Multilevel iterative solution and adaptive mesh refinement for mixed finite element discretizations

We consider the numerical solution of elliptic boundary value problems by mixed finite element discretizations on simplicial triangulations. Emphasis is on the efficient iterative solution of the discretized problems by multilevel techniques and on adaptive grid refinement. The iterative process relies on a preconditioned conjugate gradient iteration in a suitably chosen subspace with a multilevel preconditioner of hierarchical type that can be constructed by means of appropriate multilevel decompositions of the mixed ansatz spaces. Using the Dryja-Widlund theory of additive Schwarz iterations, we show that the spectral condition number of the preconditioned stiffness matrix asymptotically exhibits a quadratic growth in the refinement level as it is the case in the standard conforming approach. The adaptive grid refinement is based on an efficient and reliable a posteriori error estimator for the total error in the flux which can be established by a defect correction in higher order mixed ansatz spaces combined with a hierarchical two-level splitting.