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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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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

Abstract:

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: https://doi.org/10.1090/S0025-5718-1987-0906174-X
  • MathSciNet review: 906174