{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T07:03:28Z","timestamp":1781593408598,"version":"3.54.5"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T00:00:00Z","timestamp":1627603200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T00:00:00Z","timestamp":1627603200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100004377","name":"Hong Kong Polytechnic University","doi-asserted-by":"publisher","award":["11541"],"award-info":[{"award-number":["11541"]}],"id":[{"id":"10.13039\/501100004377","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2021,9]]},"DOI":"10.1007\/s10915-021-01588-8","type":"journal-article","created":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T23:10:42Z","timestamp":1627686642000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Second-Order Convergence of the Linearly Extrapolated Crank\u2013Nicolson Method for the Navier\u2013Stokes Equations with $$\\mathbf{H^1}$$ Initial Data"],"prefix":"10.1007","volume":"88","author":[{"given":"Buyang","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shu","family":"Ma","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Na","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,7,30]]},"reference":[{"key":"1588_CR1","volume-title":"Sobolev Spaces","author":"RA Adams","year":"1975","unstructured":"Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)"},{"key":"1588_CR2","volume-title":"Sobolev Spaces","author":"RA Adams","year":"2003","unstructured":"Adams, R.A., Fournier, J.F.: Sobolev Spaces, 2nd edn. Academic Press, Amsterdam (2003)","edition":"2"},{"issue":"2","key":"1588_CR3","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1007\/s002110050056","volume":"68","author":"AAO Ammi","year":"1994","unstructured":"Ammi, A.A.O., Marion, M.: Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations. Numer. Math. 68(2), 189\u2013213 (1994)","journal-title":"Numer. Math."},{"key":"1588_CR4","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1090\/S0025-5718-1982-0669634-0","volume":"39","author":"GA Baker","year":"1982","unstructured":"Baker, G.A., Dougalis, V., Karakashian, O.: On a higher order accurate, fully discrete Galerkin approximation to the Navier-Stokes equations. Math. Comp. 39, 339\u2013375 (1982)","journal-title":"Math. Comp."},{"issue":"5","key":"1588_CR5","doi-asserted-by":"publisher","first-page":"757","DOI":"10.1051\/m2an:2004037","volume":"38","author":"E Emmrich","year":"2004","unstructured":"Emmrich, E.: Error of the two-step BDF for the incompressible Navier-Stokes problem. ESAIM: M2AN 38(5), 757\u2013764 (2004)","journal-title":"ESAIM: M2AN"},{"key":"1588_CR6","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0063447","volume-title":"Finite Element Approximations of the Navier-Stokes Equations","author":"V Girault","year":"1979","unstructured":"Girault, V., Raviart, P.A.: Finite Element Approximations of the Navier-Stokes Equations. Springer-Verlag, New York (1979).. (Lecture Notes in Mathematics)"},{"key":"1588_CR7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61623-5","volume-title":"Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms","author":"V Girault","year":"1986","unstructured":"Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer-Verlag, Berlin (1986)"},{"key":"1588_CR8","doi-asserted-by":"publisher","first-page":"134","DOI":"10.1016\/j.camwa.2017.08.034","volume":"75","author":"Y Guo","year":"2017","unstructured":"Guo, Y., He, Y.: Unconditional convergence and optimal $$L^2$$ error estimates of the Crank-Nicolson extrapolation FEM for the nonstationary Navier-Stokes equations. Comput. Math. Appl. 75, 134\u2013152 (2017)","journal-title":"Comput. Math. Appl."},{"issue":"264","key":"1588_CR9","doi-asserted-by":"publisher","first-page":"2097","DOI":"10.1090\/S0025-5718-08-02127-3","volume":"77","author":"Y He","year":"2008","unstructured":"He, Y.: The Euler implicit\/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data. Math. Comp. 77(264), 2097\u20132124 (2008)","journal-title":"Math. Comp."},{"issue":"1","key":"1588_CR10","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1002\/num.20613","volume":"28","author":"Y He","year":"2011","unstructured":"He, Y.: The Crank-Nicolson\/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations with nonsmooth initial data. Numer. Methods PDEs 28(1), 155\u2013187 (2011)","journal-title":"Numer. Methods PDEs"},{"issue":"1","key":"1588_CR11","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1007\/s002110050332","volume":"79","author":"Y He","year":"1998","unstructured":"He, Y., Li, K.: Convergence and stability of finite element nonlinear Galerkin method for the Navier-Stokes equations. Numer. Math. 79(1), 77\u2013106 (1998)","journal-title":"Numer. Math."},{"issue":"2","key":"1588_CR12","doi-asserted-by":"publisher","first-page":"837","DOI":"10.1137\/050639910","volume":"45","author":"Y He","year":"2007","unstructured":"He, Y., Sun, W.: Stability and convergence of the Crank-Nicolson\/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 45(2), 837\u2013869 (2007)","journal-title":"SIAM J. Numer. Anal."},{"key":"1588_CR13","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1137\/0719018","volume":"19","author":"JG Heywood","year":"1982","unstructured":"Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem I: regularity of solutions and second order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275\u2013311 (1982)","journal-title":"SIAM J. Numer. Anal."},{"key":"1588_CR14","doi-asserted-by":"publisher","first-page":"489","DOI":"10.1137\/0725032","volume":"25","author":"JG Heywood","year":"1988","unstructured":"Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem, part III. Smoothing property and higher order error estimates for spatial discretization. SIAM J. Numer. Anal. 25, 489\u2013512 (1988)","journal-title":"SIAM J. Numer. Anal."},{"key":"1588_CR15","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1137\/0727022","volume":"27","author":"JG Heywood","year":"1990","unstructured":"Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem part IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353\u2013384 (1990)","journal-title":"SIAM J. Numer. Anal."},{"key":"1588_CR16","doi-asserted-by":"publisher","first-page":"633","DOI":"10.1093\/imanum\/20.4.633","volume":"20","author":"AT Hill","year":"2000","unstructured":"Hill, A.T., S\u00fcli, E.: Approximation of the global attractor for the incompressible Navier-Stokes equations. IMA J. Numer. Anal. 20, 633\u2013667 (2000)","journal-title":"IMA J. Numer. Anal."},{"issue":"4","key":"1588_CR17","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1016\/0022-1236(76)90035-5","volume":"21","author":"RB Kellogg","year":"1976","unstructured":"Kellogg, R.B., Osborn, J.E.: A regularity result for the Stokes problem in a convex polygon. J. Funct. Anal. 21(4), 397\u2013431 (1976)","journal-title":"J. Funct. Anal."},{"key":"1588_CR18","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1016\/j.apnum.2019.10.010","volume":"150","author":"W Liu","year":"2020","unstructured":"Liu, W., Hou, Y., Xue, D.: Numerical analysis of a 4th-order time parallel algorithm for the time-dependent Navier-Stokes equations. Appl. Numer. Math. 150, 361\u2013383 (2020)","journal-title":"Appl. Numer. Math."},{"issue":"2","key":"1588_CR19","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1051\/m2an\/2015047","volume":"50","author":"H Notsu","year":"2016","unstructured":"Notsu, H., Tabata, M.: Error estimates of a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equations. ESAIM: M2AN 50(2), 361\u2013380 (2016)","journal-title":"ESAIM: M2AN"},{"issue":"1","key":"1588_CR20","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1137\/0729004","volume":"29","author":"J Shen","year":"1992","unstructured":"Shen, J.: On error estimates of projection methods for Navier-Stokes equations: First-order schemes. SIAM J. Numer. Anal. 29(1), 57\u201377 (1992)","journal-title":"SIAM J. Numer. Anal."},{"issue":"215","key":"1588_CR21","doi-asserted-by":"publisher","first-page":"1039","DOI":"10.1090\/S0025-5718-96-00750-8","volume":"65","author":"J Shen","year":"1996","unstructured":"Shen, J.: On error estimates of the projection methods for the Navier-Stokes equations: Second-order schemes. Math. Comp. 65(215), 1039\u20131065 (1996)","journal-title":"Math. Comp."},{"issue":"1","key":"1588_CR22","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1137\/18M1234813","volume":"58","author":"F Sonner","year":"2020","unstructured":"Sonner, F., Richter, T.: Second order pressure estimates for the Crank-Nicolson discretization of the incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 58(1), 375\u2013409 (2020)","journal-title":"SIAM J. Numer. Anal."},{"key":"1588_CR23","doi-asserted-by":"publisher","first-page":"110","DOI":"10.1016\/j.apnum.2017.09.012","volume":"124","author":"Q Tang","year":"2018","unstructured":"Tang, Q., Huang, Y.: Stability and convergence analysis of a Crank-Nicolson leap-frog scheme for the unsteady incompressible Navier-Stokes equations. Appl. Numer. Math. 124, 110\u2013129 (2018)","journal-title":"Appl. Numer. Math."},{"key":"1588_CR24","volume-title":"Navier-Stokes Equations: Theory and Numerical Analysis","author":"R Temam","year":"1977","unstructured":"Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis. North-Holland Publishing Company, New York (1977)"},{"issue":"2","key":"1588_CR25","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1051\/m2an\/1984180201751","volume":"18","author":"R Verf\u00fcrth","year":"1984","unstructured":"Verf\u00fcrth, R.: Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO Anal. Numer. 18(2), 175\u2013182 (1984)","journal-title":"RAIRO Anal. Numer."},{"issue":"17","key":"1588_CR26","first-page":"8269","volume":"218","author":"K Wang","year":"2012","unstructured":"Wang, K., He, Y.: Convergence analysis for a higher order scheme for the time-dependent Navier-Stokes equations. Appl. Math. Comput. 218(17), 8269\u20138278 (2012)","journal-title":"Appl. Math. Comput."},{"issue":"15","key":"1588_CR27","doi-asserted-by":"publisher","first-page":"1996","DOI":"10.1080\/00207160.2012.694427","volume":"89","author":"K Wang","year":"2012","unstructured":"Wang, K., Lv, C.: Third-order temporal discrete scheme for the non-stationary Navier-Stokes equations. Int. J. Comput. Math. 89(15), 1996\u20132018 (2012)","journal-title":"Int. J. Comput. Math."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01588-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-021-01588-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01588-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,3]],"date-time":"2021-09-03T22:48:42Z","timestamp":1630709322000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-021-01588-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,30]]},"references-count":27,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,9]]}},"alternative-id":["1588"],"URL":"https:\/\/doi.org\/10.1007\/s10915-021-01588-8","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,30]]},"assertion":[{"value":"12 January 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 June 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 July 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 July 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"70"}}