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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


New convergence estimates for multigrid algorithms
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by James H. Bramble and Joseph E. Pasciak PDF
Math. Comp. 49 (1987), 311-329 Request permission


In this paper, new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems. Our theory relates the convergence of multigrid algorithms to a "regularity and approximation" parameter $\alpha \in (0,1]$ and the number of relaxations m. We show that for the symmetric and nonsymmetric $\mathcal {V}$ cycles, the multigrid iteration converges for any positive m at a rate which deteriorates no worse than $1 - c{j^{ - (1 - \alpha )/\alpha }}$, where j is the number of grid levels. We then define a generalized $\mathcal {V}$ cycle algorithm which involves exponentially increasing (for example, doubling) the number of smoothings on successively coarser grids. We show that the resulting symmetric and nonsymmetric multigrid iterations converge for any $\alpha$ with rates that are independent of the mesh size. The theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods.
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Math. Comp. 49 (1987), 311-329
  • MSC: Primary 65Nxx; Secondary 65F10
  • DOI:
  • MathSciNet review: 906174