 
Summary: PREPRINT
FINITE ELEMENTS FOR SYMMETRIC TENSORS IN THREE
DIMENSIONS
DOUGLAS N. ARNOLD, GERARD AWANOU, AND RAGNAR WINTHER
Abstract. We construct finite element subspaces of the space of symme
tric tensors with squareintegrable divergence on a threedimensional domain.
These spaces can be used to approximate the stress field in the classical
HellingerReissner mixed formulation of the elasticty equations, when stan
dard discontinuous 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 threedimensional analogue of the triangular element family for plane
elasticity previously proposed by Arnold and Winther.
1. Introduction
The classical Lagrange finite element spaces provide natural simplicial finite el
ement discretizations of the Sobolev space H1
. Similarly, various finite element
