Inexact and Preconditioned Uzawa Algorithms for Saddle Point Problems

Variants of the Uzawa algorithm for solving symmetric indefinite linear systems are developed and analyzed. Each step of this algorithm requires the solution of a symmetric positive-definite system of linear equations. It is shown that if this computation is replaced by an approximate solution produced by an arbitrary iterative method, then with relatively modest requirements on the accuracy of the approximate solution, the resulting inexact Uzawa algorithm is convergent, with a convergence rate close to that of the exact algorithm. In addition, it is shown that preconditioning can be used to improve performance. The analysis is illustrated and supplemented using several examples derived from mixed finite element discretization of the Stokes equations.