{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T16:10:34Z","timestamp":1786119034687,"version":"build-2736575974"},"reference-count":34,"publisher":"American Mathematical Society (AMS)","issue":"362","license":[{"start":{"date-parts":[[2026,7,22]],"date-time":"2026-07-22T00:00:00Z","timestamp":1784678400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2024M751547"],"award-info":[{"award-number":["2024M751547"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We construct in this paper first- and second-order (in time) decoupled and unconditionally stable schemes for the Navier\u2013Stokes equations based on the consistent splitting methods and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C Superscript 0\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">C^0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    finite elements, where spatial discontinuity of the gradient of the discrete velocity is treated using a discontinuous Galerkin framework. The challenge for their stability analysis is that the gradient of discrete velocity is discontinuous so that the estimate on the continuous Stokes pressure by Liu, Liu, and Pego [Comm. Pure Appl. Math. 60 (2007), pp.\u00a01443\u20131487] cannot be applied in the space discrete case. We first extend the estimate on the Stokes pressure to the space discrete case, and then use it to prove the unconditional stability and carry out an error analysis for the proposed fully discrete schemes. Numerical results are presented to validate our analytical results.\n                  <\/p>","DOI":"10.1090\/mcom\/4126","type":"journal-article","created":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T18:22:09Z","timestamp":1753208529000},"page":"2669-2694","source":"Crossref","is-referenced-by-count":1,"title":["Unconditionally stable consistent splitting schemes for the Navier\u2013Stokes equations with \ud835\udc36\u2070 finite elements"],"prefix":"10.1090","volume":"95","author":[{"given":"Xu","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jie","family":"Shen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,7,22]]},"reference":[{"issue":"5","key":"1","doi-asserted-by":"publisher","first-page":"1749","DOI":"10.1137\/S0036142901384162","article-title":"Unified analysis of discontinuous Galerkin methods for elliptic problems","volume":"39","author":"Arnold, Douglas N.","year":"2001","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1007\/978-3-642-23914-4_2","article-title":"\ud835\udc36\u2070 interior penalty methods","author":"Brenner, Susanne C.","year":"2012","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642239137"},{"key":"3","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":"4","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/s10915-004-4135-7","article-title":"\ud835\udc36\u2070 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains","volume":"22\/23","author":"Brenner, Susanne C.","year":"2005","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"317","DOI":"10.4310\/cms.2003.v1.n2.a6","article-title":"Gauge method for viscous incompressible flows","volume":"1","author":"E, Weinan","year":"2003","journal-title":"Commun. Math. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/1539-6746","issn-type":"print"},{"issue":"4","key":"6","doi-asserted-by":"publisher","first-page":"1017","DOI":"10.1137\/0732047","article-title":"Projection method. I. Convergence and numerical boundary layers","volume":"32","author":"E, Weinan","year":"1995","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"34","key":"7","doi-asserted-by":"publisher","first-page":"3669","DOI":"10.1016\/S0045-7825(02)00286-4","article-title":"Continuous\/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity","volume":"191","author":"Engel, G.","year":"2002","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"8","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":"44-47","key":"9","doi-asserted-by":"publisher","first-page":"6011","DOI":"10.1016\/j.cma.2005.10.010","article-title":"An overview of projection methods for incompressible flows","volume":"195","author":"Guermond, J. L.","year":"2006","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1016\/j.jcp.2003.07.009","article-title":"A new class of truly consistent splitting schemes for incompressible flows","volume":"192","author":"Guermond, J. L.","year":"2003","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"6","key":"11","doi-asserted-by":"publisher","first-page":"A2656--A2681","DOI":"10.1137\/140975231","article-title":"High-order time stepping for the incompressible Navier-Stokes equations","volume":"37","author":"Guermond, Jean-Luc","year":"2015","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"A770--A788","DOI":"10.1137\/18M1209301","article-title":"High-order adaptive time stepping for the incompressible Navier-Stokes equations","volume":"41","author":"Guermond, Jean-Luc","year":"2019","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"13","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1016\/j.cam.2016.04.033","article-title":"High-order time stepping for the Navier-Stokes equations with minimal computational complexity","volume":"310","author":"Guermond, Jean-Luc","year":"2017","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"3","key":"14","doi-asserted-by":"publisher","first-page":"917","DOI":"10.1137\/090768758","article-title":"Error analysis of a fractional time-stepping technique for incompressible flows with variable density","volume":"49","author":"Guermond, J.-L.","year":"2011","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"15","doi-asserted-by":"publisher","first-page":"112","DOI":"10.1137\/S0036142901395400","article-title":"Velocity-correction projection methods for incompressible flows","volume":"41","author":"Guermond, J. L.","year":"2003","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"248","key":"16","doi-asserted-by":"publisher","first-page":"1719","DOI":"10.1090\/S0025-5718-03-01621-1","article-title":"On the error estimates for the rotational pressure-correction projection methods","volume":"73","author":"Guermond, J. L.","year":"2004","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"17","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1137\/0719018","article-title":"Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization","volume":"19","author":"Heywood, John G.","year":"1982","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"5","key":"18","doi-asserted-by":"publisher","first-page":"2408","DOI":"10.1137\/23M1556022","article-title":"Stability and error analysis of a second-order consistent splitting scheme for the Navier-Stokes equations","volume":"61","author":"Huang, Fukeng","year":"2023","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"19","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1016\/j.jcp.2004.02.009","article-title":"Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term","volume":"199","author":"Johnston, Hans","year":"2004","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"2","key":"20","doi-asserted-by":"publisher","first-page":"414","DOI":"10.1016\/0021-9991(91)90007-8","article-title":"High-order splitting methods for the incompressible Navier-Stokes equations","volume":"97","author":"Karniadakis, George Em.","year":"1991","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"6","key":"21","doi-asserted-by":"publisher","first-page":"2105","DOI":"10.1051\/m2an\/2022073","article-title":"Analysis of fully discrete finite element methods for 2D Navier-Stokes equations with critical initial data","volume":"56","author":"Li, Buyang","year":"2022","journal-title":"ESAIM Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/2822-7840","issn-type":"print"},{"issue":"3","key":"22","doi-asserted-by":"publisher","first-page":"Paper No. 70, 20","DOI":"10.1007\/s10915-021-01588-8","article-title":"Second-order convergence of the linearly extrapolated Crank-Nicolson method for the Navier-Stokes equations with \ud835\udc3b\u00b9 initial data","volume":"88","author":"Li, Buyang","year":"2021","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"key":"23","doi-asserted-by":"publisher","first-page":"Paper No. 114892, 19","DOI":"10.1016\/j.cam.2022.114892","article-title":"Error analysis of a fully discrete consistent splitting MAC scheme for time dependent Stokes equations","volume":"421","author":"Li, Xiaoli","year":"2023","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"333","key":"24","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1090\/mcom\/3651","article-title":"New SAV-pressure correction methods for the Navier-Stokes equations: stability and error analysis","volume":"91","author":"Li, Xiaoli","year":"2021","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"10","key":"25","doi-asserted-by":"publisher","first-page":"1443","DOI":"10.1002\/cpa.20178","article-title":"Stability and convergence of efficient Navier-Stokes solvers via a commutator estimate","volume":"60","author":"Liu, Jian-Guo","year":"2007","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"issue":"6","key":"26","doi-asserted-by":"publisher","first-page":"743","DOI":"10.1007\/s11401-009-0116-3","article-title":"Error estimates for finite-element Navier-Stokes solvers without standard inf-sup conditions","volume":"30","author":"Liu, Jian-Guo","year":"2009","journal-title":"Chinese Ann. Math. Ser. B","ISSN":"https:\/\/id.crossref.org\/issn\/0252-9599","issn-type":"print"},{"issue":"250","key":"27","doi-asserted-by":"publisher","first-page":"521","DOI":"10.1090\/S0025-5718-04-01687-4","article-title":"Error estimates for semi-discrete gauge methods for the Navier-Stokes equations","volume":"74","author":"Nochetto, Ricardo H.","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"28","series-title":"Advances in Numerical Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-663-11171-9","volume-title":"Projection and quasi-compressibility methods for solving the incompressible Navier-Stokes equations","author":"Prohl, Andreas","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/3519027232"},{"issue":"1","key":"29","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1137\/0729004","article-title":"On error estimates of projection methods for Navier-Stokes equations: first-order schemes","volume":"29","author":"Shen, Jie","year":"1992","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"30","doi-asserted-by":"publisher","first-page":"386","DOI":"10.1137\/0732016","article-title":"On error estimates of the penalty method for unsteady Navier-Stokes equations","volume":"32","author":"Shen, Jie","year":"1995","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"31","series-title":"CBMS-NSF Regional Conference Series in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970050","volume-title":"Navier-Stokes equations and nonlinear functional analysis","volume":"66","author":"Temam, Roger","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0898713404","edition":"2"},{"issue":"7","key":"32","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1002\/(SICI)1097-0363(19960415)22:7<673::AID-FLD373>3.0.CO;2-O","article-title":"An approximate projection scheme for incompressible flow using spectral elements","volume":"22","author":"Timmermans, L. J. P.","year":"1996","journal-title":"Internat. J. Numer. Methods Fluids","ISSN":"https:\/\/id.crossref.org\/issn\/0271-2091","issn-type":"print"},{"issue":"232","key":"33","doi-asserted-by":"publisher","first-page":"1385","DOI":"10.1090\/S0025-5718-00-01248-5","article-title":"Convergence of gauge method for incompressible flow","volume":"69","author":"Wang, Cheng","year":"2000","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"34","doi-asserted-by":"publisher","first-page":"Paper No. 111097, 16","DOI":"10.1016\/j.jcp.2022.111097","article-title":"A new class of higher-order decoupled schemes for the incompressible Navier-Stokes equations and applications to rotating dynamics","volume":"458","author":"Wu, Ke","year":"2022","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-362\/S0025-5718-2025-04126-7\/mcom4126_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-362\/S0025-5718-2025-04126-7\/S0025-5718-2025-04126-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,6]],"date-time":"2026-08-06T17:15:34Z","timestamp":1786036534000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-362\/S0025-5718-2025-04126-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,22]]},"references-count":34,"journal-issue":{"issue":"362","published-print":{"date-parts":[[2026,11]]}},"alternative-id":["S0025-5718-2025-04126-7"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/4126","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":[[2025,7,22]]}}}