We propose a family of mixed finite elements that are robust for the nearly incompressible strain gradient model, which is a fourth-order singular perturbed elliptic system. The element is similar to [C. Taylor and P. Hood, Comput. & Fluids, 1(1973), 73-100] in the Stokes flow. Using a uniform discrete B-B inequality for the mixed finite element pairs, we show the optimal rate of convergence that is robust in the incompressible limit. By a new regularity result that is uniform in both the materials parameter and the incompressibility, we prove the method converges with $1/2$ order to the solution with strong boundary layer effects. Moreover, we estimate the convergence rate of the numerical solution to the unperturbed second-order elliptic system. Numerical results for both smooth solutions and the solutions with sharp layers confirm the theoretical prediction.

翻译：针对近不可压缩应变梯度模型（一种四阶奇异摄动椭圆系统），本文提出了一族鲁棒的混合有限元。该单元类似于斯托克斯流中的[C. Taylor and P. Hood, Comput. & Fluids, 1(1973), 73-100]。利用混合有限元对的一致离散B-B不等式，我们展示了在不可压缩极限下具有鲁棒性的最优收敛速率。借助关于材料参数和不可压缩性均一致的正则性新结果，我们证明了该方法对具有强边界层效应的解以1/2阶收敛。此外，我们估算了数值解对无扰动二阶椭圆系统的收敛速度。针对光滑解和具有尖锐层的解的数值结果均证实了理论预测。