Robust weak Galerkin finite element methods for linear elasticity with continuous displacement trace approximation

TL;DR

This paper introduces a new class of weak Galerkin finite element methods for linear elasticity that are robust, efficient, and provide optimal error estimates, even with high Lamé constants, confirmed by numerical tests.

Contribution

The paper develops a novel weak Galerkin method with continuous displacement trace approximation, achieving robustness and optimal error estimates for linear elasticity problems.

Findings

Methods produce SPD systems with only trace unknowns.

Error estimates are optimal and uniform in Lamé constant.

Numerical experiments confirm theoretical robustness and accuracy.

Abstract

This paper proposes and analyzes a class of new weak Galerkin (WG) finite element methods for 2- and 3-dimensional linear elasticity problems. The methods use discontinuous piecewise-polynomial approximations of degrees $k(\geq 0)$ for the stress, $k+1$ for the displacement, and a continuous piecewise-polynomial approximation of degree $k+1$ for the displacement trace on the inter-element boundaries, respectively. After the local elimination of unknowns defined in the interior of elements, the WG methods result in SPD systems where the unknowns are only the degrees of freedom describing the continuous trace approximation. We show that the proposed methods are robust in the sense that the derived a priori error estimates are optimal and uniform with respect to the Lam\'{e} constant $\lambda$. Numerical experiments confirm the theoretical results.