Superconvergence Analysis for the Stable Conforming Rectangular Mixed Finite Elements for the Linear Elasticity Problem

Abstract

In this paper, we consider the linear elasticity problem based on the Hellinger-Reissner variational principle. An O(h 2 ) order superclose property for the stress and displacement and a global superconvergence result of the displacement are established by employing a Clement interpolation, an integral identity and appropriate postprocessing techniques. In this paper, we consider the mixed finite element (for short MFE) approximation of a stress-displacement system derived from the Hellinger-Reissner variational principle for the linear elasticity problem. As is known to all, the MFE methods require that the pair of finite element spaces satisfying the B-B condition. Although there are a number of well-known stable MFEs for the analogous problems involving vector fields and scalar fields (1), the combination of the symmetry and continuity conditions of the stress field is a substantial additional difficulty. On the other hand, a lot of efforts, dating back four decades, have been devoted to develop stable MFEs for the linear elasticity problem, but no stable MFE scheme with polynomial shape functions are yielded. Not until the year 2002, were there some development in this direction. In (2), a sufficient condition was given and then a family of stable MFEs were constructed with respect to arbitrary triangular meshes, with 24 stress and 6 displacement degrees of freedom for the lowest order element, and an optimal order error estimate was obtained. An analogous family of conforming MFEs based on rectangular meshes were proposed in (3), involving 45 stress and 12 displacement degrees of freedom for the lowest order element. Two nonconforming triangular elements were presented in (4) with 12 degrees of freedom for the stress and 3 degrees of freedom for the displacement. Although many stable elements have been constructed for this problem, they involve too much degrees of freedom. Recently, some more simple elements have been developed. In (5), a group of nonconforming rectangular elements were introduced, with the convergence order of