Solving elliptic finite element systems in near-linear time with support preconditioners.

Journal Article · Sat Jan 01 04:00:00 UTC 2005 · Proposed for publication in the SIAM Journal on Matrix Analysis.

OSTI ID:951697

Vavasis, Stephen ^[1]; Hendrickson, Bruce Alan; Boman, Erik Gunnar

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1. Cornell University, Ithica, NY

We consider linear systems arising from the use of the finite element method for solving a certain class of linear elliptic problems. Our main result is that these linear systems, which are symmetric and positive semidefinite, are well approximated by symmetric diagonally dominant matrices. Our framework for defining matrix approximation is support theory. Significant graph theoretic work has already been developed in the support framework for preconditioners in the diagonally dominant case, and in particular it is known that such systems can be solved with iterative methods in nearly linear time. Thus, our approximation result implies that these graph theoretic techniques can also solve a class of finite element problems in nearly linear time. We show that the quality of our approximation, which controls the number of iterations in the preconditioned iterative solver, depends primarily on a mesh quality measure but not on the problem size or shape of the domain.

• Research Organization: Sandia National Laboratories
• Sponsoring Organization: USDOE
• DOE Contract Number: AC04-94AL85000
• OSTI ID: 951697
• Report Number(s): SAND2005-0668J
• Journal Information: Proposed for publication in the SIAM Journal on Matrix Analysis., Journal Name: Proposed for publication in the SIAM Journal on Matrix Analysis.
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
99 GENERAL AND MISCELLANEOUS
APPROXIMATIONS
ELLIPTICAL CONFIGURATION
FINITE ELEMENT METHOD
ITERATIVE METHODS
MATHEMATICAL SOLUTIONS