Preconditioned iterative methods applied to singularly perturbed elliptic boundary value problems
A class of multigrid preconditioners is developed, analyzed, and tested in connection with the iterative solution of multi-dimensional boundary value problems for which the magnitude of the coefficients of the lower order terms are large compared to the coefficients of the highest order terms. In particular, this occurs in connection with singularly perturbed elliptic boundary value problems, where the coefficients are characterized by a parameter, epsilon, that varies over a specified parameter range. Conjugate gradient type iterative methods are applied to the large, sparse system of linear equations arising from a finite element or finite difference discretization. It is proved for variable coefficient partial differential operators that the condition number of the preconditioned matrix is uniformly bounded for all epsilon, provided the number of grid levels in a ''partial'' multigrid cycle is suitably chosen as a function of epsilon. The multigrid cycle is applied to the five-point scheme for the Dirichlet problem, even when more complicated differential operators and higher order discretizations are considered. This preconditioner is well suited for implementation on vector and parallel computers.
- Research Organization:
- Brookhaven National Lab., Upton, NY (USA)
- DOE Contract Number:
- AC02-76CH00016
- OSTI ID:
- 6364627
- Report Number(s):
- BNL-51916; ON: DE86003125
- Country of Publication:
- United States
- Language:
- English
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