{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:13:10Z","timestamp":1776802390054,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"307","license":[{"start":{"date-parts":[[2017,10,27]],"date-time":"2017-10-27T00:00:00Z","timestamp":1509062400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1418805"],"award-info":[{"award-number":["DMS-1418805"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1418805"],"award-info":[{"award-number":["DMS-1418805"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1418805"],"award-info":[{"award-number":["DMS-1418805"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1418805"],"award-info":[{"award-number":["DMS-1418805"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>The scalar and vector Laplacians are basic operators in physics and engineering. In applications, they frequently show up perturbed by lower-order terms. The effect of such perturbations on mixed finite element methods in the scalar case is well understood, but that in the vector case is not. In this paper, we first show that, surprisingly, for certain elements there is degradation of the convergence rates with certain lower-order terms even when both the solution and the data are smooth. We then give a systematic analysis of lower-order terms in mixed methods by extending the Finite Element Exterior Calculus (FEEC) framework, which contains the scalar, vector Laplacian, and many other elliptic operators as special cases. We prove that stable mixed discretization remains stable with lower-order terms for sufficiently fine discretization. Moreover, we derive sharp improved error estimates for each individual variable. In particular, this yields new results for the vector Laplacian problem which are useful in applications such as electromagnetism and acoustics modeling. Further, our results imply many previous results for the scalar problem and thus unify them all under the FEEC framework.<\/p>","DOI":"10.1090\/mcom\/3158","type":"journal-article","created":{"date-parts":[[2016,3,23]],"date-time":"2016-03-23T14:52:55Z","timestamp":1458744775000},"page":"2193-2212","source":"Crossref","is-referenced-by-count":6,"title":["Finite element exterior calculus with lower-order terms"],"prefix":"10.1090","volume":"86","author":[{"given":"Douglas","family":"Arnold","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lizao","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,10,27]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492906210018","article-title":"Finite element exterior calculus, homological techniques, and applications","volume":"15","author":"Arnold, Douglas N.","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/0521868157","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1090\/S0273-0979-10-01278-4","article-title":"Finite element exterior calculus: from Hodge theory to numerical stability","volume":"47","author":"Arnold, Douglas N.","year":"2010","journal-title":"Bull. Amer. Math. Soc. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0273-0979","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"322","DOI":"10.1007\/BF02165003","article-title":"Error-bounds for finite element method","volume":"16","author":"Babu\u0161ka, Ivo","year":"1970","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1007\/BF01396752","article-title":"Mixed finite elements for second order elliptic problems in three variables","volume":"51","author":"Brezzi, Franco","year":"1987","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1007\/BF01389710","article-title":"Two families of mixed finite elements for second order elliptic problems","volume":"47","author":"Brezzi, Franco","year":"1985","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"6","key":"6","doi-asserted-by":"publisher","first-page":"1938","DOI":"10.1137\/S0036142900376900","article-title":"Suboptimal and optimal convergence in mixed finite element methods","volume":"39","author":"Demlow, Alan","year":"2002","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"7","first-page":"91","article-title":"Mixed finite element methods for second order elliptic problems","volume":"1","author":"Douglas, J., Jr.","year":"1982","journal-title":"Mat. Apl. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0101-8205","issn-type":"print"},{"issue":"169","key":"8","doi-asserted-by":"publisher","first-page":"39","DOI":"10.2307\/2007791","article-title":"Global estimates for mixed methods for second order elliptic equations","volume":"44","author":"Douglas, Jim, Jr.","year":"1985","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"9","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1051\/m2an\/1980140302491","article-title":"Error estimates for mixed methods","volume":"14","author":"Falk, R. S.","year":"1980","journal-title":"RAIRO Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0399-0516","issn-type":"print"},{"key":"10","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1073\/pnas.37.1.48","article-title":"The harmonic operator for exterior differential forms","volume":"37","author":"Gaffney, Matthew P.","year":"1951","journal-title":"Proc. Nat. Acad. Sci. U.S.A.","ISSN":"https:\/\/id.crossref.org\/issn\/0027-8424","issn-type":"print"},{"key":"11","series-title":"Numerical Mathematics and Scientific Computation","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198566656.001.0001","volume-title":"Mathematical methods for the magnetohydrodynamics of liquid metals","author":"Gerbeau, Jean-Fr\u00e9d\u00e9ric","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9780198566656"},{"key":"12","series-title":"Lecture Notes in Computational Science and Engineering","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-23099-8","volume-title":"Automated solution of differential equations by the finite element method","volume":"84","year":"2012","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642230981"},{"issue":"3","key":"13","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1007\/BF01396415","article-title":"Mixed finite elements in \ud835\udc45\u00b3","volume":"35","author":"N\u00e9d\u00e9lec, J.-C.","year":"1980","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"14","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/BF01389668","article-title":"A new family of mixed finite elements in \ud835\udc45\u00b3","volume":"50","author":"N\u00e9d\u00e9lec, J.-C.","year":"1986","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"15","doi-asserted-by":"publisher","first-page":"346","DOI":"10.1007\/BF02166687","article-title":"Ein Kriterium f\u00fcr die Quasi-Optimalit\u00e4t des Ritzschen Verfahrens","volume":"11","author":"Nitsche, J.","year":"1968","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"2","key":"16","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1007\/BF01161700","article-title":"An elementary proof for a compact imbedding result in generalized electromagnetic theory","volume":"187","author":"Picard, R.","year":"1984","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"key":"17","first-page":"292","article-title":"A mixed finite element method for 2nd order elliptic problems","author":"Raviart, P.-A.","year":"1977"},{"key":"18","doi-asserted-by":"publisher","first-page":"959","DOI":"10.2307\/2005357","article-title":"An observation concerning Ritz-Galerkin methods with indefinite bilinear forms","volume":"28","author":"Schatz, Alfred H.","year":"1974","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-307\/S0025-5718-2016-03158-0\/mcom3158_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"http:\/\/www.ams.org\/mcom\/2017-86-307\/S0025-5718-2016-03158-0\/S0025-5718-2016-03158-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-307\/S0025-5718-2016-03158-0\/S0025-5718-2016-03158-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:18:06Z","timestamp":1776799086000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-307\/S0025-5718-2016-03158-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,10,27]]},"references-count":18,"journal-issue":{"issue":"307","published-print":{"date-parts":[[2017,9]]}},"alternative-id":["S0025-5718-2016-03158-0"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3158","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2016,10,27]]}}}