Efficient numerical solution of mixed finite element discretizations by adaptive multilevel methods

We consider mixed finite element discretizations of second order elliptic boundary value problems. Emphasis is on the efficient iterative solution by multilevel techniques with respect to an adaptively generated hierarchy of nonuniform triangulations. In particular, we present two multilevel solvers, the first one relying on ideas from domain decomposition and the second one resulting from mixed hybridization. Local refinement of the underlying triangulations is done by efficient and reliable a posteriori error estimators which can be derived by a defect correction in higher order ansatz spaces or by taking advantage of superconvergence results. The performance of the algorithms is illustrated by several numerical examples.