A robust lower order mixed finite element method for a strain gradient elasticity model

TL;DR

This paper introduces a robust mixed finite element method for strain gradient elasticity models, providing stable discretization and uniform error estimates validated by numerical experiments.

Contribution

A new lower order $C^0$-continuous $H^2$-nonconforming finite element for SGE models is developed, with proven stability and uniform error bounds.

Findings

Established robust discrete inf-sup condition.

Achieved sharp, uniform error estimates.

Validated results through numerical experiments.

Abstract

A robust nonconforming mixed finite element method is developed for a strain gradient elasticity (SGE) model. In two and three dimensional cases, a lower order $C^0$-continuous $H^2$-nonconforming finite element is constructed for the displacement field through enriching the quadratic Lagrange element with bubble functions. This together with the linear Lagrange element is exploited to discretize a mixed formulation of the SGE model. The robust discrete inf-sup condition is established. The sharp and uniform error estimates with respect to both the small size parameter and the Lam\'{e} coefficient are achieved, which is also verified by numerical results. In addition, the uniform regularity of the SGE model is derived under two reasonable assumptions.