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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
DOI: https://doi.org/10.1090/S0025-5718-00-01222-9
Published electronically: February 21, 2000
MathSciNet review: 1709148
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Abstract: 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
Email: bramble@math.tamu.edu

Xuejun Zhang
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843
Email: xzhang@math.tamu.edu

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