{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T06:26:54Z","timestamp":1774506414835,"version":"3.50.1"},"reference-count":45,"publisher":"American Mathematical Society (AMS)","issue":"312","license":[{"start":{"date-parts":[[2019,2,6]],"date-time":"2019-02-06T00:00:00Z","timestamp":1549411200000},"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-1719829"],"award-info":[{"award-number":["DMS-1719829"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>Nearly all classical inf-sup stable mixed finite element methods for the incompressible Stokes equations are not pressure-robust, i.e., the velocity error is dependent on the pressure. However, recent results show that pressure-robustness can be recovered by a nonstandard discretization of the right-hand side alone. This variational crime introduces a consistency error in the method which can be estimated in a straightforward manner provided that the exact velocity solution is sufficiently smooth. The purpose of this paper is to analyze the pressure-robust scheme with low regularity. The numerical analysis applies divergence-free <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>-conforming Stokes finite element methods as a theoretical tool. As an example, pressure-robust velocity and pressure a priori error estimates will be presented for the (first-order) nonconforming Crouzeix\u2013Raviart element. A key feature in the analysis is the dependence of the errors on the Helmholtz projector of the right-hand side data, and not on the entire data term. Numerical examples illustrate the theoretical results.<\/p>","DOI":"10.1090\/mcom\/3344","type":"journal-article","created":{"date-parts":[[2017,12,27]],"date-time":"2017-12-27T14:19:14Z","timestamp":1514384354000},"page":"1543-1566","source":"Crossref","is-referenced-by-count":21,"title":["Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes-problem"],"prefix":"10.1090","volume":"87","author":[{"given":"A.","family":"Linke","sequence":"first","affiliation":[]},{"given":"C.","family":"Merdon","sequence":"additional","affiliation":[]},{"given":"M.","family":"Neilan","sequence":"additional","affiliation":[]},{"given":"F.","family":"Neumann","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2018,2,6]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"N. Ahmed, A. Linke, and C. Merdon, On really locking-free mixed methods for the transient incompressible Stokes equations, SIAM Journal on Numerical Analysis (2017), 1\u201324, accepted. WIAS Preprint 2368.","DOI":"10.1137\/17M1112017"},{"key":"2","doi-asserted-by":"publisher","first-page":"98","DOI":"10.1016\/j.aml.2015.01.018","article-title":"A note on the importance of mass conservation in long-time stability of Navier-Stokes simulations using finite elements","volume":"45","author":"Belenli, Mine Akbas","year":"2015","journal-title":"Appl. Math. Lett.","ISSN":"https:\/\/id.crossref.org\/issn\/0893-9659","issn-type":"print"},{"key":"3","unstructured":"D. N. Arnold and J. Qin, Quadratic velocity\/linear pressure Stokes elements, Advances in Computer Methods for Partial Differential Equations VII (R. Vichnevetsky, D. Knight, and G. Richter, eds.), IMACS, 1992, pp. 28\u201334."},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"191","DOI":"10.4208\/jcm.1411-m4499","article-title":"Optimal and pressure-independent \ud835\udc3f\u00b2 velocity error estimates for a modified Crouzeix-Raviart Stokes element with BDM reconstructions","volume":"33","author":"Brennecke, C.","year":"2015","journal-title":"J. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0254-9409","issn-type":"print"},{"key":"5","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"},{"issue":"2","key":"6","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"},{"key":"7","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and hybrid finite element methods","volume":"15","author":"Brezzi, Franco","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0387975829"},{"issue":"11-12","key":"8","doi-asserted-by":"publisher","first-page":"1407","DOI":"10.1002\/fld.2337","article-title":"IsoGeometric Analysis: stable elements for the 2D Stokes equation","volume":"65","author":"Buffa, A.","year":"2011","journal-title":"Internat. J. Numer. Methods Fluids","ISSN":"https:\/\/id.crossref.org\/issn\/0271-2091","issn-type":"print"},{"key":"9","doi-asserted-by":"crossref","unstructured":"S. H. Christiansen, J. Hu, and K. Hu, Nodal finite element de Rham complexes, arXiv:1611.02558, 2016.","DOI":"10.1007\/s00211-017-0939-x"},{"issue":"8","key":"10","doi-asserted-by":"publisher","first-page":"1421","DOI":"10.1142\/S0218202513500139","article-title":"Isogeometric divergence-conforming B-splines for the steady Navier-Stokes equations","volume":"23","author":"Evans, John A.","year":"2013","journal-title":"Math. Models Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-2025","issn-type":"print"},{"issue":"2","key":"11","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"},{"key":"12","series-title":"Springer Monographs in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-09620-9","volume-title":"An introduction to the mathematical theory of the Navier-Stokes equations","author":"Galdi, G. P.","year":"2011","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387096193","edition":"2"},{"key":"13","doi-asserted-by":"publisher","first-page":"166","DOI":"10.1016\/j.cma.2012.05.008","article-title":"Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection","volume":"237\/240","author":"Galvin, Keith J.","year":"2012","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"14","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":"272","key":"15","doi-asserted-by":"publisher","first-page":"2169","DOI":"10.1090\/S0025-5718-10-02360-4","article-title":"A new error analysis for discontinuous finite element methods for linear elliptic problems","volume":"79","author":"Gudi, Thirupathi","year":"2010","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"16","doi-asserted-by":"publisher","first-page":"1484","DOI":"10.1093\/imanum\/drr040","article-title":"A family of nonconforming elements for the Brinkman problem","volume":"32","author":"Guzm\u00e1n, Johnny","year":"2012","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"285","key":"17","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":"4","key":"18","doi-asserted-by":"publisher","first-page":"1489","DOI":"10.1093\/imanum\/drt053","article-title":"Conforming and divergence-free Stokes elements in three dimensions","volume":"34","author":"Guzm\u00e1n, Johnny","year":"2014","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"3","key":"19","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\/0036-1445","issn-type":"print"},{"key":"20","unstructured":"P. Lederer, Pressure-robust discretizations for Navier\u2013Stokes equations: Divergence-free reconstruction for Taylor\u2013Hood elements and high order Hybrid Discontinuous Galerkin methods, Master\u2019s thesis, Vienna Technical University, 2016, diploma thesis."},{"issue":"3","key":"21","doi-asserted-by":"publisher","first-page":"1291","DOI":"10.1137\/16M1089964","article-title":"Divergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements","volume":"55","author":"Lederer, Philip L.","year":"2017","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"7","key":"22","doi-asserted-by":"publisher","first-page":"937","DOI":"10.1002\/mma.2849","article-title":"New error estimates of nonconforming mixed finite element methods for the Stokes problem","volume":"37","author":"Li, Mingxia","year":"2014","journal-title":"Math. Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0170-4214","issn-type":"print"},{"issue":"41-44","key":"23","doi-asserted-by":"publisher","first-page":"3278","DOI":"10.1016\/j.cma.2009.06.016","article-title":"Collision in a cross-shaped domain\u2014a steady 2d Navier-Stokes example demonstrating the importance of mass conservation in CFD","volume":"198","author":"Linke, Alexander","year":"2009","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"24","doi-asserted-by":"publisher","first-page":"782","DOI":"10.1016\/j.cma.2013.10.011","article-title":"On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime","volume":"268","author":"Linke, Alexander","year":"2014","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"25","doi-asserted-by":"publisher","first-page":"654","DOI":"10.1016\/j.jcp.2016.02.070","article-title":"On velocity errors due to irrotational forces in the Navier-Stokes momentum balance","volume":"313","author":"Linke, A.","year":"2016","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"26","doi-asserted-by":"publisher","first-page":"304","DOI":"10.1016\/j.cma.2016.08.018","article-title":"Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier-Stokes equations","volume":"311","author":"Linke, A.","year":"2016","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"1","key":"27","doi-asserted-by":"publisher","first-page":"354","DOI":"10.1093\/imanum\/drw019","article-title":"Optimal \ud835\udc3f\u00b2 velocity error estimate for a modified pressure-robust Crouzeix-Raviart Stokes element","volume":"37","author":"Linke, A.","year":"2017","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"1","key":"28","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1051\/m2an\/2015044","article-title":"Robust arbitrary order mixed finite element methods for the incompressible Stokes equations with pressure independent velocity errors","volume":"50","author":"Linke, Alexander","year":"2016","journal-title":"ESAIM Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"2","key":"29","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1051\/m2an\/2010042","article-title":"Enabling numerical accuracy of Navier-Stokes-\ud835\udefc through deconvolution and enhanced stability","volume":"45","author":"Manica, Carolina C.","year":"2011","journal-title":"ESAIM Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"1","key":"30","doi-asserted-by":"publisher","first-page":"3","DOI":"10.2478\/cmam-2011-0001","article-title":"On an efficient finite element method for Navier-Stokes-\\overline{\ud835\udf14} with strong mass conservation","volume":"11","author":"Manica, Carolina C.","year":"2011","journal-title":"Comput. Methods Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1609-4840","issn-type":"print"},{"issue":"11","key":"31","doi-asserted-by":"publisher","first-page":"2917","DOI":"10.1007\/s11425-010-3120-x","article-title":"On the error bounds of nonconforming finite elements","volume":"53","author":"Mao, ShiPeng","year":"2010","journal-title":"Sci. China Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1674-7283","issn-type":"print"},{"issue":"5","key":"32","doi-asserted-by":"publisher","first-page":"1605","DOI":"10.1137\/S0036142901383910","article-title":"A robust finite element method for Darcy-Stokes flow","volume":"40","author":"Mardal, Kent Andre","year":"2002","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"33","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1007\/s10092-015-0148-x","article-title":"Stokes elements on cubic meshes yielding divergence-free approximations","volume":"53","author":"Neilan, Michael","year":"2016","journal-title":"Calcolo","ISSN":"https:\/\/id.crossref.org\/issn\/0008-0624","issn-type":"print"},{"key":"34","unstructured":"F. Neumann, Quasi-Optimalit\u00e4t bei einer druckrobusten nichtkonformen Finite-Elemente-Methode f\u00fcr das Stokes Problem, Master\u2019s thesis, Humboldt-Universit\u00e4t zu Berlin, 2016."},{"issue":"3","key":"35","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":"36","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":"37","volume-title":"On the convergence of some low order mixed finite elements for incompressible fluids","author":"Qin, Jinshui","year":"1994"},{"issue":"5","key":"38","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1002\/fld.1407","article-title":"Stability and approximability of the \ud835\udcab\u2081-\ud835\udcab\u2080 element for Stokes equations","volume":"54","author":"Qin, Jinshui","year":"2007","journal-title":"Internat. J. Numer. Methods Fluids","ISSN":"https:\/\/id.crossref.org\/issn\/0271-2091","issn-type":"print"},{"issue":"1","key":"39","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"},{"key":"40","series-title":"Modern Birkh\\\"{a}user Classics","isbn-type":"print","volume-title":"The Navier-Stokes equations","author":"Sohr, Hermann","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/9783034805506"},{"issue":"1","key":"41","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1007\/s00211-009-0272-0","article-title":"A nonstandard mixed finite element family","volume":"115","author":"Stenberg, Rolf","year":"2010","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"42","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1007\/BF01396304","article-title":"An analysis of the \ud835\udc5d-version of the finite element method for nearly incompressible materials. Uniformly valid, optimal error estimates","volume":"41","author":"Vogelius, Michael","year":"1983","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"43","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"},{"issue":"250","key":"44","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":"3","key":"45","first-page":"456","article-title":"On the P1 Powell-Sabin divergence-free finite element for the Stokes equations","volume":"26","author":"Zhang, Shangyou","year":"2008","journal-title":"J. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0254-9409","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2018-87-312\/S0025-5718-2018-03344-0\/mcom3344_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"http:\/\/www.ams.org\/mcom\/2018-87-312\/S0025-5718-2018-03344-0\/S0025-5718-2018-03344-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2018-87-312\/S0025-5718-2018-03344-0\/S0025-5718-2018-03344-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T07:26:46Z","timestamp":1627630006000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2018-87-312\/S0025-5718-2018-03344-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,2,6]]},"references-count":45,"journal-issue":{"issue":"312","published-print":{"date-parts":[[2018,7]]}},"alternative-id":["S0025-5718-2018-03344-0"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3344","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,2,6]]}}}