In this paper, we construct two lower order mixed elements for the linear elasticity problem in the Hellinger-Reissner formulation, one for the 2D problem and one for the 3D problem, both on macro-element meshes. The discrete stress spaces enrich the analogous $P_k$ stress spaces in [J. Hu and S. Zhang, arxiv, 2014, J. Hu and S. Zhang, Sci. China Math., 2015] with simple macro-element bubble functions, and the discrete displacement spaces are discontinuous piecewise $P_{k-1}$ polynomial spaces, with $k=2,3$, respectively. Discrete stability and optimal convergence is proved by using the macro-element technique. As a byproduct, the discrete stability and optimal convergence of the $P_2-P_1$ mixed element in [L. Chen and X. Huang, SIAM J. Numer. Anal., 2022] in 3D is proved on another macro-element mesh. For the mixed element in 2D, an $H^2$-conforming composite element is constructed and an exact discrete elasticity sequence is presented. Numerical experiments confirm the theoretical results.

翻译：本文针对Hellinger-Reissner变分形式下的线性弹性问题，在宏单元网格上分别构造了适用于二维与三维问题的两种低阶混合元。离散应力空间通过在[J. Hu and S. Zhang, arxiv, 2014; J. Hu and S. Zhang, Sci. China Math., 2015]中类$P_k$应力空间的基础上添加简单宏单元气泡函数进行增强，离散位移空间采用分片间断的$P_{k-1}$多项式空间，其中$k$分别取2和3。通过宏单元技术证明了离散格式的稳定性与最优收敛性。作为推论，在另一类宏单元网格上证明了[L. Chen and X. Huang, SIAM J. Numer. Anal., 2022]中三维$P_2-P_1$混合元的离散稳定性与最优收敛性。针对二维混合元，构造了$H^2$相容的复合单元并给出了精确的离散弹性序列。数值实验验证了理论结果。