An Optimal and Robust Nonconforming Finite Element Method for the Strain Gradient Elasticity

TL;DR

This paper introduces an optimal, robust low-order nonconforming finite element method for strain gradient elasticity, applicable in any dimension, and confirms its effectiveness through numerical results.

Contribution

It develops a new $H^2$-nonconforming quadratic vector finite element for SGE and applies Nitsche's technique, enhancing robustness and optimality in arbitrary dimensions.

Findings

Method is optimal and robust with respect to key parameters.

Numerical results confirm theoretical properties.

Discretization of the smooth Stokes complex is achieved in 2D and 3D.

Abstract

An optimal and robust low-order nonconforming finite element method is developed for the strain gradient elasticity (SGE) model in arbitrary dimension. An $H^2$-nonconforming quadratic vector-valued finite element in arbitrary dimension is constructed, which together with the Nitsche's technique, is applied for solving the SGE model. The resulting nonconforming finite element method is optimal and robust with respect to the Lam\'{e} coefficient $\lambda$ and the size parameter $\iota$, as confirmed by numerical results. Additionally, nonconforming finite element discretization of the smooth Stokes complex in two and three dimensions is devised.