  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.
