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

## TL;DR

This paper introduces a mixed discontinuous Galerkin method for linear elasticity that achieves optimal error estimates and demonstrates sharp convergence orders through numerical validation.

## Contribution

The paper develops a novel mixed DG scheme with arbitrary order discontinuous finite element spaces for linear elasticity, providing rigorous stability and error analysis.

## Key findings

- Optimal error estimates for stress and displacement are established.
- Numerical results confirm the sharpness of convergence orders.
- The method is well-posed for any polynomial degree k ≥ 0.

## Abstract

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 ($k\geq 0$). The mixed DG scheme is proved to be well-posed under proper norms. Specifically, we prove that, for any $k \geq 0$, the $H({\rm div})$-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 $k \geq d$. We also provide numerical results of MDG showing that the orders of convergence are actually sharp.