Analysis of iterative methods for saddle point problems: a unified approach

Author:
Walter Zulehner

Journal:
Math. Comp. **71** (2002), 479-505

MSC (2000):
Primary 65N22, 65F10

DOI:
https://doi.org/10.1090/S0025-5718-01-01324-2

Published electronically:
May 14, 2001

MathSciNet review:
1885611

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**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**

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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:
https://doi.org/10.1090/S0025-5718-01-01324-2

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

Published electronically:
May 14, 2001

Article copyright:
© Copyright 2001
American Mathematical Society