  We address fundamental aspects in the approximation theory of vector-valued
finite element methods, using finite element exterior calculus as a unifying
framework. We generalize the Cl\'ement interpolant and the Scott-Zhang
interpolant to finite element differential forms, and we derive a broken
Bramble-Hilbert Lemma. Our interpolants require only minimal smoothness
assumptions and respect partial boundary conditions. This permits us to state
local error estimates in terms of the mesh size. Our theoretical results apply
to curl-conforming and divergence-conforming finite element methods over
simplicial triangulations.
