{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:38:59Z","timestamp":1776836339509,"version":"3.51.2"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"338","license":[{"start":{"date-parts":[[2023,7,12]],"date-time":"2023-07-12T00:00:00Z","timestamp":1689120000000},"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":["1913083"],"award-info":[{"award-number":["1913083"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1913083"],"award-info":[{"award-number":["1913083"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2011733"],"award-info":[{"award-number":["2011733"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2011733"],"award-info":[{"award-number":["2011733"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1913083"],"award-info":[{"award-number":["1913083"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1913083"],"award-info":[{"award-number":["1913083"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2011733"],"award-info":[{"award-number":["2011733"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2011733"],"award-info":[{"award-number":["2011733"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1913083"],"award-info":[{"award-number":["1913083"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1913083"],"award-info":[{"award-number":["1913083"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2011733"],"award-info":[{"award-number":["2011733"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2011733"],"award-info":[{"award-number":["2011733"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We construct several smooth finite element spaces defined on three-dimensional Worsey\u2013Farin splits. In particular, we construct\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">C^1<\/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=\"upper H Superscript 1 Baseline left-parenthesis c u r l right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>curl<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H^1(\\operatorname {curl})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -conforming finite element spaces and show the discrete spaces satisfy local exactness properties. A feature of the spaces is their low polynomial degree and lack of extrinsic supersmoothness at subsimplices of the mesh. In the lowest order case, the last two spaces in the sequence consist of piecewise linear and piecewise constant spaces, and are suitable for the discretization of the (Navier-)Stokes equation.\n                  <\/p>","DOI":"10.1090\/mcom\/3746","type":"journal-article","created":{"date-parts":[[2022,7,12]],"date-time":"2022-07-12T09:50:12Z","timestamp":1657619412000},"page":"2571-2608","source":"Crossref","is-referenced-by-count":3,"title":["Exact sequences on Worsey\u2013Farin splits"],"prefix":"10.1090","volume":"91","author":[{"given":"Johnny","family":"Guzm\u00e1n","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anna","family":"Lischke","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Neilan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2022,7,12]]},"reference":[{"key":"1","unstructured":"D. N. Arnold and J. Qin, Quadratic velocity\/linear pressure Stokes elements, Advances in computer methods for partial differential equations, 7:28\u201334, 1992."},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/s00211-018-0970-6","article-title":"Generalized finite element systems for smooth differential forms and Stokes\u2019 problem","volume":"140","author":"Christiansen, Snorre H.","year":"2018","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"3","series-title":"Classics in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719208","volume-title":"The finite element method for elliptic problems","volume":"40","author":"Ciarlet, Philippe G.","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/0898715148"},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1051\/m2an\/1979130302271","article-title":"A family of \ud835\udc36\u00b9 finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems","volume":"13","author":"Douglas, Jim, Jr.","year":"1979","journal-title":"RAIRO Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0399-0516","issn-type":"print"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"1308","DOI":"10.1137\/120888132","article-title":"Stokes complexes and the construction of stable finite elements with pointwise mass conservation","volume":"51","author":"Falk, Richard S.","year":"2013","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"5","key":"6","doi-asserted-by":"publisher","first-page":"Paper No. 70, 15","DOI":"10.1007\/s10444-020-09813-y","article-title":"A characterization of supersmoothness of multivariate splines","volume":"46","author":"Floater, Michael S.","year":"2020","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"issue":"323","key":"7","doi-asserted-by":"publisher","first-page":"1059","DOI":"10.1090\/mcom\/3520","article-title":"Exact smooth piecewise polynomial sequences on Alfeld splits","volume":"89","author":"Fu, Guosheng","year":"2020","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"Paper No. 13, 25","DOI":"10.1007\/s10092-020-00361-x","article-title":"Exact sequences on Powell-Sabin splits","volume":"57","author":"Guzm\u00e1n, J.","year":"2020","journal-title":"Calcolo","ISSN":"https:\/\/id.crossref.org\/issn\/0008-0624","issn-type":"print"},{"issue":"285","key":"9","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1090\/S0025-5718-2013-02753-6","article-title":"Conforming and divergence-free Stokes elements on general triangular meshes","volume":"83","author":"Guzm\u00e1n, Johnny","year":"2014","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"5","key":"10","doi-asserted-by":"publisher","first-page":"2826","DOI":"10.1137\/17M1153467","article-title":"inf-sup stable finite elements on barycentric refinements producing divergence-free approximations in arbitrary dimensions","volume":"56","author":"Guzm\u00e1n, Johnny","year":"2018","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"11","doi-asserted-by":"publisher","first-page":"492","DOI":"10.1137\/15M1047696","article-title":"On the divergence constraint in mixed finite element methods for incompressible flows","volume":"59","author":"John, Volker","year":"2017","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"key":"12","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511721588","volume-title":"Spline functions on triangulations","volume":"110","author":"Lai, Ming-Jun","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521875929"},{"key":"13","unstructured":"A. Lischke, Exact smooth piecewise polynomials on Powell\u2013Sabin and Worsey\u2013Farin splits, PhD thesis, Division of Applied Mathematics, Brown University, 2020."},{"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"},{"issue":"295","key":"15","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":"16","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":"1","key":"17","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"},{"key":"18","doi-asserted-by":"publisher","first-page":"334","DOI":"10.1016\/0021-9045(73)90036-1","article-title":"Polynomial approximation on tetrahedrons in the finite element method","volume":"7","author":"\u017den\u00ed\u0161ek, Alexander","year":"1973","journal-title":"J. Approximation Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"issue":"3","key":"19","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1016\/0167-8396(88)90001-5","article-title":"A trivariate Powell-Sabin interpolant","volume":"5","author":"Worsey, A. J.","year":"1988","journal-title":"Comput. Aided Geom. Design","ISSN":"https:\/\/id.crossref.org\/issn\/0167-8396","issn-type":"print"},{"issue":"250","key":"20","doi-asserted-by":"publisher","first-page":"543","DOI":"10.1090\/S0025-5718-04-01711-9","article-title":"A new family of stable mixed finite elements for the 3D Stokes equations","volume":"74","author":"Zhang, Shangyou","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"21","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1016\/j.apnum.2008.02.002","article-title":"A family of 3D continuously differentiable finite elements on tetrahedral grids","volume":"59","author":"Zhang, Shangyou","year":"2009","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"3","key":"22","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1007\/s10092-010-0035-4","article-title":"Quadratic divergence-free finite elements on Powell-Sabin tetrahedral grids","volume":"48","author":"Zhang, Shangyou","year":"2011","journal-title":"Calcolo","ISSN":"https:\/\/id.crossref.org\/issn\/0008-0624","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2022-91-338\/S0025-5718-2022-03746-7\/mcom3746_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2022-91-338\/S0025-5718-2022-03746-7\/S0025-5718-2022-03746-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:41:31Z","timestamp":1776832891000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2022-91-338\/S0025-5718-2022-03746-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,12]]},"references-count":22,"journal-issue":{"issue":"338","published-print":{"date-parts":[[2022,11]]}},"alternative-id":["S0025-5718-2022-03746-7"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3746","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":[[2022,7,12]]}}}