The weak Galerkin (WG) finite element method has shown great potential in solving various type of partial differential equations. In this paper, we propose an arbitrary order locking-free WG method for solving linear elasticity problems, with the aid of an appropriate $H(div)$-conforming displacement reconstruction operator. Optimal order locking-free error estimates in both the $H^1$-norm and the $L^2$-norm are proved, i.e., the error is independent of the $Lam\acute{e}$ constant $\lambda$. Moreover, the term $\lambda\|\nabla\cdot \mathbf{u}\|_k$ does not need to be bounded in order to achieve these estimates. We validate the accuracy and the robustness of the proposed locking-free WG algorithm by numerical experiments.

翻译：弱Galerkin（WG）有限元方法在求解各类偏微分方程方面展现出巨大潜力。本文借助合适的$H(div)$-一致位移重构算子，提出了一种任意阶无闭锁WG方法，用于求解线弹性问题。我们证明了$H^1$范数和$L^2$范数下的最优阶无闭锁误差估计，即误差与$Lam\acute{e}$常数$\lambda$无关。此外，为实现这些估计，无需对项$\lambda\|\nabla\cdot \mathbf{u}\|_k$进行有界性约束。通过数值实验，验证了所提出的无闭锁WG算法的准确性和鲁棒性。