A Mixed Discontinuous Galerkin Method for Linear Elasticity with Strongly Imposed Symmetry

In this paper, we study a mixed discontinuous Galerkin (MDG) method to solve linear elasticity problem with arbitrary order discontinuous finite element spaces in $d$-dimension ($d=2,3$). This method uses polynomials of degree $k+1$ for the stress and of degree $k$ for the displacement ($k\ge 0$). The mixed DG scheme is proved to be well-posed under proper norms. Specifically, we prove that, for any $k\ge 0$, the $H(\mathrm{d}\mathrm{i}\mathrm{v})$-like error estimate for the stress and $L^2$ error estimate for the displacement are optimal. We further establish the optimal $L^2$ error estimate for the stress provided that the ${\mathcal{P}}_{k+2}-{\mathcal{P}}_{k+1}^{-1}$ Stokes pair is stable and $k\ge d$. We also provide numerical results of MDG showing that the orders of convergence are actually sharp.