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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 -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

DOI:
https://doi.org/10.1090/S0025-5718-00-01222-9

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