A Locking-Free Weak Galerkin Finite Element Method for Linear Elasticity Problems

TL;DR

This paper presents a new locking-free weak Galerkin finite element method for linear elasticity that achieves optimal error estimates independent of the Lamé parameter, validated through numerical experiments.

Contribution

The paper introduces a novel locking-free weak Galerkin scheme with an $H(div)$-conforming displacement reconstruction for linear elasticity problems.

Findings

Method is locking-free and effective.

Achieves optimal error estimates independent of Lamé parameter.

Numerical experiments confirm theoretical results.

Abstract

In this paper, we introduce and analyze a lowest-order locking-free weak Galerkin (WG) finite element scheme for the grad-div formulation of linear elasticity problems. The scheme uses linear functions in the interior of mesh elements and constants on edges (2D) or faces (3D), respectively, to approximate the displacement. An $H(div)$-conforming displacement reconstruction operator is employed to modify test functions in the right-hand side of the discrete form, in order to eliminate the dependence of the $Lam\acute{e}$ parameter $\lambda$ in error estimates, i.e., making the scheme locking-free. The method works without requiring $\lambda \|\nabla\cdot \mathbf{u}\|_1$ to be bounded. We prove optimal error estimates, independent of $\lambda$, in both the $H^1$-norm and the $L^2$-norm. Numerical experiments validate that the method is effective and locking-free.