An interior penalty nonconforming finite element method for strain gradient elasticity

Abstract

An interior penalty nonconforming finite element method is developed for a strain gradient elasticity (SGE) model in arbitrary dimensions, using an $H^1$-nonconforming displacement element built from vector-valued quadratic polynomials enriched by divergence-free face bubbles. Rigorous analysis establishes optimal and robust error estimates with respect to both the Lam\'{e} coefficient $\lambda$ and the size parameter $\iota$. The underlying divergence-commuting structure further leads to nonconforming finite element Stokes complexes in two and three dimensions. Numerical experiments support the predicted convergence behavior and parameter robustness, while a circular-hole benchmark under uniaxial tension illustrates the applicability of the method to a multiply connected domain.