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Schwarz methods of neumann‐neumann type for three‐dimensional elliptic finite element problems
Author(s) -
Dryja Maksymilian,
Widlund Olof B.
Publication year - 1995
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160480203
Subject(s) - preconditioner , domain decomposition methods , mathematics , finite element method , degrees of freedom (physics and chemistry) , von neumann architecture , domain (mathematical analysis) , neumann boundary condition , elliptic operator , type (biology) , boundary (topology) , operator (biology) , rank (graph theory) , mathematical analysis , pure mathematics , linear system , ecology , biochemistry , physics , chemistry , repressor , quantum mechanics , combinatorics , biology , gene , transcription factor , thermodynamics
Several domain decomposition methods of Neumann‐Neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements. We will only consider problems in three dimensions. Several new variants of the basic algorithm are introduced in a Schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multi‐level methods. The Neumann‐Neumann algorithms have several advantages over other domain decomposition methods. The subregions, which define the subproblems, only share the boundary degrees of freedom with their neighbors. The subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries. In its original form, however, the algorithm lacks a mechanism for global transportation of information and its performance therefore suffers when the number of subregions increases. In the new variants of the algorithms, considered in this paper, the preconditioners include global components, of low rank, to overcome this difficulty. Bounds are established for the condition number of the iteration operator, which are independent of the number of subregions, and depend only polylogarithmically on the number of degrees of freedom of individual local subproblems. Results are also given for problems with arbitrarily large jumps in the coefficients across the interfaces separating the subregions. ©1995 John Wiley & Sons, Inc.