A Locking-Free Weak Galerkin Finite Element 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 help 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 Lamé 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.