  We construct finite element de~Rham complexes of higher and possibly
non-uniform polynomial order in finite element exterior calculus (FEEC).
Starting from the finite element differential complex of lowest-order, known as
the complex of Whitney forms, we incrementally construct the higher-order
complexes by adjoining exact local complexes associated to simplices. We define
a commuting canonical interpolant. On the one hand, this research provides a
base for studying $hp$-adaptive methods in finite element exterior calculus. On
the other hand, our construction of higher-order spaces enables a new tool in
numerical analysis which we call "partially localized flux reconstruction". One
major application of this concept is in the area of equilibrated a~posteriori
error estimators: we generalize the Braess-Sch\"oberl error estimator to edge
elements of higher and possibly non-uniform order.
