{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:44:12Z","timestamp":1776836652820,"version":"3.51.2"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"341","license":[{"start":{"date-parts":[[2023,11,30]],"date-time":"2023-11-30T00:00:00Z","timestamp":1701302400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001711","name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung","doi-asserted-by":"publisher","award":["P500PT_203082"],"award-info":[{"award-number":["P500PT_203082"]}],"id":[{"id":"10.13039\/501100001711","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>We consider discretizations of the stationary Stokes equation in three spatial dimensions by non-conforming Crouzeix-Raviart elements. The original definition in the seminal paper by M. Crouzeix and P.-A. Raviart in 1973 [Rev. Fran\u00e7aise Automat. Informat. Recherche Op\u00e9rationnelle S\u00e9r. Rouge 7 (1973), pp. 33\u201375] is implicit and also contains substantial freedom for a concrete choice.<\/p>\n                  <p>\n                    In this paper, we introduce\n                    <italic>basic<\/italic>\n                    Crouzeix-Raviart spaces in 3D in analogy to the 2D case in a fully explicit way. We prove that this basic Crouzeix-Raviart element for the Stokes equation is inf-sup stable for polynomial degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k equals 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k=2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (quadratic velocity approximation). We identify spurious pressure modes for the conforming\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis k comma k minus 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\left ( k,k-1\\right )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    3D Stokes element and show that these are eliminated by using the basic Crouzeix-Raviart space.\n                  <\/p>","DOI":"10.1090\/mcom\/3793","type":"journal-article","created":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T14:11:02Z","timestamp":1669817462000},"page":"1033-1059","source":"Crossref","is-referenced-by-count":9,"title":["On the inf-sup stability of Crouzeix-Raviart Stokes elements in 3D"],"prefix":"10.1090","volume":"92","author":[{"given":"Stefan","family":"Sauter","sequence":"first","affiliation":[]},{"given":"C\u00e9line","family":"Torres","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2022,11,30]]},"reference":[{"issue":"6","key":"1","doi-asserted-by":"publisher","first-page":"3207","DOI":"10.1137\/07070838X","article-title":"Fully computable bounds for the error in nonconforming finite element approximations of arbitrary order on triangular elements","volume":"46","author":"Ainsworth, Mark","year":"2008","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00607-007-0219-1","article-title":"Gauss-Legendre elements: a stable, higher order non-conforming finite element family","volume":"79","author":"Baran, \u00c1.","year":"2007","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"key":"3","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4338-8","volume-title":"The mathematical theory of finite element methods","volume":"15","author":"Brenner, Susanne C.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0387941932"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1002\/num.21892","article-title":"Forty years of the Crouzeix-Raviart element","volume":"31","author":"Brenner, Susanne C.","year":"2015","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"key":"5","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1016\/j.camwa.2021.12.010","article-title":"Critical functions and inf-sup stability of Crouzeix-Raviart elements","volume":"108","author":"Carstensen, C.","year":"2022","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"issue":"337","key":"6","doi-asserted-by":"publisher","first-page":"2041","DOI":"10.1090\/mcom\/3742","article-title":"Crouzeix-Raviart triangular elements are inf-sup stable","volume":"91","author":"Carstensen, Carsten","year":"2022","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2-3","key":"7","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/S0096-3003(99)00109-5","article-title":"Stable nonconforming methods for the Stokes problem","volume":"114","author":"Cha, Youngjoon","year":"2000","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"issue":"5","key":"8","doi-asserted-by":"publisher","first-page":"649","DOI":"10.1142\/S0219530518500070","article-title":"A family of Crouzeix-Raviart finite elements in 3D","volume":"16","author":"Ciarlet, Patrick","year":"2018","journal-title":"Anal. Appl. (Singap.)","ISSN":"https:\/\/id.crossref.org\/issn\/0219-5305","issn-type":"print"},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1007\/s00211-015-0730-9","article-title":"Intrinsic finite element methods for the computation of fluxes for Poisson\u2019s equation","volume":"132","author":"Ciarlet, P. G.","year":"2016","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"186","key":"10","doi-asserted-by":"publisher","first-page":"437","DOI":"10.2307\/2008475","article-title":"Nonconforming finite elements for the Stokes problem","volume":"52","author":"Crouzeix, Michel","year":"1989","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"11","first-page":"33","article-title":"Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I","volume":"7","author":"Crouzeix, M.","year":"1973","journal-title":"Rev. Fran\\c{c}aise Automat. Informat. Recherche Op\\'{e}rationnelle S\\'{e}r. Rouge"},{"key":"12","isbn-type":"print","volume-title":"NIST handbook of mathematical functions","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521140638"},{"key":"13","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107786134","volume-title":"Orthogonal polynomials of several variables","volume":"155","author":"Dunkl, Charles F.","year":"2014","ISBN":"https:\/\/id.crossref.org\/isbn\/9781107071896","edition":"2"},{"issue":"2","key":"14","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1007\/BF01390424","article-title":"A three-dimensional quadratic nonconforming element","volume":"46","author":"Fortin, M.","year":"1985","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"4","key":"15","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1002\/nme.1620190405","article-title":"A nonconforming piecewise quadratic finite element on triangles","volume":"19","author":"Fortin, M.","year":"1983","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"key":"16","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61623-5","volume-title":"Finite element methods for Navier-Stokes equations","volume":"5","author":"Girault, Vivette","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/3540157964"},{"issue":"316","key":"17","doi-asserted-by":"publisher","first-page":"515","DOI":"10.1090\/mcom\/3346","article-title":"The Scott-Vogelius finite elements revisited","volume":"88","author":"Guzm\u00e1n, Johnny","year":"2019","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"295","key":"18","doi-asserted-by":"publisher","first-page":"2059","DOI":"10.1090\/S0025-5718-2015-02958-5","article-title":"Discrete and conforming smooth de Rham complexes in three dimensions","volume":"84","author":"Neilan, Michael","year":"2015","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"19","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1051\/m2an\/1985190101111","article-title":"Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials","volume":"19","author":"Scott, L. R.","year":"1985","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"165","key":"20","doi-asserted-by":"publisher","first-page":"9","DOI":"10.2307\/2007557","article-title":"Analysis of mixed finite elements methods for the Stokes problem: a unified approach","volume":"42","author":"Stenberg, Rolf","year":"1984","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"190","key":"21","doi-asserted-by":"publisher","first-page":"495","DOI":"10.2307\/2008498","article-title":"Error analysis of some finite element methods for the Stokes problem","volume":"54","author":"Stenberg, Rolf","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"6-7","key":"22","doi-asserted-by":"publisher","first-page":"967","DOI":"10.1016\/j.camwa.2005.10.011","article-title":"Crouzeix-Velte decompositions for higher-order finite elements","volume":"51","author":"Stoyan, G.","year":"2006","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-341\/S0025-5718-2022-03793-5\/mcom3793_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-341\/S0025-5718-2022-03793-5\/S0025-5718-2022-03793-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:52:38Z","timestamp":1776833558000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-341\/S0025-5718-2022-03793-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,30]]},"references-count":22,"journal-issue":{"issue":"341","published-print":{"date-parts":[[2023,5]]}},"alternative-id":["S0025-5718-2022-03793-5"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3793","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":[[2022,11,30]]}}}