A new elasticity element made for enforcing weak stress symmetry

Abstract:

We introduce a new mixed method for linear elasticity. The novelty is a simplicial element for the approximate stress. For every positive integer $k$, 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 approximate stress. This is achieved using certain “bubble matrices”, which are special divergence-free matrix-valued polynomials. We prove that the approximation 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.