Simplified Weak Galerkin Methods for Linear Elasticity on Nonconvex Domains

TL;DR

This paper introduces a simplified, stabilizer-free weak Galerkin finite element method for linear elasticity applicable to nonconvex domains, achieving optimal error estimates and confirmed by numerical experiments.

Contribution

The paper develops a novel weak Galerkin method that is symmetric, positive definite, stabilizer-free, and applicable to general polygonal meshes for linear elasticity.

Findings

Achieves optimal-order error estimates in discrete $H^1$-norm.

Demonstrates efficiency and accuracy through numerical experiments.

Applicable to nonconvex domains without convexity constraints.

Abstract

This paper presents a weak Galerkin (WG) finite element method for linear elasticity on general polygonal and polyhedral meshes, free from convexity constraints, by leveraging bubble functions as central analytical tools. The proposed method eliminates the need for stabilizers commonly used in traditional WG methods, resulting in a simplified formulation. The method is symmetric, positive definite, and straightforward to implement. Optimal-order error estimates are established for the WG approximations in the discrete $H^1$-norm, assuming sufficient smoothness of the exact solution, and in the standard $L^2$-norm under regularity assumptions for the dual problem. Numerical experiments confirm the efficiency and accuracy of the proposed stabilizer-free WG method.