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Uniform convergence of the multigrid V-cycle for an anisotropic problem
Author(s):
James
H.
Bramble;
Xuejun
Zhang.
Journal:
Math. Comp.
70
(2001),
453-470.
MSC (2000):
Primary 65N30;
Secondary 65F10
Posted:
February 21, 2000
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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  -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:
10.1090/S0025-5718-00-01222-9
PII:
S 0025-5718(00)01222-9
Received by editor(s):
December 4, 1997
Received by editor(s) in revised form:
June 23, 1998 and April 6, 1999
Posted:
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.
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Copyright
2000,
American Mathematical Society
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