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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Article copyright:
© Copyright 1987
American Mathematical Society