New convergence estimates for multigrid algorithms
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 ..cap alpha.. epsilon (0, 1) and the number of relaxations m. We show that for the symmetric and nonsymmetric ..nu.. cycles, the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1-cj/sup -(1-//sup ..cap alpha..//sup )///sup ..cap alpha../, where j is the number of grid levels. We then define a generalized ..nu.. 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 ..cap 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.
- Authors:
- Publication Date:
- Research Org.:
- Itek Optical Systems, Litton Industries, 10 Maguire Road, Lexington, Massachusetts 02173
- OSTI Identifier:
- 6134267
- Resource Type:
- Journal Article
- Journal Name:
- Math. Comput.; (United States)
- Additional Journal Information:
- Journal Volume: 49:180
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; PARTIAL DIFFERENTIAL EQUATIONS; NUMERICAL SOLUTION; ALGORITHMS; CONVERGENCE; DIFFERENTIAL EQUATIONS; EQUATIONS; MATHEMATICAL LOGIC; 990230* - Mathematics & Mathematical Models- (1987-1989)
Citation Formats
Bramble, J H, and Pasciak, J E. New convergence estimates for multigrid algorithms. United States: N. p., 1987.
Web. doi:10.1090/S0025-5718-1987-0906174-X.
Bramble, J H, & Pasciak, J E. New convergence estimates for multigrid algorithms. United States. https://doi.org/10.1090/S0025-5718-1987-0906174-X
Bramble, J H, and Pasciak, J E. 1987.
"New convergence estimates for multigrid algorithms". United States. https://doi.org/10.1090/S0025-5718-1987-0906174-X.
@article{osti_6134267,
title = {New convergence estimates for multigrid algorithms},
author = {Bramble, J H and Pasciak, J E},
abstractNote = {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 ..cap alpha.. epsilon (0, 1) and the number of relaxations m. We show that for the symmetric and nonsymmetric ..nu.. cycles, the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1-cj/sup -(1-//sup ..cap alpha..//sup )///sup ..cap alpha../, where j is the number of grid levels. We then define a generalized ..nu.. 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 ..cap 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.},
doi = {10.1090/S0025-5718-1987-0906174-X},
url = {https://www.osti.gov/biblio/6134267},
journal = {Math. Comput.; (United States)},
number = ,
volume = 49:180,
place = {United States},
year = {Thu Oct 01 00:00:00 EDT 1987},
month = {Thu Oct 01 00:00:00 EDT 1987}
}