  In this paper we present a unified framework for constructing spectrally
equivalent low-order-refined discretizations for the high-order finite element
de Rham complex. This theory covers diffusion problems in $H^1$, $H({\rm
curl})$, and $H({\rm div})$, and is based on combining a low-order
discretization posed on a refined mesh with a high-order basis for N\'ed\'elec
and Raviart-Thomas elements that makes use of the concept of polynomial
histopolation (polynomial fitting using prescribed mean values over certain
regions). This spectral equivalence, coupled with algebraic multigrid methods
constructed using the low-order discretization, results in highly scalable
matrix-free preconditioners for high-order finite element problems in the full
de Rham complex. Additionally, a new lowest-order (piecewise constant)
preconditioner is developed for high-order interior penalty discontinuous
Galerkin (DG) discretizations, for which spectral equivalence results and
convergence proofs for algebraic multigrid methods are provided. In all cases,
the spectral equivalence results are independent of polynomial degree and mesh
size; for DG methods, they are also independent of the penalty parameter. These
new solvers are flexible and easy to use; any "black-box" preconditioner for
low-order problems can be used to create an effective and efficient
preconditioner for the corresponding high-order problem. A number of numerical
experiments are presented, based on an implmentation in the finite element
library MFEM. The theoretical properties of these preconditioners are
corroborated, and the flexibility and scalability of the method are
demonstrated on a range of challenging three-dimensional problems.
