An arbitrary order locking-free weak Galerkin method for linear elasticity problems based on a reconstruction operator

Abstract: 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.