This paper constructs a lower order mixed finite element for the linear elasticity problem in 3D. The discrete stresses are piecewise cubic polynomials, and the discrete displacements are discontinuous piecewise quadratic polynomials. The continuity of the discrete stress space is characterized by moving all the edge degrees of freedom of the analogous Hu-Zhang stress element for $P_3$ [Hu, Zhang, Sci. Math. China, 2015, Hu, J. Comput. Math., 2015] to the faces. The macro-element technique is used to define an interpolation operator for proving the discrete stability.

翻译：本文构建了一种用于三维线性弹性问题的低阶混合有限元。离散应力为分片三次多项式，离散位移为间断分片二次多项式。离散应力空间的连续性通过将类比于$P_3$的Hu-Zhang应力单元[Hu, Zhang, Sci. Math. China, 2015; Hu, J. Comput. Math., 2015]的所有边自由度移至面来实现。采用宏单元技术定义插值算子以证明离散稳定性。