Lori Badea
Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania

We introduce and analyze a multigrid inexact Uzawa algorithm for solving general saddle point problems. For the definition of the problem and that of the Uzawa algorithm, we adopt the framework introduced in I. Ekeland and R. Temam, Convex analysis and variational problems, North-Holland Publishing Company, Amsterdam, Oxford, 1976, where the saddle point is defined by optimizations on convex sets. The results are obtained for Hilbert spaces and therefore they can be applied to obtain convergence results for the multigrid methods in finite element spaces. We prove the convergence of the algorithm and, also, we give new convergence proofs for the Uzawa algorithm itself in order to better characterize its convergence. The numerical experiments performed for the driven-cavity Stokes problem showed a very good convergence of the proposed multigrid method, even better than that theoretically obtained.