|
Analysis of iterative methods for saddle point problems: a unified approach
Author(s):
Walter
Zulehner.
Journal:
Math. Comp.
71
(2002),
479-505.
MSC (2000):
Primary 65N22, 65F10
Posted:
May 14, 2001
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
In this paper two classes of iterative methods for saddle point problems are considered: inexact Uzawa algorithms and a class of methods with symmetric preconditioners. In both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices, a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated eigenvalue problems. The obtained estimates apply for a wider range of situations and are partially sharper than the known estimates in literature. A few numerical tests are given which confirm the sharpness of the estimates.
References:
-
- 1.
- K. Arrow, L. Hurwicz, and H. Uzawa, Studies in nonlinear programming, Stanford University Press, Stanford, CA, 1958. MR 21:7115
- 2.
- R. E. Bank, B. D. Welfert, and H. Yserentant, A class of iterative methods for solving saddle point problems, Numer. Math. 56 (1990), 645 - 666. MR 91b:65035
- 3.
- J. H. Bramble and J. E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. Comp. 50 (1988), 1 - 17. MR 89m:65097a
- 4.
- J. H. Bramble, J. E. Pasciak, and A. T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal. 34 (1997), 1072 - 1092. MR 98c:65182
- 5.
- F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer-Verlag, 1991. MR 92d:65187
- 6.
- H. C. Elman and G. H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal. 31 (1994), 1645 - 1661. MR 95f:65065
- 7.
- M. Fortin and R. Glowinski, Augmented Lagrangian methods: Applications to the numerical solution of boundary value problems, North-Holland, Amsterdam, 1983. MR 85a:49004
- 8.
- W. Hackbusch, Iterative solutions of large sparse systems of equations, Springer Verlag, New York, 1994. MR 94k:65002
- 9.
- J. Iliash, T. Rossi, and J. Toivanen, Two iterative methods for solving the Stokes problem, Tech. Report 2, University of Jyväskylä, Department of Mathematics, Laboratory of Scientific Computing, 1993.
- 10.
- W. Queck, The convergence factor of preconditioned algorithms of the Arrow-Hurwicz type, SIAM J. Numer. Anal. 26 (1989), 1016 - 1030. MR 90m:65071
- 11.
- D. Silvester and A. Wathen, Fast iterative solutions of stabilized Stokes systems. Part II: Using general block preconditioners, SIAM J. Numer. Anal. 31 (1994), 1352 - 1367. MR 95g:65132
- 12.
- R. Verfürth, A combined conjugate gradient-multigrid algorithm for the numerical solution fo the Stokes problem, IMA J. Numer. Anal. 4 (1984), 441 - 455. MR 86f:65200
Similar Articles:
Retrieve articles in Mathematics of Computation
with MSC
(2000):
65N22, 65F10
Retrieve articles in all Journals with MSC
(2000):
65N22, 65F10
Additional Information:
Walter
Zulehner
Affiliation:
Institute of Analysis and Computational Mathematics, Johannes Kepler University, A-4040 Linz, Austria
Email:
zulehner@numa.uni-linz.ac.at
DOI:
10.1090/S0025-5718-01-01324-2
PII:
S 0025-5718(01)01324-2
Keywords:
Indefinite systems,
iterative methods,
preconditioners,
saddle point problems,
Uzawa algorithm
Received by editor(s):
March 3, 1998
Received by editor(s) in revised form:
February 11, 1999 and May 30, 2000
Posted:
May 14, 2001
Copyright of article:
Copyright
2001,
American Mathematical Society
|
Comments: webmaster@ams.org