New convergence estimates for multigrid algorithms

Journal Article · Thu Oct 01 04:00:00 EDT 1987 · Math. Comput.; (United States)

DOI:https://doi.org/10.1090/S0025-5718-1987-0906174-X· OSTI ID:6134267

Bramble, J H; Pasciak, J E

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.

• Research Organization: Itek Optical Systems, Litton Industries, 10 Maguire Road, Lexington, Massachusetts 02173
• OSTI ID: 6134267
• Journal Information: Math. Comput.; (United States), Journal Name: Math. Comput.; (United States) Vol. 49:180; ISSN MCMPA
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990230* -- Mathematics & Mathematical Models-- (1987-1989)
ALGORITHMS
CONVERGENCE
DIFFERENTIAL EQUATIONS
EQUATIONS
MATHEMATICAL LOGIC
NUMERICAL SOLUTION
PARTIAL DIFFERENTIAL EQUATIONS