The simplest mixed finite element method for linear elasticity in the symmetric formulation on $n$-rectangular grids

A family of mixed finite elements is proposed for solving the first order system of linear elasticity equations in any space dimension, where the stress field is approximated by symmetric finite element tensors. This family of elements has a perfect matching between the stress components and the displacement. The discrete spaces for the normal stress $\sigma_{ii}$, the shear stress $\sigma_{ij}$ and the displacement $u_i$ are $\operatorname{span}\{1,x_i\}$, $\operatorname{span}\{1,x_i,x_j\}$ and $\operatorname{span}\{1\}$, respectively, on rectangular grids. In particular, the definition remains the same for all space dimensions. As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. In 1D, this element is nothing else but the 1D Raviart-Thomas element, which is the only conforming element in this family. In 2D and higher dimensions, they are new elements but of the minimal degrees of freedom. The total degrees of freedom per element is 2 plus 1 in 1D, 7 plus 2 in 2D, and 15 plus 3 in 3D. The previous record of the least degrees of freedom is, 13 plus 4 in 2D, and 54 plus 12 in 3D, on the rectangular grid. These elements are the simplest element for any space dimension. The well-posedness condition and the optimal a priori error estimate of the family of finite elements are proved for both pure displacement and traction problems. Numerical tests in 2D and 3D are presented to show a superiority of the new element over others, as a superconvergence is surprisingly exhibited.