Low-degree robust Hellinger-Reissner finite element schemes for planar linear elasticity with symmetric stress tensors

Abstract

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 $$\mathrm{span}\left\{\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right),\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right),\left(\begin{array}{cc}0& x\\ x& 0\end{array}\right),\left(\begin{array}{cc}0& y\\ y& 0\end{array}\right),\left(\begin{array}{cc}0& x^2-y^2\\ x^2-y^2& 0\end{array}\right)\right\},$$ 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 $\mathbf{H}(\mathrm{div})$ 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 $\mathbf{H}(\mathrm{div};\mathbb{S})$, and the space for the displacement is piecewise constants. Robust error estimations in ${\mathbb{L}}^2$ and broken $\mathbf{H}(\mathrm{div})$ norms are presented for regular solutions and data.