We construct finite element subspaces of the space of symme- tric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger-Reissner mixed formulation of the elasticty equations, when stan- dard discontinous finite element spaces are used to approximate the displace- ment field. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and there is one for each positive value of the polynomial degree used for the displacements. For each degree, these provide a stable finite element discretization. The construction of the spaces is closely tied to discretizations of the elasticity complex, and can be viewed as the three-dimensional analogue of the triangular element family for plane elasticity previously proposed by Arnold and Winther.
['Douglas N. Arnold', 'Gerard Awanou', 'Ragnar Winther']
FINITE ELEMENTS FOR SYMMETRIC TENSORS IN THREE DIMENSIONS