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Mathematics of Computation

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New convergence estimates for multigrid algorithms


Authors: James H. Bramble and Joseph E. Pasciak
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
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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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DOI: https://doi.org/10.1090/S0025-5718-1987-0906174-X
Article copyright: © Copyright 1987 American Mathematical Society