In this talk, we consider the discretization of the strain gradient elasticity (SGE) model in arbitrary dimensions. To obtain optimal and robust error estimates with respect to the Lam\'e coefficient $\lambda$ and the size parameter $\iota$, we propose and analyze two low-order nonconforming finite element methods within a unified framework.
The first method is based on a novel quadratic vector-valued $H^2$-nonconforming finite element, combined with Nitsche's technique for the discretization of the SGE model. Building on this element, we further construct nonconforming finite element discretizations of the smooth Stokes complex in two and three dimensions.
The second method is an interior penalty nonconforming scheme employing a quadratic $H^1$-nonconforming element. This element also leads to corresponding nonconforming finite element Stokes complexes in two and three dimensions.
For both methods, we establish optimal and robust error estimates that are uniform with respect to $\lambda$ and $\iota$. Numerical experiments are presented to confirm the theoretical results.