise in finite element or finite difference approximations of partial differential equations. The preconditioners are constructed from exact or approximate solvers for the same partial differential equations restricted to a set of subregions into which the given region has been divided. The iterative substructuring methods based on decompositions of the region into nonoverlapping subregions form one of the main families of such algorithms. The paper presents a number of possibilities on how a variety of fast algorithms can be designed and analyzed.