Schwarz Analysis of Iterative Substructuring Algorithms for Elliptic Problems in Three Dimensions

• Published in: SIAM journal on numerical analysis Vol. 31; no. 6; pp. 1662 - 1694
• Main Authors: Dryja, Maksymilian, Smith, Barry F., Widlund, Olof B.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.12.1994
• Subjects: Algebra; ALGORITHMS; Applied mathematics; CALCULATION METHODS; Coefficients; Constant coefficients; CONVERGENCE; Decomposition; Decomposition methods; Design; DIFFERENTIAL EQUATIONS; EQUATIONS; Exact sciences and technology; GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; Interpolation; ITERATIVE METHODS; Laboratories; Linear and multilinear algebra, matrix theory; Mathematical constants; MATHEMATICAL LOGIC 990200 > Mathematics & Computers; Mathematics; Methods; Numerical analysis; Numerical analysis. Scientific computation; PARTIAL DIFFERENTIAL EQUATIONS; Partial differential equations, boundary value problems; Sciences and techniques of general use; THREE-DIMENSIONAL CALCULATIONS; Vertices; Numerical approximation; Finite element method; Elliptic equation; Algebraic equation; Convergence rate; Iterative method; Algorithm; Domain decomposition; Partial differential equation; Finite difference method; Equation system
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/0731086

Domain decomposition methods provide powerful preconditioners for the iterative solution of the 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 equation restricted to a set of subregions into which the given region has been divided. In addition, the preconditioner is often augmented by a coarse, second-level approximation that provides additional, global exchange of information that can enhance the rate of convergence considerably. The iterative substructuring methods, based on decompositions of the region 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 fully defined in terms of a set of subspaces and auxiliary bilinear forms. A general theoretical framework has previously been developed. In this paper, these techniques are used in an analysis of iterative substructuring 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 differential equation. Domain decomposition algorithms can conveniently be built from modules that represent local and global components of the preconditioner. 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.