Uzawa-type iterative solution methods for constrained saddle point problems
The paper deals with the iterative solution of saddle point problems in finite dimensional real valued spaces. It summarizes the general theory developed by the author with co-authors in the last years. For finite-dimensional saddle point problem with a nonlinear monotone operator and constraints, iterative methods are knows, which can be viewed as preconditioned Uzawa methods or as Uzawa-block relaxation methods. In the first section the author gives some general results on the solvability of problem the considered problem and the convergence of iterative methods. These results are based on three articles from the author and coworkers from the last years. In the following sections some application on the finite element/finete differences discretization of elliptic and parabolic PDEs optimal control problems and the resulting (optimality) conditions, i.e. saddle point problems are considered. When applying iterative method to such problems, one problem is to construct suitable preconditioners that ensure the convergence and effective implementation of iterative methods and to obtain the admissible intervals of iterative parameters which do not depend on mesh parameters. The choice of preconditioners is handled case by case for the considered example problems.
scientific article; zbMATH DE number 6945066
scientific article