Adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems

We consider mixed finite element discretizations of linear second-order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. By a well-known postprocessing technique the discrete problem is equivalent to a modified nonconforming discretization which is solved by preconditioned CG iterations using a multilevel preconditioner in the spirit of Bramble, Pasciak, and Xu designed for standard nonconforming approximations. Local refinement of the triangulations is based on an a posteriori error estimator which can be easily derived from superconvergence results. The performance of the preconditioner and the error estimator is illustrated by several numerical examples.