Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

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Abstract : We derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second-order elliptic problems. It is based on the classical primal weak formulation, the post processing of the potential proposed in [T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for second-order elliptic problems, Math. Comp. 64 (1995), 943-972], and the discrete Friedrichs inequality. Our analysis in particular avoids any notion of the (discrete) inf-sup condition and in a straightforward manner and under minimal necessary assumptions, all known convergence and super convergence results are recovered. The same framework then turns out to lead to optimal a posteriori error bounds. In particular, estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived. They give a guaranteed and fully computable upper bound on the energy error, represent error local lower bounds, and are robust under some conditions on the diffusion-dispersion tensors. They are thus suitable for both overall error control and adaptive mesh refinement. Moreover, the developed abstract framework and a posteriori error estimates are very general and apply to any locally conservative method. We finally prove that in parallel and simultaneously in converse to Galerkin finite element methods, under some circumstances, the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the H10 (Ω) space and also establish several links between mixed finite element approximations and some generalized weak solutions.

Mots Clés: Mixed finite element methods; A priori error estimates; Inf-sup condition; Postprocessing; Discrete Friedrichs inequality; Locally conservative methods; A posteriori error estimates; Guaranteed upper bound; Orthogonal projection; Generalized weak solution

Date: 2008-07-08