A Mixed Discontinuous Galerkin Method for Linear Elasticity with Strongly Imposed Symmetry
It provides a rigorous theoretical foundation and optimal error bounds for a mixed DG method in linear elasticity, addressing a known challenge of stress symmetry imposition.
This paper develops a mixed discontinuous Galerkin method for linear elasticity that strongly imposes symmetry of the stress tensor, achieving optimal error estimates for stress and displacement with arbitrary polynomial orders. Numerical experiments confirm the sharpness of the convergence rates.
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.