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Sci. Comput."],"published-print":{"date-parts":[[2023,4,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>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 [Formula: see text], [Formula: see text], and [Formula: see text] and is based on combining a low-order discretization posed on a refined mesh with a high-order basis for N\u00e9d\u00e9lec and Raviart\u2013Thomas 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 \u201cblack-box\u201d 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 implementation in the finite element library MFEM. A range of challenging three-dimensional problems are used to corroborate the theoretical properties and demonstrate the flexibility and scalability of the method.<\/jats:p>","DOI":"10.1137\/22m1486534","type":"journal-article","created":{"date-parts":[[2023,4,13]],"date-time":"2023-04-13T04:49:42Z","timestamp":1681361382000},"page":"A675-A702","source":"Crossref","is-referenced-by-count":24,"title":["Low-Order Preconditioning for the High-Order Finite Element de Rham Complex"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4885-2934","authenticated-orcid":true,"given":"Will","family":"Pazner","sequence":"first","affiliation":[{"name":"Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR 97201, USA."},{"name":"Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore CA 94550, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2810-3090","authenticated-orcid":true,"given":"Tzanio","family":"Kolev","sequence":"additional","affiliation":[{"name":"Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore CA 94550, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Clark R.","family":"Dohrmann","sequence":"additional","affiliation":[{"name":"Computational Solid Mechanics and Structural Dynamics, Sandia National Laboratories, Albuquerque, Tullahoma, TN 37388, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,4,26]]},"reference":[{"key":"ref1","author":"Anderson R.","year":"2020","journal-title":"Comput. 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