  Robust mixed finite element methods are developed for a quad-curl singular
perturbation problem. Lower order H(grad curl)-nonconforming but
H(curl)-conforming finite elements are constructed, which are extended to
nonconforming finite element Stokes complexes and the associated commutative
diagrams. Then H(grad curl)-nonconforming finite elements are employed to
discretize the quad-curl singular perturbation problem, which possess the sharp
and uniform error estimates with respect to the perturbation parameter. The
Nitsche's technique is exploited to achieve the optimal convergence rate in the
case of the boundary layers. Numerical results are provided to verify the
theoretical convergence rates. In addition, the regularity of the quad-curl
singular perturbation problem is established.
