Inexact Uzawa Algorithms for Nonsymmetric Saddle Point Problems

Joseph Pasciak

Dept. of Mathematics, Texas A&M University, College Station, TX 77840


In this talk I shall consider iterative algorithms of Uzawa type for solving nonsymmetric linear block saddle-point problems of the form

	| A   B | |U|    |F|
	|       | | |  = | |
	| B^t 0 | |P|    |G|.

Specifically, I will consider the case when the upper left block A is nonsymmetric linear operator with positive definite symmetric part. Such systems arise, for example, in certain discretizations of Navier-Stokes equations. The main results of the talk are for an inexact Uzawa algorithm. The classical Uzawa algorithm requires the action of the inverse of the operator A . The inexact Uzawa methods replace the action of the exact inverse of A with an ``incomplete'' or ``approximate'' evaluation of its action. In practice, this is provided by a preconditioner for the symmetric part of A such as one multigrid V-cycle. A convergence result for the inexact algorithm will be reported which shows that the iterative algorithm is a contraction in an appropriate norm. This norm convergence is achieved without the assumption of a sufficiently accurate approximation to the inverse of A . Applications of the inexact Uzawa method to the numerical solution of steady state Navier-Stokes equations will also be discussed.