Two robust nonconforming $$\hbox {H}^2$$-elements for linear strain gradient elasticity

We propose two finite elements to approximate a boundary value problem arising from strain gradient elasticity, which is a high order perturbation of the linearized elastic system. Our elements are \(\hbox {H}^2\)-nonconforming while \(\hbox {H}^1\)-conforming. We show both elements converge in the energy norm uniformly with respect to the perturbation parameter.