A Posteriori Error Analysis of Finite Element Methods for Reissner-Mindlin Plates

This paper establishes a very general theory for a posteriori error analysis of finite element methods of the Reissner-Mindlin plate problem in the literature. The theory assures reliability of explicit residual error estimates. The conclusion of this theory is sparsity in the mathematical research of uniform a posteriori error control. Indeed, the a posteriori error estimate for various finite element methods of the Reissner-Mindlin plate problem is reduced to three parts: (1) Check the four conditions (H1)-(H4); (2) Design free functions φ̃h, w̃h, and γ̃h and the free parameter α; (3) Estimate the last six terms of the abstract estimator η̃h. As examples, we apply the present theory to four classes of finite element methods for the Reissner-Mindlin plate problem: the methods based on the linked technique, the Arnold-Falk type methods, the MITC methods, and the discontinuous Galerkin methods. For all these methods, it is proved that the error can be estimated by a computable error estimator from above and below up to multiplicative constants that are independent of both the meshsize and the plate thickness. The error is bounded in norms that are analyzed in the a priori error analysis for all the methods under consideration. Among aforementioned methods, the first class of methods has been analyzed in literature under the saturation assumption. The estimator of this paper improves that result by abandoning that constraint. For the second class of methods, only the Arnold-Falk element has been estimated under the condition t . hK for any element K of the mesh Th with the element diameter hK and the plate thickness t. However, this assumption is removed by the present theory. For other methods of the Arnold-Falk type, there is no a posteriori analysis in literature. For the MITC methods, our theory recovers the results in literature. For the discontinuous Galerkin methods, no a posteriori analysis can be found. It is stressed that, for all the methods which have already been analyzed, the frameworks of the analysis herein are completely different from those used for them in the literature.