Multilevel Iterative Solution and Adaptive Mesh Reenement for Mixed Nite Element Discretizations

We consider the numerical solution of elliptic boundary value problems by mixed nite element discretizations on simplicial triangulations. Emphasis is on the eecient iterative solution of the discretized problems by multilevel techniques and on adaptive grid reenement. 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 de-compositions of the mixed ansatz spaces. Using the Dryja-Widlund theory of additive Schwarz iterations, we show that the spectral condition number of the preconditioned stiiness matrix asymptotically exhibits aquadratic growth in the reenement level as it is the case in the standard conforming approach. The adaptive grid reenement is based on an eecient and reliable a posteriori error estimator for the total error in the ux which can be established by a defect correction in higher order mixed ansatz spaces combined with a hierarchical two-level splitting.