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

Abstract

An optimal and robust low-order nonconforming finite element method is developed for the strain gradient elasticity (SGE) model in arbitrary dimensions. An $H^2$-nonconforming quadratic vector-valued finite element in arbitrary dimensions 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 Lamé coefficient $λ$ and the size parameter $ι$, as confirmed by numerical results. Additionally, nonconforming finite element discretization of the smooth Stokes complex in two and three dimensions is devised.