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Finite element approximations of symmetric tensors on simplicial grids in Rn: the lower order case

In this paper, we construct, in a unified fashion, lower order finite element subspaces of spaces of symmetric tensors with square-integrable divergence on a domain in any dimension. These subspaces are essentially the symmetric H(div)-Pk (1=<k<=n) tensor spaces, enriched, for each n-1 dimensional simplex, by (n+1)n/2 H(div)-Pn+1 bubble functions when 1=< k<= n-1, and by (n-1)n/2 H(div)-P n+1 bubble functions when k= n. These spaces can be used to approximate the symmetric matrix field in a mixed formulation problem where the other variable is approximated by discontinuous piecewise Pk-1 polynomials. This in particular leads to first order mixed elements on simplicial grids with total degrees of freedom per element $18$ plus $3$ in 2D, 48 plus 6 in 3D. The previous record of the degrees of freedom of first order mixed elements is, 21 plus 3 in 2D, and 156 plus 6 in 3D, on simplicial grids. We also derive, in a unified way and without using any tools like differential forms, a family of auxiliary mixed finite elements in any dimension. One example in this family is the Raviart-Thomas elements in one dimension, the second example is the mixed finite elements for linear elasticity in two dimensions due to Arnold and Winther, the third example is the mixed finite elements for linear elasticity in three dimensions due to Arnold, Awanou and Winther.