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

Journal Article · Thu Dec 01 04:00:00 EST 1994 · SIAM Journal on Numerical Analysis (Society for Industrial and Applied Mathematics); (United States)

DOI:https://doi.org/10.1137/0731086· OSTI ID:6835567

Dryja, M ^[1]; Smith, B F ^[2]; Widlund, O B ^[3]

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1. Warsaw Univ. (Poland). Dept. of Mathematics
2. Univ. of California, Los Angeles, CA (United States). Dept. of Mathematics
3. Courant Inst. of Mathematical Sciences, New York, NY (United States)

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.

• DOE Contract Number: FG02-92ER25127; FG02-88ER25053
• OSTI ID: 6835567
• Journal Information: SIAM Journal on Numerical Analysis (Society for Industrial and Applied Mathematics); (United States), Journal Name: SIAM Journal on Numerical Analysis (Society for Industrial and Applied Mathematics); (United States) Vol. 31:6; ISSN 0036-1429; ISSN SJNAAM
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
ALGORITHMS
CALCULATION METHODS
CONVERGENCE
DIFFERENTIAL EQUATIONS
EQUATIONS
ITERATIVE METHODS
MATHEMATICAL LOGIC
PARTIAL DIFFERENTIAL EQUATIONS
THREE-DIMENSIONAL CALCULATIONS