  We consider new degrees of freedom for higher order differential forms on
cubical meshes. The approach is inspired by the idea of Rapetti and Bossavit to
define higher order Whitney forms and their degrees of freedom using small
simplices. We show that higher order differential forms on cubical meshes can
be defined analogously using small cubes and prove that these small cubes yield
unisolvent degrees of freedom. Significantly, this approach is compatible with
discrete exterior calculus and expands the framework to cover higher order
methods on cubical meshes, complementing the earlier strategy based on
simplices.
