New lower order mixed finite element methods for linear elasticity

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.