Two-Level Local Refinement Preconditioners (Ewing et al.)
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SIAM Journal on Scientific Computing
Volume 15-1, January 1994, pp. 149-163
(C) 1994 by Society for Industrial and Applied Mathematics
All rights reserved
Title: Two-Level Local Refinement Preconditioners for
Nonsymmetric and Indefinite Elliptic Problems
Author: Richard E. Ewing, Svetozara I. Petrova, and
Panayot S. Vassilevski
AMS Subject
Classifications: 65F10, 65N20, 65N30
Key words: indefinite problems, nonsymmetric problems, optimal
order preconditioner, local refinement, two-level
ethod, generalized conjugate gradients, elliptic
problems
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ABSTRACT
Preconditioners of optimal order for nonselfadjoint and indefinite
elliptic boundary value problems discretized on grids with local
refinement are constructed. The proposed technique utilizes solution
of a discrete problem on a uniform coarse grid; then, the reduced
problem is handled by a generalized conjugate gradient (GCG)
method. The reduced problem is coercive if the initial coarse
mesh is sufficiently fine and is local, solving only for the
unknowns on the subdomains where local refinement has been introduced.
The reduced problem can be preconditioned by a preconditioner for the
symmetric positive definite matrix arising from the symmetric and
coercive principal part of the original bilinear form restricted to
the subdomains containing local refinement. This problem also utilizes
a uniform grid. In the numerical tests, the recent algebraic
multilevel (AMLI) preconditioners [Axelsson and Vassilevski,
{\it SIAM J.\ Numer.\Anal}., 27 (1990), pp.\ 1569--1590;
Saad and Schultz, {\it SIAM J.\ Sci.\ Statist.\ Comput}., 7 (1986),
pp.\ 856--869], which are of optimal order for selfadjoint and
coercive elliptic problems, were used.
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