Preconditioned iterative methods applied to singularly perturbed elliptic boundary value problems: Volume 2
It is the purpose of this report to modify and extend results in a previous report, concerning a class of multigrid preconditioners developed for use in the iterative solution of singularly perturbed elliptic boundary value problems. Typically, the low order derivative terms are multiplied by a large parameter, K. The preconditioner was based on a ''partial'' multigrid V cycle applied to the discrete LaPlacian, -..delta../sup h/, where h is the grid size. By this, we mean that the coarse grid residual problem is not solved. At most, one or a few relaxation sweeps are applied on this grid level. The multigrid preconditioner is now extended to include the W cycle applied to -..delta../sup h/ as well as -..delta../sup h/ + K/sup 2/. It is shown how to choose the coarsest grid size as a function of K, h, and the type of multigrid cycle in order to obtain optimal convergence of the preconditioned iterative method. It is proved that the low frequency subspace is scaled by 0(K/sup -2/) for both the V and W cycles. For the W cycle, it is proved that the multigrid preconditioner is norm equivalent to (-..delta../sup h/ + K/sup 2/)/sup -1/, uniformly in both K and h. The results are proved using Fourier analysis.
- Research Organization:
- Brookhaven National Lab., Upton, NY (USA)
- DOE Contract Number:
- AC02-76CH00016
- OSTI ID:
- 6003236
- Report Number(s):
- BNL-51916-Vol.2; ON: DE87014337
- Country of Publication:
- United States
- Language:
- English
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