A Family of Conforming Mixed Finite Elements for Linear Elasticity on Triangular Grids

This paper presents a family of mixed finite elements on triangular grids for solving the classical Hellinger-Reissner mixed problem of the elasticity equations. In these elements, the matrix-valued stress field is approximated by the full C0-Pk space enriched by (k − 1) H(div) edge bubble functions on each internal edge, while the displacement field by the full discontinuous Pk−1 vector-valued space, for the polynomial degree k ≥ 3. As a result, compared with most of mixed elements for linear elasticity in the literature, the basis of the stress space is surprisingly easy to construct. The main challenge is to find the correct stress finite element space matching the full C−1-Pk−1 displacement space. The discrete stability analysis for the inf-sup condition does not rely on the usual Fortin operator, which is difficult to construct. It is done by characterizing the divergence of local stress space which covers the Pk−1 space of displacement orthogonal to the local rigid-motion. The wellposedness condition and the optimal a priori error estimate are proved for this family of finite elements. Numerical tests are presented to confirm the theoretical results.