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Mathematics of Computation

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Uzawa type algorithms
for nonsymmetric saddle point problems

Authors: James H. Bramble, Joseph E. Pasciak and Apostol T. Vassilev
Journal: Math. Comp. 69 (2000), 667-689
MSC (1991): Primary 65N22, 65N30, 65F10
Published electronically: May 17, 1999
MathSciNet review: 1659867
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Abstract: In this paper, we consider iterative algorithms of Uzawa type for solving linear nonsymmetric saddle point problems. Specifically, we consider systems, written as usual in block form, where the upper left block is an invertible linear operator with positive definite symmetric part. Such saddle point problems arise, for example, in certain finite element and finite difference discretizations of Navier-Stokes equations, Oseen equations, and mixed finite element discretization of second order convection-diffusion problems. We consider two algorithms, each of which utilizes a preconditioner for the operator in the upper left block. Convergence results for the algorithms are established in appropriate norms. The convergence of one of the algorithms is shown assuming only that the preconditioner is spectrally equivalent to the inverse of the symmetric part of the operator. The other algorithm is shown to converge provided that the preconditioner is a sufficiently accurate approximation of the inverse of the upper left block. Applications to the solution of steady-state Navier-Stokes equations are discussed, and, finally, the results of numerical experiments involving the algorithms are presented.

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Additional Information

James H. Bramble
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Joseph E. Pasciak
Affiliation: Schlumberger, 8311 N. FM 620, Austin, Texas 78726

Keywords: Indefinite systems, iterative methods, preconditioners, saddle point problems, nonsymmetric saddle point systems, Navier-Stokes equations, Oseen equations, Uzawa algorithm
Received by editor(s): July 8, 1997
Received by editor(s) in revised form: June 23, 1998
Published electronically: May 17, 1999
Additional Notes: This manuscript was written under contract number DE-AC02-76CH00016 with the U.S. Department of Energy. Accordingly, the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. This work was also supported in part under the National Science Foundation Grant No. DMS-9626567 and by the Schlumberger Technology Corporation.
Article copyright: © Copyright 2000 American Mathematical Society

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