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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Uniform convergence of the multigrid V-cycle for an anisotropic problem
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by James H. Bramble and Xuejun Zhang PDF
Math. Comp. 70 (2001), 453-470 Request permission

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.
  • © Copyright 2000 American Mathematical Society
  • Journal: Math. Comp. 70 (2001), 453-470
  • MSC (2000): Primary 65N30; Secondary 65F10
  • DOI: https://doi.org/10.1090/S0025-5718-00-01222-9
  • MathSciNet review: 1709148