Finite elements for symmetric tensors in three dimensions

Abstract

Abstract. We construct finite element subspaces of the space of symmetric 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 standard discontinuous finite element spaces are used to approximate the displacement 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.

Symmetric Matrix Fields in the Finite Element Method

“…has the required dimension 6 was proved in [21]. An analogous element was obtained in [22], where the last set of degrees of freedom was replaced by ∫ e T t · t dx for each of the six edges.…”

“…The above sequences are exact if the domain Ω is contractible [20,21]. In an attempt to mimic the polynomial sequence at the discrete level,…”

“…The degrees of freedom are defined in a fashion similar to the lowest order case [21]. We obtained O(h k+2 ) convergence rate for the stress and O(h k+1 ) convergence rate for the displacement.…”

Inf–sup conditions for twofold saddle point problems

“…First Inf-Sup Condition: The issue of finding conforming finite elements for symmetric tensors satisfying an inf-sup condition of the form (4.2) is well-documented ( [14,32,4,5]). We consider the finite element pairs (S h , U h ) of symmetric tensors and vectors constructed by Arnold and Winther [7,1] which satisfy the inf-sup condition.…”

“…Figure 4.1 gives a diagram of the degrees of freedom on each triangle for the lowest order Arnold-Winther (S h , u h ) pair (k = 1) in two dimensions. In [7,1] an interpolation operator…”