A robust nonconforming mixed finite element method is developed for a strain gradient elasticity (SGE) model. In two and three dimensional cases, a lower order $C^0$-continuous $H^2$-nonconforming finite element 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.

翻译：针对应变梯度弹性（SGE）模型，提出了一种稳健的非协调混合有限元方法。在二维和三维情形下，通过用泡函数对二次拉格朗日单元进行富集，为位移场构造了一个低阶$C^0$连续$H^2$非协调有限元。这一单元与线性拉格朗日单元相结合，用于离散SGE模型的混合形式。建立了稳健的离散inf-sup条件。关于小尺寸参数和Lamé系数均获得了尖锐且一致的误差估计，数值结果也验证了这一点。此外，在两个合理假设下推导了SGE模型的均匀正则性。