This paper constructs the first mixed finite element for the linear elasticity problem in 3D using $P_3$ polynomials for the stress and discontinuous $P_2$ polynomials for the displacement on tetrahedral meshes under some mild mesh conditions. The degrees of freedom of the stress space as well as the corresponding nodal basis are established by characterizing a space of some piecewise constant symmetric matrices on a patch around each edge. Macro-element techniques are used to define a stable interpolation to prove the discrete inf-sup condition. Optimal convergence is obtained theoretically.

翻译：本文在四面体网格上构建了首个用于三维线性弹性问题的混合有限元，该元在温和网格条件下，对应力采用$P_3$多项式，对位移采用不连续的$P_2$多项式。通过刻画每条边周围区域内分片常数对称矩阵空间，建立了应力空间的自由度及其相应的节点基函数。采用宏元技术定义稳定的插值算子，以证明离散inf-sup条件。理论上获得了最优收敛性。