A family of conforming mixed finite elements for linear elasticity on triangular grids

Abstract

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 \(C^0\)-\(P_k\) space enriched by \((k-1)\) \(H(\d)\) edge bubble functions on each internal edge, while the displacement field by the full discontinuous \(P_{k-1}\) vector-valued space, for the polynomial degree \(k\ge 3\). The main challenge is to find the correct stress finite element space matching the full \(C^{-1}\)-\(P_{k-1}\) 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 \(P_{k-1}\) space of displacement orthogonal to the local rigid-motion. The well-posedness 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.