Low-Order Preconditioning for the High-Order Finite Element de Rham Complex

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