We consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points. We are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations. We present an iterative method involving two levels of iteration, similar in some respects to the Uzawa algorithm. We relate the rates of convergence of the outer and inner iterations, proving that, under natural hypotheses, the outer iteration achieves the rate of convergence of the inner iteration. The technique is applied to finite element approximations of the Stokes equations.

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We derive and analyze an a posteriori error estimate for the mini-element discretization of the Stokes equations. The estimate is based on the solution of a local Stokes problem in each element of the nite element mesh, using spaces of quadratic bump functions for both velocity and pressure errors. This results in solving a 9 9 system which reduces to two 3 3 systems easily invertible. Comparisons with other estimates based on a Petrov-Galerkin solution are used in our analysis, which shows that it provides a reasonable approximation of the actual discretization error. Numerical experiments clearly show the e ciency of such an estimate in the solution of self adaptive mesh re nement procedures.

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