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Uniform convergence of the multigrid V-cycle for an anisotropic problem

Authors: James H. Bramble and Xuejun Zhang
Journal: Math. Comp. 70 (2001), 453-470
MSC (2000): Primary 65N30; Secondary 65F10
Published electronically: February 21, 2000
MathSciNet review: 1709148
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In this paper, we consider the linear systems arising from the standard finite element discretizations of certain second order anisotropic problems with variable coefficients on a rectangle. We study the performance of a V-cycle multigrid method applied to the finite element equations. Since the usual ``regularity and approximation'' assumption does not hold for the anisotropic finite element problems, the standard multigrid convergence theory cannot be applied directly. In this paper, a modification of the theory of Braess and Hackbusch will be presented. We show that the V-cycle multigrid iteration with a line smoother is a uniform contraction in the energy norm. In the verification of the hypotheses in our theory, we use a weighted $L^2$-norm estimate for the error in the Galerkin finite element approximation and a smoothing property of the line smoothers which is proved in this paper.

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Additional Information

James H. Bramble
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843

Xuejun Zhang
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843

Received by editor(s): December 4, 1997
Received by editor(s) in revised form: June 23, 1998, and April 6, 1999
Published electronically: February 21, 2000
Additional Notes: The work of the first author was partially supported by the National Science Foundation under grant #DMS-9626567, and the work of the second author was partially supported by the National Science Foundation under Grant #DMS-9805590.
Article copyright: © Copyright 2000 American Mathematical Society

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