In this thesis we study the obstacle problem. It is a free boundary problem and the computation of approximate solution can be difficult and expensive. This thesis addresses some aspects of this issue.

We discretize the gradient with lowest order Raviart-Thomas elements and functional values by piecewise constant elements. Existence and uniqueness of solutions to the discrete problems is studied and error estimates are obtained. We develop the Uzawa-type algorithms for the discrete system of linear equations and inequalities that results from the discretization of the mixed formulation of the obstacle problem.

The convergence of classical Uzawa method is analyzed and we display numerical results that agree with theoretical results.