A new elasticity element made for enforcing weak stress symmetry

Abstract. We introduce a new mixed method for linear elasticity. The nov-elty is a simplicial element for the approximate stress. For every positive inte-ger k, the row-wise divergence of the element space spans the set of polynomials of total degree k. The degrees of freedom are suited to achieve continuity of the normal stresses. What makes the element distinctive is that its dimension is the smallest required for enforcing a weak symmetry condition on the ap-proximate stress. This is achieved using certain “bubble matrices”, which are special divergence-free matrix-valued polynomials. We prove that the approx-imation error is of order k + 1 in both the displacement and the stress, and that a postprocessed displacement approximation converging at order k + 2 can be computed element by element. We also show that the globally coupled degrees of freedom can be reduced by hybridization to those of a displacement approximation on the element boundaries. 1