Lower order rectangular nonconforming mixed finite element for the three-dimensional elasticity problem

In this paper, we propose a first-order rectangular nonconforming element for the stress-displacement system derived from the Hellinger-Reissner variational principle for the three-dimensional elasticity problem. We show that the discrete inf-sup condition holds for this scheme. Based on some superconvergence of the consistency error, we prove the optimal error estimate of $\mathcal{O}(h)$ for both the displacement and stress.