Strong vs. Weak Symmetry in Stress-Based Mixed Finite Element Methods for Linear Elasticity

Abstract

Based on the Hellinger-Reissner principle, accurate stress approximations can be computed directly in suitable H(div)-like finite element spaces treating conservation of momentum and the symmetry of the stress tensor as constraints. Two stress finite element spaces of polynomial degree 2 which were proposed in this context will be compared and relations between the two will be established. The first approach uses Raviart-Thomas spaces of next-to-lowest degree and is therefore H(div)-conforming but produces only weakly symmetric stresses. The stresses obtained from the second approach satisfy symmetry exactly but are nonconforming with respect to H(div). It is shown how the latter finite element space can be derived by augmenting the componentwise next-to-lowest Raviart-Thomas space with suitable bubbles. However, the convergence order of the resulting stress approximation is reduced from two to one as will be confirmed by numerical results. Finally, the weak stress symmetry property of the first approach is discussed in more detail and a post-processing procedure for the construction of stresses which are element-wise symmetric on average is proposed.