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

A robust nonconforming mixed finite element method is developed for a strain gradient elastic (SGE) model. A lower-order C0-continuous H2-nonconforming finite element in arbitrary dimension 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 Lame 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.