{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T07:34:24Z","timestamp":1777620864970,"version":"3.51.4"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,3,31]],"date-time":"2020-03-31T00:00:00Z","timestamp":1585612800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,3,31]],"date-time":"2020-03-31T00:00:00Z","timestamp":1585612800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math.Comput.Sci."],"published-print":{"date-parts":[[2021,3]]},"DOI":"10.1007\/s11786-020-00473-1","type":"journal-article","created":{"date-parts":[[2020,3,31]],"date-time":"2020-03-31T11:03:10Z","timestamp":1585652590000},"page":"135-154","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["On the $${ H}^{1}$$ Conforming Virtual Element Method for Time Dependent Stokes Equation"],"prefix":"10.1007","volume":"15","author":[{"given":"Dibyendu","family":"Adak","sequence":"first","affiliation":[]},{"given":"Sundararajan","family":"Natarajan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,3,31]]},"reference":[{"issue":"1","key":"473_CR1","doi-asserted-by":"publisher","first-page":"222","DOI":"10.1002\/num.22298","volume":"35","author":"D Adak","year":"2019","unstructured":"Adak, D., Natarajan, E., Kumar, S.: Convergence analysis of virtual element methods for semilinear parabolic problems on polygonal meshes. Numer. Methods Partial Differ. Equ. 35(1), 222\u2013245 (2019)","journal-title":"Numer. Methods Partial Differ. Equ."},{"issue":"5","key":"473_CR2","doi-asserted-by":"publisher","first-page":"971","DOI":"10.1080\/00207160.2018.1475651","volume":"96","author":"D Adak","year":"2019","unstructured":"Adak, D., Natarajan, E., Kumar, S.: Virtual element method for semilinear hyperbolic problems on polygonal meshes. Int. J. Comput. Math. 96(5), 971\u2013991 (2019)","journal-title":"Int. J. Comput. Math."},{"issue":"3","key":"473_CR3","doi-asserted-by":"publisher","first-page":"376","DOI":"10.1016\/j.camwa.2013.05.015","volume":"66","author":"B Ahmad","year":"2013","unstructured":"Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L.D., Russo, A.: Equivalent projectors for virtual element methods. Comput. Math. Appl. 66(3), 376\u2013391 (2013)","journal-title":"Comput. Math. Appl."},{"issue":"1","key":"473_CR4","doi-asserted-by":"publisher","first-page":"386","DOI":"10.1137\/13091141X","volume":"52","author":"PF Antonietti","year":"2014","unstructured":"Antonietti, P.F., Beir\u00e3o da Veiga, L., Mora, D., Verani, M.: A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 52(1), 386\u2013404 (2014)","journal-title":"SIAM J. Numer. Anal."},{"key":"473_CR5","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1016\/S0045-7949(00)00123-1","volume":"79","author":"K Bathe","year":"2001","unstructured":"Bathe, K.: The inf-sup condition and its evaluation for mixed finite element methods. Comput. Struct. 79, 243\u2013252 (2001)","journal-title":"Comput. Struct."},{"issue":"01","key":"473_CR6","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1142\/S0218202512500492","volume":"23","author":"L Beir\u00e3o da Veiga","year":"2013","unstructured":"Beir\u00e3o da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L., Russo, A.: Basic principles of virtual element methods. Math. Model Methods Appl. Sci. 23(01), 199\u2013214 (2013)","journal-title":"Math. Model Methods Appl. Sci."},{"issue":"04","key":"473_CR7","doi-asserted-by":"publisher","first-page":"729","DOI":"10.1142\/S0218202516500160","volume":"26","author":"L Beir\u00e3o da Veiga","year":"2016","unstructured":"Beir\u00e3o da Veiga, L., Brezzi, F., Marini, L., Russo, A.: Virtual element method for general second-order elliptic problems on polygonal meshes. Math. Model Methods Appl. Sci. 26(04), 729\u2013750 (2016)","journal-title":"Math. Model Methods Appl. Sci."},{"issue":"2","key":"473_CR8","doi-asserted-by":"publisher","first-page":"794","DOI":"10.1137\/120874746","volume":"51","author":"L Beir\u00e3o da Veiga","year":"2013","unstructured":"Beir\u00e3o da Veiga, L., Brezzi, F., Marini, L.D.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794\u2013812 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"473_CR9","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1016\/j.cma.2015.07.013","volume":"295","author":"L Beir\u00e3o da Veiga","year":"2015","unstructured":"Beir\u00e3o da Veiga, L., Lovadina, C., Mora, D.: A virtual element method for elastic and inelastic problems on polytope meshes. Comput. Methods Appl. Mech. Eng. 295, 327\u2013346 (2015)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"2","key":"473_CR10","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1051\/m2an\/2016032","volume":"51","author":"L Beir\u00e3o da Veiga","year":"2017","unstructured":"Beir\u00e3o da Veiga, L., Lovadina, C., Vacca, G.: Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM Math. Model. Numer. 51(2), 509\u2013535 (2017)","journal-title":"ESAIM Math. Model. Numer."},{"key":"473_CR11","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1016\/j.cma.2016.07.043","volume":"311","author":"MF Benedetto","year":"2016","unstructured":"Benedetto, M.F., Berrone, S., Borio, A., Pieraccini, S., Scial\u00f2, S.: Order preserving SUPG stabilization for the virtual element formulation of advection\u2013diffusion problems. Comput. Methods Appl. Mech. Eng. 311, 18\u201340 (2016)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"1","key":"473_CR12","doi-asserted-by":"publisher","first-page":"82","DOI":"10.1137\/S0036142905444482","volume":"44","author":"PB Bochev","year":"2006","unstructured":"Bochev, P.B., Dohrmann, C.R., Gunzburger, M.D.: Stabilization of low-order mixed finite elements for the Stokes equations. SIAM J. Numer. Anal. 44(1), 82\u2013101 (2006)","journal-title":"SIAM J. Numer. Anal."},{"key":"473_CR13","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-36519-5","volume-title":"Mixed finite element methods and applications","author":"D Boffi","year":"2013","unstructured":"Boffi, D., Brezzi, F., Fortin, M., et al.: Mixed finite element methods and applications, vol. 44. Springer, Berlin (2013)"},{"key":"473_CR14","volume-title":"The mathematical theory of finite element methods","author":"S Brenner","year":"2007","unstructured":"Brenner, S., Scott, R.: The mathematical theory of finite element methods, vol. 15. Springer, Berlin (2007)"},{"issue":"1","key":"473_CR15","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1016\/0045-7825(92)90102-P","volume":"96","author":"F Brezzi","year":"1992","unstructured":"Brezzi, F., Bristeau, M.O., Franca, L.P., Mallet, M., Rog\u00e9, G.: A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Comput. Methods Appl. Mech. Eng. 96(1), 117\u2013129 (1992)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"473_CR16","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1016\/j.cma.2012.09.012","volume":"253","author":"F Brezzi","year":"2013","unstructured":"Brezzi, F., Marini, L.D.: Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Eng. 253, 455\u2013462 (2013)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"473_CR17","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1007\/978-3-663-14169-3_2","volume-title":"Efficient solutions of elliptic systems","author":"F Brezzi","year":"1984","unstructured":"Brezzi, F., Pitk\u00e4ranta, J.: On the stabilization of finite element approximations of the Stokes equations. In: Hackbusch, W. (ed.) Efficient solutions of elliptic systems, pp. 11\u201319. Springer, Berlin (1984)"},{"issue":"1\u20133","key":"473_CR18","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/0045-7825(82)90071-8","volume":"32","author":"AN Brooks","year":"1982","unstructured":"Brooks, A.N., Hughes, T.J.: Streamline upwind\/Petrov\u2013Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier\u2013Stokes equations. Comput. Methods Appl. Mech. Eng. 32(1\u20133), 199\u2013259 (1982)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"6","key":"473_CR19","doi-asserted-by":"publisher","first-page":"3411","DOI":"10.1137\/15M1049531","volume":"54","author":"A Cangiani","year":"2016","unstructured":"Cangiani, A., Gyrya, V., Manzini, G.: The nonconforming virtual element method for the Stokes equations. SIAM J. Numer. Anal. 54(6), 3411\u20133435 (2016)","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"473_CR20","first-page":"1317","volume":"37","author":"A Cangiani","year":"2016","unstructured":"Cangiani, A., Manzini, G., Sutton, O.J.: Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. 37(3), 1317\u20131354 (2016)","journal-title":"IMA J. Numer. Anal."},{"issue":"186","key":"473_CR21","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1090\/S0025-5718-1989-0958871-X","volume":"52","author":"J Douglas","year":"1989","unstructured":"Douglas, J., Wang, J.P.: An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52(186), 495\u2013508 (1989)","journal-title":"Math. Comput."},{"issue":"2","key":"473_CR22","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1016\/0045-7825(93)90126-I","volume":"105","author":"LP Franca","year":"1993","unstructured":"Franca, L.P., Hughes, T.J.: Convergence analyses of Galerkin least-squares methods for symmetric advective\u2013diffusive forms of the stokes and incompressible Navier\u2013Stokes equations. Comput. Methods Appl. Mech. Eng. 105(2), 285\u2013298 (1993)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"257","key":"473_CR23","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1090\/S0025-5718-06-01886-2","volume":"76","author":"Y He","year":"2007","unstructured":"He, Y., Sun, W.: Stabilized finite element method based on the Crank\u2013Nicolson extrapolation scheme for the time-dependent Navier\u2013Stokes equations. Math. Comput. 76(257), 115\u2013136 (2007)","journal-title":"Math. Comput."},{"issue":"2","key":"473_CR24","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\u2013Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19(2), 275\u2013311 (1982)","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"473_CR25","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\u2013Stokes problem III. Smoothing property and higher order error estimates for spatial discretization. SIAM J. Numer. Anal. 25(3), 489\u2013512 (1988)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"473_CR26","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\u2013Stokes problem. Part IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27(2), 353\u2013384 (1990)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"473_CR27","doi-asserted-by":"publisher","first-page":"58","DOI":"10.1016\/j.cam.2007.02.015","volume":"214","author":"J Li","year":"2008","unstructured":"Li, J., He, Y.: A stabilized finite element method based on two local gauss integrations for the Stokes equations. J. Comput. Appl. Math. 214(1), 58\u201365 (2008)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"473_CR28","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.cma.2007.06.029","volume":"197","author":"J Li","year":"2007","unstructured":"Li, J., He, Y., Chen, Z.: A new stabilized finite element method for the transient Navier\u2013Stokes equations. Comput. Methods Appl. Mech. Eng. 197(1), 22\u201335 (2007)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"473_CR29","doi-asserted-by":"publisher","first-page":"1163","DOI":"10.1016\/j.jcp.2013.07.031","volume":"257","author":"K Lipnikov","year":"2014","unstructured":"Lipnikov, K., Manzini, G., Shashkov, M.: Mimetic finite difference method. J. Comput. Phys. 257, 1163\u20131227 (2014)","journal-title":"J. Comput. Phys."},{"issue":"2","key":"473_CR30","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1002\/fld.2010","volume":"62","author":"Y Shang","year":"2010","unstructured":"Shang, Y.: New stabilized finite element method for time-dependent incompressible flow problems. Int. J. Numer. Methods Fluids 62(2), 166\u2013187 (2010)","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"473_CR31","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1016\/0045-7930(73)90027-3","volume":"1","author":"C Taylor","year":"1973","unstructured":"Taylor, C., Hood, P.: A numerical solution of the Navier\u2013Stokes equations using the finite element technique. Comput Fluids 1, 73\u2013100 (1973)","journal-title":"Comput Fluids"},{"key":"473_CR32","doi-asserted-by":"publisher","first-page":"882","DOI":"10.1016\/j.camwa.2016.04.029","volume":"74","author":"G Vacca","year":"2017","unstructured":"Vacca, G.: Virtual element methods for hyperbolic problems on polygonal meshes. Comput. Math. Appl. 74, 882\u2013898 (2017)","journal-title":"Comput. Math. Appl."},{"issue":"01","key":"473_CR33","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1142\/S0218202518500057","volume":"28","author":"G Vacca","year":"2018","unstructured":"Vacca, G.: An $$H^1$$-conforming virtual element for Darcy and Brinkman equations. Math. Models Methods Appl. Sci. 28(01), 159\u2013194 (2018)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"6","key":"473_CR34","doi-asserted-by":"publisher","first-page":"2110","DOI":"10.1002\/num.21982","volume":"31","author":"G Vacca","year":"2015","unstructured":"Vacca, G., Beir\u00e3o da Veiga, L.: Virtual element methods for parabolic problems on polygonal meshes. Numer. Methods Partial Differ. Equ. 31(6), 2110\u20132134 (2015)","journal-title":"Numer. Methods Partial Differ. Equ."}],"container-title":["Mathematics in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11786-020-00473-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11786-020-00473-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11786-020-00473-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,3]],"date-time":"2024-08-03T03:55:50Z","timestamp":1722657350000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11786-020-00473-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,31]]},"references-count":34,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,3]]}},"alternative-id":["473"],"URL":"https:\/\/doi.org\/10.1007\/s11786-020-00473-1","relation":{},"ISSN":["1661-8270","1661-8289"],"issn-type":[{"value":"1661-8270","type":"print"},{"value":"1661-8289","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,3,31]]},"assertion":[{"value":"17 July 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 February 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 March 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 March 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}