  In mixed finite element approximations of Hodge Laplace problems associated
with the de Rham complex, the exterior derivative operators are computed
exactly, so the spatial locality is preserved. However, the numerical
approximations of the associated coderivatives are nonlocal and it can be
regarded as an undesired effect of standard mixed methods. For numerical
methods with local coderivatives a perturbation of low order mixed methods in
the sense of variational crimes has been developed for simplicial and cubical
meshes.
  In this paper we extend the low order method to all high orders on cubical
meshes using a new family of finite element differential forms on cubical
meshes. The key theoretical contribution is a generalization of the linear
degree, in the construction of the serendipity family of differential forms,
and the generalization is essential in the unisolvency proof of the new family
of finite element differential forms.
