{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T13:39:03Z","timestamp":1762522743824,"version":"3.37.3"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,5,21]],"date-time":"2022-05-21T00:00:00Z","timestamp":1653091200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,5,21]],"date-time":"2022-05-21T00:00:00Z","timestamp":1653091200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"crossref","award":["NRF-2018R1D1A1B0705058314 and NRF-2021R1F1A106243411"],"award-info":[{"award-number":["NRF-2018R1D1A1B0705058314 and NRF-2021R1F1A106243411"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"crossref","award":["NRF-2015R1A5A1009350"],"award-info":[{"award-number":["NRF-2015R1A5A1009350"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"crossref","award":["NRF-2020R1I1A1A01070361"],"award-info":[{"award-number":["NRF-2020R1I1A1A01070361"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2022,10]]},"DOI":"10.1007\/s11075-022-01288-w","type":"journal-article","created":{"date-parts":[[2022,5,21]],"date-time":"2022-05-21T12:02:40Z","timestamp":1653134560000},"page":"959-996","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Error analysis for the pseudostress formulation of unsteady Stokes problem"],"prefix":"10.1007","volume":"91","author":[{"given":"Dongho","family":"Kim","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3836-1595","authenticated-orcid":false,"given":"Eun-Jae","family":"Park","sequence":"additional","affiliation":[]},{"given":"Boyoon","family":"Seo","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,5,21]]},"reference":[{"issue":"1-2","key":"1288_CR1","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1007\/BF01395876","volume":"53","author":"DN Arnold","year":"1988","unstructured":"Arnold, D. N., Falk, R. S.: A new mixed formulation for elasticity. Numer. Math. 53(1-2), 13\u201330 (1988)","journal-title":"Numer. Math."},{"key":"1288_CR2","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1007\/BF02417101","volume":"101","author":"JH Bramble","year":"1974","unstructured":"Bramble, J. H., Thomee, V.: Discrete time Galerkin methods for a parabolic boundary value problem. Ann. Mat. Pura Appl. 101, 115\u2013152 (1974)","journal-title":"Ann. Mat. Pura Appl."},{"key":"1288_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and hybrid finite element methods","author":"F Brezzi","year":"1991","unstructured":"Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer, New York (1991)"},{"key":"1288_CR4","doi-asserted-by":"publisher","first-page":"843","DOI":"10.1137\/S0036142903422673","volume":"42","author":"Z Cai","year":"2004","unstructured":"Cai, Z., Lee, B., Wang, P.: Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems. SIAM J. Numer. Anal. 42, 843\u2013859 (2004)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"1288_CR5","doi-asserted-by":"publisher","first-page":"957","DOI":"10.1002\/num.20467","volume":"26","author":"Z Cai","year":"2010","unstructured":"Cai, Z., Tong, C., Vassilevski, P. S., Wang, C.: Mixed finite element methods for incompressible flow: stationary Stokes equations. Numer. Methods Part. Diff. Eqs. 26(4), 957\u2013978 (2010)","journal-title":"Numer. Methods Part. Diff. Eqs."},{"issue":"5","key":"1288_CR6","doi-asserted-by":"publisher","first-page":"2078","DOI":"10.1137\/060661429","volume":"29","author":"Z Cai","year":"2007","unstructured":"Cai, Z., Wang, Y.: A multigrid method for the pseudostress formulation of Stokes problems. SIAM J. Sci. Comput. 29(5), 2078\u20132095 (2007)","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"1288_CR7","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1002\/fld.2077","volume":"63","author":"Z Cai","year":"2010","unstructured":"Cai, Z., Wang, Y.: Pseudostress-velocity formulation for incompressible Navier-Stokes equations. Int. J. Numer. Mech. Fluids 63(3), 341\u2013356 (2010)","journal-title":"Int. J. Numer. Mech. Fluids"},{"issue":"1","key":"1288_CR8","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1137\/080718413","volume":"48","author":"Z Cai","year":"2010","unstructured":"Cai, Z., Wang, C., Zhang, S.: Mixed finite element methods for incompressible flow: stationary Navier-Stokes equations. S.AM J. Numer. Anal. 48(1), 79\u201394 (2010)","journal-title":"S.AM J. Numer. Anal."},{"issue":"3","key":"1288_CR9","doi-asserted-by":"publisher","first-page":"1715","DOI":"10.1137\/110852346","volume":"51","author":"C Carstensen","year":"2013","unstructured":"Carstensen, C., Gallistl, D., Schedensack, M.: Quasi-optimal adaptive pseudostress approximation of the Stokes equations. SIAM J. Numer. Anal. 51(3), 1715\u20131734 (2013)","journal-title":"SIAM J. Numer. Anal."},{"issue":"6","key":"1288_CR10","doi-asserted-by":"publisher","first-page":"2501","DOI":"10.1137\/100816237","volume":"49","author":"C Carstensen","year":"2011","unstructured":"Carstensen, C., Kim, D., Park, E. -J., priori, A: a posteriori pseudostress-velocity mixed finite element error analysis for the Stokes problem. SIAM J. Numer Anal. 49(6), 2501\u20132523 (2011)","journal-title":"SIAM J. Numer Anal."},{"key":"1288_CR11","doi-asserted-by":"crossref","unstructured":"Caucao, S., Yotov, I.: A Banach space mixed formulation for the unsteady Brinkman\u2013Forchheimer equations, IMA J. Numer. Anal. 41(4), 2708\u20132743 (2021)","DOI":"10.1093\/imanum\/draa035"},{"issue":"1","key":"1288_CR12","doi-asserted-by":"publisher","first-page":"296","DOI":"10.1093\/imanum\/drw002","volume":"37","author":"E C\u00e1ceres","year":"2017","unstructured":"C\u00e1ceres, E., Gatica, G. N.: A mixed virtual element method for the pseudostress-velocity formulation of the Stokes problem. IMA J. Numer. Anal. 37(1), 296\u2013331 (2017)","journal-title":"IMA J. Numer. Anal."},{"key":"1288_CR13","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1016\/j.apnum.2018.09.003","volume":"135","author":"E C\u00e1ceres","year":"2019","unstructured":"C\u00e1ceres, E., Gatica, G. N., Sequeira, F. A.: A mixed virtual element method for a pseudostress-based formulation of linear elasticity. Appl. Numer. Math. 135, 423\u2013442 (2019)","journal-title":"Appl. Numer. Math."},{"key":"1288_CR14","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4355-5","volume-title":"Theory and practice of finite elements applied mathematical sciences, vol. 159","author":"A Ern","year":"2004","unstructured":"Ern, A., Guermond, J. L.: Theory and practice of finite elements applied mathematical sciences, vol. 159. Springer, New York (2004)"},{"issue":"17\u201320","key":"1288_CR15","doi-asserted-by":"publisher","first-page":"1064","DOI":"10.1016\/j.cma.2009.11.024","volume":"199","author":"GN Gatica","year":"2010","unstructured":"Gatica, G. N., M\u00e1rquez, A., S\u00e1nchez, M. A.: Analysis of a velocity-pressure-pseudostress formulation for the stationary Stokes equation. Comput. Methods Appl. Mech. Eng. 199(17\u201320), 1064\u20131079 (2010)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1288_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61623-5","volume-title":"Finite element methods for Navier-Stokes equations","author":"V Girault","year":"1986","unstructured":"Girault, V., Raviart, P. A.: Finite element methods for Navier-Stokes equations. Springer, New York (1986)"},{"key":"1288_CR17","unstructured":"Johnson, C.: Numerical solution of partial differential equations by the finite element method. Cambridge Univercity Press (1987)"},{"key":"1288_CR18","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1051\/m2an\/1981150100411","volume":"15","author":"C Johnson","year":"1981","unstructured":"Johnson, C., Thom\u00e9e, V.: Error estimates for some mixed finite element methods for parabolic type problems. R.A.I.R.O. Anal. Num\u00e9r. 15, 41\u201378 (1981)","journal-title":"R.A.I.R.O. Anal. Num\u00e9r."},{"key":"1288_CR19","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1016\/j.aml.2017.10.017","volume":"78","author":"D Kim","year":"2018","unstructured":"Kim, D., Park, E. -J., Seo, B.: Optimal error estimates for the pseudostress formulation of the Navier-Stokes equations. Appl. Math. Lett. 78, 24\u201330 (2018)","journal-title":"Appl. Math. Lett."},{"issue":"4","key":"1288_CR20","first-page":"873","volume":"10","author":"D Kim","year":"2008","unstructured":"Kim, D., Park, E. -J.: Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems. Discret. Contin. Dyn. Syst. Ser. B 10(4), 873\u2013886 (2008)","journal-title":"Discret. Contin. Dyn. Syst. Ser. B"},{"issue":"2","key":"1288_CR21","doi-asserted-by":"publisher","first-page":"267","DOI":"10.4134\/JKMS.2014.51.2.267","volume":"51","author":"D Kim","year":"2014","unstructured":"Kim, D., Park, E. -J., Seo, B.: Two-scale product approximation for semilinear parabolic problems in mixed methods. J. Korean Math. Soc. 51(2), 267\u2013288 (2014)","journal-title":"J. Korean Math. Soc."},{"issue":"3","key":"1288_CR22","doi-asserted-by":"publisher","first-page":"1725","DOI":"10.1007\/s10915-017-0514-8","volume":"74","author":"D Kim","year":"2018","unstructured":"Kim, D., Park, E. -J., Seo, B.: Space-time adaptive methods for the mixed formulation of a linear parabolic problem. J. Sci. Comput. 74(3), 1725\u20131756 (2018)","journal-title":"J. Sci. Comput."},{"issue":"4","key":"1288_CR23","doi-asserted-by":"publisher","first-page":"A2537","DOI":"10.1137\/20M1322170","volume":"42","author":"L Dohyun Kim","year":"2020","unstructured":"Dohyun Kim, L., Zhao, E. -J.: Park, staggered DG methods for the pseudostress-velocity formulation of the Stokes equations on general meshes. SIAM J. Sci. Comput. 42(4), A2537\u2013A2560 (2020)","journal-title":"SIAM J. Sci. Comput."},{"key":"1288_CR24","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/0893-9659(95)00106-9","volume":"9","author":"M-Y Kim","year":"1996","unstructured":"Kim, M. -Y., Milner, F. A., Park, E. -J.: Some observations on mixed methods for fully nonlinear parabolic problems in divergence form. Appl. Math. Lett. 9, 75\u201381 (1996)","journal-title":"Appl. Math. Lett."},{"issue":"9","key":"1288_CR25","doi-asserted-by":"publisher","first-page":"1061","DOI":"10.1016\/S0898-1221(00)85016-6","volume":"40","author":"M-Y Kim","year":"2000","unstructured":"Kim, M. -Y., Park, E. -J., Park, J.: Mixed finite element domain decomposition for nonlinear parabolic problems. Comput. Math. Appl 40(9), 1061\u20131070 (2000)","journal-title":"Comput. Math. Appl"},{"issue":"2","key":"1288_CR26","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1002\/num.20035","volume":"21","author":"E-J Park","year":"2005","unstructured":"Park, E. -J.: Mixed finite element methods for generalized Forchheimer flow in porous media. Numer. Methods Partial Differ. Equ. 21(2), 213\u2013228 (2005)","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"1288_CR27","doi-asserted-by":"crossref","unstructured":"Raviart, P. A., Thomas, J.: A mixed finite element method for 2-nd order elliptic problems, Mathematical aspects of the Finite Elements Method, Lectures Notes in Math. vol. 606, pp. 292\u2013315. Springer, Berlin (1977)","DOI":"10.1007\/BFb0064470"},{"key":"1288_CR28","unstructured":"Temam, R.: Navier-Stokes equations, North Holland. Amsterdam (1977)"},{"key":"1288_CR29","doi-asserted-by":"crossref","unstructured":"Thom\u00e9e, V.: Galerkin element methods for parabolic problems. Springer (1997)","DOI":"10.1007\/978-3-662-03359-3"},{"key":"1288_CR30","unstructured":"Wang, C.: Mixed finite element methods for the Stokes and Navier-Stokes equations. Thesis (Ph.D.)\u2013Purdue University, pp. 1\u2013112 (2007)"},{"key":"1288_CR31","doi-asserted-by":"publisher","first-page":"854","DOI":"10.1016\/j.cma.2018.11.016","volume":"345","author":"L Zhao","year":"2019","unstructured":"Zhao, L., Park, E. -J., Shin, D.-W.: A staggered DG method of minimal dimension for the Stokes equations on general meshes. Comput. Methods Appl. Mech. Eng. 345, 854\u2013875 (2019)","journal-title":"Comput. Methods Appl. Mech. Eng."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-022-01288-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-022-01288-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-022-01288-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,9]],"date-time":"2022-09-09T07:41:51Z","timestamp":1662709311000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-022-01288-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,21]]},"references-count":31,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,10]]}},"alternative-id":["1288"],"URL":"https:\/\/doi.org\/10.1007\/s11075-022-01288-w","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2022,5,21]]},"assertion":[{"value":"24 July 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 February 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 May 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interest"}}]}}