A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids

A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed, where the stress is approximated by symmetric H(div)-Pk polynomial tensors and the displacement is approximated by C−1-Pk−1 polynomial vectors, for all k ⩽ 4. The main ingredients for the analysis are a new basis of the space of symmetric matrices, an intrinsic H(div) bubble function space on each element, and a new technique for establishing the discrete inf-sup condition. In particular, they enable us to prove that the divergence space of the H(div) bubble function space is identical to the orthogonal complement space of the rigid motion space with respect to the vector-valued Pk−1 polynomial space on each tetrahedron. The optimal error estimate is proved, verified by numerical examples.