New lower order $H(\textrm{div})$-conforming finite elements for symmetric tensors are constructed in arbitrary dimension. The space of shape functions is defined by enriching the symmetric quadratic polynomial space with the $(d+1)$-order normal-normal face bubble space. The reduced counterpart has only $d(d+1)^2$ degrees of freedom. In two dimensions, basis functions are explicitly given in terms of barycentric coordinates. Lower order conforming finite element elasticity complexes starting from the Bell element, are developed in two dimensions. These finite elements for symmetric tensors are applied to devise robust mixed finite element methods for the linear elasticity problem, which possess the uniform error estimates with respect to the Lam\'{e} coefficient $\lambda$, and superconvergence for the displacement. Numerical results are provided to verify the theoretical convergence rates.

翻译：本文在任意维度上构造了适用于对称张量的新型低阶$H(\textrm{div})$相容有限元。通过将对称二次多项式空间与$(d+1)$阶法向-法向面泡空间进行扩充来定义形函数空间，其简化形式仅具有$d(d+1)^2$个自由度。在二维情形下，基于重心坐标显式给出了基函数。从Bell元出发，发展了二维低阶相容有限元弹性复形。这些对称张量有限元被应用于设计针对线性弹性问题的鲁棒混合有限元方法，该方法具有关于Lame系数$\lambda$的一致误差估计以及位移的超收敛性。数值结果验证了理论收敛阶。