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

Author: Martin Vohralík
Journal: Math. Comp. 79 (2010), 2001-2032
MSC (2010): Primary 65N15, 65N30, 76S05
Published electronically: May 26, 2010
MathSciNet review: 2684353
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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 postprocessing 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 explicit use of the uniform discrete $ \inf$-$ \sup$ condition and in a straightforward manner and under minimal necessary assumptions, known convergence and superconvergence results are recovered. The same framework then turns out to lead to optimal a posteriori energy 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 tensor. They are thus suitable for both overall error control and adaptive mesh refinement. Moreover, the developed abstract framework and a posteriori error estimates are quite 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 $ H^1_0(\Omega)$ space and also establish several links between mixed finite element approximations and some generalized weak solutions.

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Additional Information

Martin Vohralík
Affiliation: UPMC Université Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France –and– CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France

Keywords: 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
Received by editor(s): July 7, 2008
Received by editor(s) in revised form: August 6, 2009
Published electronically: May 26, 2010
Additional Notes: This work was supported by the GNR MoMaS project “Numerical Simulations and Mathematical Modeling of Underground Nuclear Waste Disposal”, PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN, France.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.