Abstract:
We are concerned with the development, analysis and implementation
of adaptive mixed and nonconforming finite element methods (MFEM
and NFEM) for second order elliptic PDEs. In case of standard
conforming P1 approximations, such methods have been considered
previously in [3]. The methods presented in this
contribution provide an error reduction and thus guarantee
convergence of the adaptive loop which consists of the essential
steps 'SOLVE', 'ESTIMATE', 'MARK', and 'REFINE'. Here, 'SOLVE'
stands for the efficient solution of the finite element
discretized problems. The following step 'ESTIMATE' is devoted to
the a posteriori error estimation of the global discretization
error. A greedy algorithm is the core of the step 'MARK' to
indicate selected elements for refinement, whereas the final step
'REFINE' deals with the technical
realization of the refinement process itself.
The analysis is carried out for the Poisson equation with
homogeneous Dirichlet boundary data as a model problem and
discretization by the lowest order Raviart-Thomas and
Crouzeix-Raviart finite elements. Important tools in the
convergence proof are the reliability of the estimator, a strong
discrete local efficiency, and quasi-orthogonality properties. The
proof does not require regularity of the solution nor does it make
use of duality arguments.