{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:36:31Z","timestamp":1776836191157,"version":"3.51.2"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"337","license":[{"start":{"date-parts":[[2023,6,8]],"date-time":"2023-06-08T00:00:00Z","timestamp":1686182400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    The Crouzeix-Raviart triangular finite elements are\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"inf\">\n                        <mml:semantics>\n                          <mml:mo movablelimits=\"true\" form=\"prefix\">inf<\/mml:mo>\n                          <mml:annotation encoding=\"application\/x-tex\">\\inf<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sup\">\n                        <mml:semantics>\n                          <mml:mo movablelimits=\"true\" form=\"prefix\">sup<\/mml:mo>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sup<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    stable for the Stokes equations for any mesh with at least one interior vertex. This result affirms a\n                    <italic>conjecture of Crouzeix-Falk<\/italic>\n                    from 1989 for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p equals 3\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">p=3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Our proof applies to\n                    <italic>any odd degree<\/italic>\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p greater-than-or-equal-to 3\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">p\\ge 3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and concludes the overall stability analysis: Crouzeix-Raviart triangular finite elements of degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in two dimensions and the piecewise polynomials of degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p minus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">p-1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with vanishing integral form a stable Stokes pair\n                    <italic>for all positive integers<\/italic>\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/mcom\/3742","type":"journal-article","created":{"date-parts":[[2022,3,9]],"date-time":"2022-03-09T09:21:09Z","timestamp":1646817669000},"page":"2041-2057","source":"Crossref","is-referenced-by-count":7,"title":["Crouzeix-Raviart triangular elements are inf-sup stable"],"prefix":"10.1090","volume":"91","author":[{"given":"Carsten","family":"Carstensen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stefan","family":"Sauter","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2022,6,8]]},"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"},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"722","DOI":"10.1137\/0720048","article-title":"Stability of finite elements under divergence constraints","volume":"20","author":"Boland, J. M.","year":"1983","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"4","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0","volume-title":"The mathematical theory of finite element methods","volume":"15","author":"Brenner, Susanne C.","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387759333","edition":"3"},{"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":"2-3","key":"6","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":"3","key":"7","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":"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":"186","key":"9","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":"10","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":"11","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-56341-7","volume-title":"Finite elements I---Approximation and interpolation","volume":"72","author":"Ern, Alexandre","year":"[2021] \\copyright2021","ISBN":"https:\/\/id.crossref.org\/isbn\/9783030563400"},{"issue":"4","key":"12","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":"13","series-title":"Graduate Texts in Mathematics, No. 23","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-9446-4","volume-title":"Linear algebra","author":"Greub, Werner","year":"1975","edition":"4"},{"issue":"316","key":"14","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"},{"key":"15","isbn-type":"print","volume-title":"Matrix analysis","author":"Horn, Roger A.","year":"2013","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521548236","edition":"2"},{"key":"16","series-title":"Numerical Mathematics and Scientific Computation","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198508885.001.0001","volume-title":"Finite element methods for Maxwell's equations","author":"Monk, Peter","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0198508883"},{"issue":"295","key":"17","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":"18","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":"19","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":"6-7","key":"20","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"},{"issue":"9","key":"21","doi-asserted-by":"publisher","first-page":"848","DOI":"10.2307\/2324660","article-title":"The fundamental theorem of linear algebra","volume":"100","author":"Strang, Gilbert","year":"1993","journal-title":"Amer. Math. Monthly","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9890","issn-type":"print"},{"key":"22","doi-asserted-by":"publisher","first-page":"672","DOI":"10.2307\/2305561","article-title":"A recurring theorem on determinants","volume":"56","author":"Taussky, Olga","year":"1949","journal-title":"Amer. Math. Monthly","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9890","issn-type":"print"},{"issue":"1","key":"23","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1007\/BF01396303","article-title":"A right-inverse for the divergence operator in spaces of piecewise polynomials. Application to the \ud835\udc5d-version of the finite element method","volume":"41","author":"Vogelius, Michael","year":"1983","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2022-91-337\/S0025-5718-2022-03742-X\/S0025-5718-2022-03742-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:37:13Z","timestamp":1776832633000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2022-91-337\/S0025-5718-2022-03742-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,8]]},"references-count":23,"journal-issue":{"issue":"337","published-print":{"date-parts":[[2022,9]]}},"alternative-id":["S0025-5718-2022-03742-X"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3742","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":[[2022,6,8]]}}}