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

A double-parameter robust nonconforming mixed finite element method is developed for a strain gradient elastic (SGE) model. A lower order $C^0$-continuous $H^2$-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 Lam\'{e} 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.