A $C^0$-continuous nonconforming virtual element method for linear strain gradient elasticity

TL;DR

This paper introduces a new $C^0$-continuous nonconforming virtual element method for strain gradient elasticity, capable of handling polygonal meshes with robust stability and accurate error estimates.

Contribution

It develops a novel virtual element method for strain gradient elasticity that ensures stability, robustness, and optimal error estimates on general polygonal meshes.

Findings

Stable and robust VEM developed for 2D strain gradient elasticity.

Achieved sharp, uniform error estimates with respect to microscopic parameters.

Numerical results verify theoretical error bounds.

Abstract

A robust $C^0$-continuous nonconforming virtual element method (VEM) is developed for a boundary value problem arising from strain gradient elasticity in two dimensions, with the family of polygonal meshes satisfying a very general geometric assumption given in Brezzi et al. (2009) and Chen and Huang (2018). The stability condition of the VEMs is derived by establishing Korn-type inequalities and inverse inequalities. Some crucial commutative relations for locking-free analysis as in elastic problems are derived. The sharp and uniform error estimates with respect to both the microscopic parameter and the Lam\'e coefficient are achieved in the lowest-order case, which is also verified by numerical results.