  In this paper, we study the construction of low-degree robust finite element
schemes for planar linear elasticity on general triangulations.
  Firstly, we present a low-degree nonconforming Helling-Reissner finite
element scheme. For the stress tensor space, the piecewise polynomial shape
function space is $$ {\rm span}\left\{\left(\begin{array}{cc}1&0\\ 0 &
0\end{array}\right),\left(\begin{array}{cc}0&1\\ 1 &
0\end{array}\right),\left(\begin{array}{cc}0&0\\ 0 &
1\end{array}\right),\left(\begin{array}{cc}0&x\\ x &
0\end{array}\right),\left(\begin{array}{cc}0&y\\ y &
0\end{array}\right),\left(\begin{array}{cc}0&x^2-y^2\\ x^2-y^2 &
0\end{array}\right)\right\}, $$ the dimension of the total space is
asymptotically 8 times of the number of vertices, and the supports of the basis
functions are each a patch of an edge. The piecewise rigid body space is used
for the displacement. Robust error estimations in $\mathbb{L}^2$ and broken
$\mathbf{H}({\rm div})$ norms are presented.
  Secondly, we investigate the theoretical construction of schemes with
lowest-degree polynomial shape function spaces. Specifically, a
Hellinger-Reissner finite element scheme is constructed, with the local shape
function space for the stress tensor being 5-dimensional which is of the lowest
degree for the local approximation of $\mathbf{H}({\rm div};\mathbb{S})$, and
the space for the displacement is piecewise constants. Robust error estimations
in $\mathbb{L}^2$ and broken $\mathbf{H}({\rm div})$ norms are presented for
regular solutions and data.
