Finite element approximations of symmetric tensors on simplicial grids in Rn: the lower order case
For researchers in computational mechanics and numerical analysis, this provides more efficient mixed finite elements for linear elasticity, significantly reducing computational cost.
This paper constructs lower-order finite element subspaces for symmetric tensors with square-integrable divergence on simplicial grids in any dimension, reducing total degrees of freedom per element from 21+3 to 18+3 in 2D and from 156+6 to 48+6 in 3D for first-order mixed elements.
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.