The simplest conforming anisotropic rectangular and cubic mixed finite elements for elasticity

In this paper, we construct two simplest conforming rectangular elements for the linear elasticity problem under the Hellinger–Reissner variational principle. One is a rectangular element in 2D with only 8 degrees of freedom for the stress and 2 degrees of freedom for the displacement. Another one is a cubic element in 3D with only 18 + 3 degrees of freedom. We prove that the two elements are stable and anisotropic convergent. Numerical test is presented to illustrate the element is stable and effective.