{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T04:47:46Z","timestamp":1747198066633,"version":"3.40.5"},"reference-count":48,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,4,1]]},"abstract":"Abstract<\/jats:title>\n We present in this paper a rigorous error analysis of the vector penalty-projection method for solving the time-dependent incompressible Stokes equations with open boundary conditions on part of the boundary.\nFirst, we prove the stability of the scheme.\nThen we provide an error analysis for the second-order vector penalty-projection method which shows that the convergence rate of the error on the velocity and the pressure is of order 2 in \n \n \n \n \n l<\/m:mi>\n \u221e<\/m:mi>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n l^{\\infty}(\\mathbf{L}^{2}(\\Omega))<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n \n \n \n l<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n l^{2}(L^{2}(\\Omega))<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> respectively.\nIn addition, it is shown that the splitting errors of the method varies as \n \n \n \n O<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03b5<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n \\mathcal{O}(\\varepsilon)<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, where \ud835\udf00 is a penalty parameter chosen as small as desired.\nSeveral numerical tests in agreement with the theoretical results are presented.\nTo the best of our knowledge, this paper provides the first rigorous proof of optimal error estimates for second-order splitting schemes with open boundary conditions.<\/jats:p>","DOI":"10.1515\/cmam-2023-0261","type":"journal-article","created":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T17:26:37Z","timestamp":1731518797000},"page":"397-414","source":"Crossref","is-referenced-by-count":0,"title":["Error Analysis of the Vector Penalty-Projection Methods for the Time-Dependent Stokes Equations with Open Boundary Conditions"],"prefix":"10.1515","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1260-3135","authenticated-orcid":false,"given":"Rima","family":"Cheaytou","sequence":"first","affiliation":[{"name":"Mathematics Division, School of Arts and Sciences , American University in Dubai , Sheikh Zayed Road, PO Box 28282 , Dubai , United Arab Emirates"}]},{"given":"Philippe","family":"Angot","sequence":"additional","affiliation":[{"name":"Institut de Math\u00e9matiques de Marseille (I2M) \u2013 CNRS UMR7373 , Aix-Marseille Universit\u00e9 , Centrale Marseille, 13453 Marseille cedex 13 , France"}]}],"member":"374","published-online":{"date-parts":[[2024,11,14]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"N. Ahmed, T. Chac\u00f3n Rebollo, V. John and S. Rubino,\nAnalysis of a full space-time discretization of the Navier\u2013Stokes equations by a local projection stabilization method,\nIMA J. Numer. Anal. 37 (2017), no. 3, 1437\u20131467.","key":"2025032910202776849_j_cmam-2023-0261_ref_001","DOI":"10.1093\/imanum\/drw048"},{"doi-asserted-by":"crossref","unstructured":"P. Angot, F. Boyer and F. Hubert,\nAsymptotic and numerical modelling of flows in fractured porous media,\nM2AN Math. Model. Numer. Anal. 43 (2009), no. 2, 239\u2013275.","key":"2025032910202776849_j_cmam-2023-0261_ref_002","DOI":"10.1051\/m2an\/2008052"},{"unstructured":"P. Angot, J.-P. Caltagirone and P. Fabrie,\nVector penalty-projection methods for the solution of unsteady incompressible flows,\nFinite Volumes for Complex Applications V,\nISTE, London (2008), 169\u2013176.","key":"2025032910202776849_j_cmam-2023-0261_ref_003"},{"doi-asserted-by":"crossref","unstructured":"P. Angot, J.-P. Caltagirone and P. Fabrie,\nA spectacular vector penalty-projection method for Darcy and Navier\u2013Stokes problems,\nFinite Volumes for Complex Applications VI. Problems & Perspectives. Volume 1, 2,\nSpringer Proc. Math. 4,\nSpringer, Heidelberg (2011), 39\u201347.","key":"2025032910202776849_j_cmam-2023-0261_ref_004","DOI":"10.1007\/978-3-642-20671-9_5"},{"doi-asserted-by":"crossref","unstructured":"P. Angot, J.-P. Caltagirone and P. Fabrie,\nA fast vector penalty-projection method for incompressible non-homogeneous or multiphase Navier\u2013Stokes problems,\nAppl. Math. Lett. 25 (2012), no. 11, 1681\u20131688.","key":"2025032910202776849_j_cmam-2023-0261_ref_005","DOI":"10.1016\/j.aml.2012.01.037"},{"doi-asserted-by":"crossref","unstructured":"P. Angot, J.-P. Caltagirone and P. Fabrie,\nA new fast method to compute saddle-points in constrained optimization and applications,\nAppl. Math. Lett. 25 (2012), no. 3, 245\u2013251.","key":"2025032910202776849_j_cmam-2023-0261_ref_006","DOI":"10.1016\/j.aml.2011.08.015"},{"unstructured":"P. Angot, J.-P. Caltagirone and P. Fabrie,\nAnalysis for the fast vector penalty-projection solver of incompressible multiphase Navier\u2013Stokes\/Brinkman problems, preprint (2015), https:\/\/hal.science\/hal-01194345; to appear in Numer. Math.","key":"2025032910202776849_j_cmam-2023-0261_ref_007"},{"doi-asserted-by":"crossref","unstructured":"P. Angot, J.-P. Caltagirone and P. Fabrie,\nA kinematic vector penalty-projection method for incompressible flow with variable density,\nC. R. Math. Acad. Sci. Paris 354 (2016), no. 11, 1124\u20131131.","key":"2025032910202776849_j_cmam-2023-0261_ref_008","DOI":"10.1016\/j.crma.2016.06.007"},{"unstructured":"P. Angot and R. Cheaytou,\nVector penalty-projection methods for incompressible fluid flows with open boundary conditions,\nAlgoritmy 2012,\nPublishing House of STU, Bratislava (2012), 219\u2013229.","key":"2025032910202776849_j_cmam-2023-0261_ref_009"},{"doi-asserted-by":"crossref","unstructured":"P. Angot and R. Cheaytou,\nOn the error estimates of the vector penalty-projection methods: Second-order scheme,\nMath. Comp. 87 (2018), no. 313, 2159\u20132187.","key":"2025032910202776849_j_cmam-2023-0261_ref_010","DOI":"10.1090\/mcom\/3309"},{"doi-asserted-by":"crossref","unstructured":"P. Angot and R. Cheaytou,\nVector penalty-projection methods for open boundary conditions with optimal second-order accuracy,\nCommun. Comput. Phys. 26 (2019), no. 4, 1008\u20131038.","key":"2025032910202776849_j_cmam-2023-0261_ref_011","DOI":"10.4208\/cicp.OA-2018-0016"},{"unstructured":"P. Angot, M. Jobelin and J.-C. Latch\u00e9,\nError analysis of the penalty-projection method for the time dependent Stokes equations,\nInt. J. Finite Vol. 6 (2009), no. 1, 1\u201326.","key":"2025032910202776849_j_cmam-2023-0261_ref_012"},{"doi-asserted-by":"crossref","unstructured":"F. Boyer and P. Fabrie,\nMathematical Tools for the Study of the Incompressible Navier\u2013Stokes Equations and Related Models,\nAppl. Math. Sci. 183,\nSpringer, New York, 2013.","key":"2025032910202776849_j_cmam-2023-0261_ref_013","DOI":"10.1007\/978-1-4614-5975-0"},{"doi-asserted-by":"crossref","unstructured":"J.-P. Caltagirone and J. Breil,\nSur une m\u00e9thode de projection vectorielle pour la r\u00e9solution des \u00e9quations de Navier\u2013Stokes,\nC. R. Math. Acad. Sci. Paris 327 (1999), no. 11, 1179\u20131184.","key":"2025032910202776849_j_cmam-2023-0261_ref_014","DOI":"10.1016\/S1287-4620(00)88522-1"},{"unstructured":"R. Cheaytou,\nEtude des m\u00e9thodes de p\u00e9nalit\u00e9-projection vectorielle pour les \u00e9quations de Navier\u2013Stokes avec conditions aux limites ouvertes,\nPh.D. Thesis, Universit\u00e9 d\u2019Aix-Marseille, 2014.","key":"2025032910202776849_j_cmam-2023-0261_ref_015"},{"doi-asserted-by":"crossref","unstructured":"A. J. Chorin,\nNumerical solution of the Navier\u2013Stokes equations,\nMath. Comp. 22 (1968), 745\u2013762.","key":"2025032910202776849_j_cmam-2023-0261_ref_016","DOI":"10.1090\/S0025-5718-1968-0242392-2"},{"doi-asserted-by":"crossref","unstructured":"J. H. Ferziger and M. Peri\u0107,\nComputational Methods for Fluid Dynamics,\nSpringer, Berlin, 1996.","key":"2025032910202776849_j_cmam-2023-0261_ref_017","DOI":"10.1007\/978-3-642-97651-3"},{"doi-asserted-by":"crossref","unstructured":"C. F\u00e9vri\u00e8re, J. Laminie, P. Poullet and P. Angot,\nOn the penalty-projection method for the Navier\u2013Stokes equations with the MAC mesh,\nJ. Comput. Appl. Math. 226 (2009), no. 2, 228\u2013245.","key":"2025032910202776849_j_cmam-2023-0261_ref_018","DOI":"10.1016\/j.cam.2008.08.014"},{"doi-asserted-by":"crossref","unstructured":"B. Garc\u00eda-Archilla, V. John and J. Novo,\nOn the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows,\nComput. Methods Appl. Mech. Engrg. 385 (2021), Article ID 114032.","key":"2025032910202776849_j_cmam-2023-0261_ref_019","DOI":"10.1016\/j.cma.2021.114032"},{"doi-asserted-by":"crossref","unstructured":"K. Goda,\nA multistep technique with implicit difference schemes for calculating two-or three-dimensional cavity flows,\nJ. Comput. Phys. 30 (1979), no. 1, 76\u201395.","key":"2025032910202776849_j_cmam-2023-0261_ref_020","DOI":"10.1016\/0021-9991(79)90088-3"},{"doi-asserted-by":"crossref","unstructured":"J.-L. Guermond,\nUn r\u00e9sultat de convergence d\u2019ordre deux en temps pour l\u2019approximation des \u00e9quations de Navier\u2013Stokes par une technique de projection incr\u00e9mentale,\nM2AN Math. Model. Numer. Anal. 33 (1999), no. 1, 169\u2013189.","key":"2025032910202776849_j_cmam-2023-0261_ref_021","DOI":"10.1051\/m2an:1999101"},{"doi-asserted-by":"crossref","unstructured":"J. L. Guermond, P. Minev and J. Shen,\nError analysis of pressure-correction schemes for the time-dependent Stokes equations with open boundary conditions,\nSIAM J. Numer. Anal. 43 (2005), no. 1, 239\u2013258.","key":"2025032910202776849_j_cmam-2023-0261_ref_022","DOI":"10.1137\/040604418"},{"doi-asserted-by":"crossref","unstructured":"J. L. Guermond, P. Minev and J. Shen,\nAn overview of projection methods for incompressible flows,\nComput. Methods Appl. Mech. Engrg. 195 (2006), no. 44\u201347, 6011\u20136045.","key":"2025032910202776849_j_cmam-2023-0261_ref_023","DOI":"10.1016\/j.cma.2005.10.010"},{"doi-asserted-by":"crossref","unstructured":"J.-L. Guermond and J. Shen,\nQuelques r\u00e9sultats nouveaux sur les m\u00e9thodes de projection,\nC. R. Acad. Sci. Paris S\u00e9r. I Math. 333 (2001), no. 12, 1111\u20131116.","key":"2025032910202776849_j_cmam-2023-0261_ref_024","DOI":"10.1016\/S0764-4442(01)02157-7"},{"doi-asserted-by":"crossref","unstructured":"J. L. Guermond and J. Shen,\nA new class of truly consistent splitting schemes for incompressible flows,\nJ. Comput. Phys. 192 (2003), no. 1, 262\u2013276.","key":"2025032910202776849_j_cmam-2023-0261_ref_025","DOI":"10.1016\/j.jcp.2003.07.009"},{"doi-asserted-by":"crossref","unstructured":"J. L. Guermond and J. Shen,\nVelocity-correction projection methods for incompressible flows,\nSIAM J. Numer. Anal. 41 (2003), no. 1, 112\u2013134.","key":"2025032910202776849_j_cmam-2023-0261_ref_026","DOI":"10.1137\/S0036142901395400"},{"doi-asserted-by":"crossref","unstructured":"J. L. Guermond and J. Shen,\nOn the error estimates for the rotational pressure-correction projection methods,\nMath. Comp. 73 (2004), no. 248, 1719\u20131737.","key":"2025032910202776849_j_cmam-2023-0261_ref_027","DOI":"10.1090\/S0025-5718-03-01621-1"},{"doi-asserted-by":"crossref","unstructured":"J. L. Guermond, J. Shen and X. Yang,\nError analysis of fully discrete velocity-correction methods for incompressible flows,\nMath. Comp. 77 (2008), no. 263, 1387\u20131405.","key":"2025032910202776849_j_cmam-2023-0261_ref_028","DOI":"10.1090\/S0025-5718-08-02109-1"},{"doi-asserted-by":"crossref","unstructured":"F. H. Harlow and J. E. Welch,\nNumerical calculation of time-dependent viscous incompressible flow of fluid with free surface,\nPhys. Fluids 8 (1965), no. 12, 2182\u20132189.","key":"2025032910202776849_j_cmam-2023-0261_ref_029","DOI":"10.1063\/1.1761178"},{"doi-asserted-by":"crossref","unstructured":"N. Hasan, S. Anwer and S. Sanghi,\nOn the outflow boundary condition for external incompressible flows: A new approach,\nJ. Comput. Phys. 206 (2005), no. 2, 661\u2013683.","key":"2025032910202776849_j_cmam-2023-0261_ref_030","DOI":"10.1016\/j.jcp.2004.12.025"},{"doi-asserted-by":"crossref","unstructured":"S. M. Hosseini and J. J. Feng,\nPressure boundary conditions for computing incompressible flows with SPH,\nJ. Comput. Phys. 230 (2011), no. 19, 7473\u20137487.","key":"2025032910202776849_j_cmam-2023-0261_ref_031","DOI":"10.1016\/j.jcp.2011.06.013"},{"doi-asserted-by":"crossref","unstructured":"M. Jobelin, C. Lapuerta, J.-C. Latch\u00e9, P. Angot and B. Piar,\nA finite element penalty-projection method for incompressible flows,\nJ. Comput. Phys. 217 (2006), no. 2, 502\u2013518.","key":"2025032910202776849_j_cmam-2023-0261_ref_032","DOI":"10.1016\/j.jcp.2006.01.019"},{"doi-asserted-by":"crossref","unstructured":"M. Jobelin, B. Piar, P. Angot and J. C. Latch\u00e9,\nUne methode de penalite-projection vectorielle pour les ecoulements dilatables,\nEur. J. Comput. Mech. 17 (2008), no. 4, 502\u2013518.","key":"2025032910202776849_j_cmam-2023-0261_ref_033","DOI":"10.3166\/remn.17.153-480"},{"doi-asserted-by":"crossref","unstructured":"H. Johnston and J.-G. Liu,\nAccurate, stable and efficient Navier\u2013Stokes solvers based on explicit treatment of the pressure term,\nJ. Comput. Phys. 199 (2004), no. 1, 221\u2013259.","key":"2025032910202776849_j_cmam-2023-0261_ref_034","DOI":"10.1016\/j.jcp.2004.02.009"},{"doi-asserted-by":"crossref","unstructured":"J. Liu,\nOpen and traction boundary conditions for the incompressible Navier\u2013Stokes equations,\nJ. Comput. Phys. 228 (2009), no. 19, 7250\u20137267.","key":"2025032910202776849_j_cmam-2023-0261_ref_035","DOI":"10.1016\/j.jcp.2009.06.021"},{"doi-asserted-by":"crossref","unstructured":"M. A. Olshanskii, A. Sokolov and S. Turek,\nError analysis of a projection method for the Navier\u2013Stokes equations with Coriolis force,\nJ. Math. Fluid Mech. 12 (2010), no. 4, 485\u2013502.","key":"2025032910202776849_j_cmam-2023-0261_ref_036","DOI":"10.1007\/s00021-009-0299-0"},{"doi-asserted-by":"crossref","unstructured":"S. A. Orszag, M. Israeli and M. O. Deville,\nBoundary conditions for incompressible flows,\nJ. Sci. Comput. 1 (1986), 75\u2013111.","key":"2025032910202776849_j_cmam-2023-0261_ref_037","DOI":"10.1007\/BF01061454"},{"doi-asserted-by":"crossref","unstructured":"A. Poux, S. Glockner, E. Ahusborde and M. Aza\u00efez,\nOpen boundary conditions for the velocity-correction scheme of the Navier\u2013Stokes equations,\nComput. & Fluids 70 (2012), 29\u201343.","key":"2025032910202776849_j_cmam-2023-0261_ref_038","DOI":"10.1016\/j.compfluid.2012.08.028"},{"doi-asserted-by":"crossref","unstructured":"A. Poux, S. Glockner and M. Aza\u00efez,\nImprovements on open and traction boundary conditions for Navier\u2013Stokes time-splitting methods,\nJ. Comput. Phys. 230 (2011), no. 10, 4011\u20134027.","key":"2025032910202776849_j_cmam-2023-0261_ref_039","DOI":"10.1016\/j.jcp.2011.02.024"},{"doi-asserted-by":"crossref","unstructured":"J.-H. Pyo and J. Shen,\nNormal mode analysis of second-order projection methods for incompressible flows,\nDiscrete Contin. Dyn. Syst. Ser. B 5 (2005), no. 3, 817\u2013840.","key":"2025032910202776849_j_cmam-2023-0261_ref_040","DOI":"10.3934\/dcdsb.2005.5.817"},{"doi-asserted-by":"crossref","unstructured":"J. Shen,\nOn error estimates of projection methods for Navier\u2013Stokes equations: First-order schemes,\nSIAM J. Numer. Anal. 29 (1992), no. 1, 57\u201377.","key":"2025032910202776849_j_cmam-2023-0261_ref_041","DOI":"10.1137\/0729004"},{"doi-asserted-by":"crossref","unstructured":"J. Shen,\nOn error estimates of some higher order projection and penalty-projection methods for Navier\u2013Stokes equations,\nNumer. Math. 62 (1992), no. 1, 49\u201373.","key":"2025032910202776849_j_cmam-2023-0261_ref_042","DOI":"10.1007\/BF01396220"},{"doi-asserted-by":"crossref","unstructured":"J. Shen,\nOn error estimates of the penalty method for unsteady Navier\u2013Stokes equations,\nSIAM J. Numer. Anal. 32 (1995), no. 2, 386\u2013403.","key":"2025032910202776849_j_cmam-2023-0261_ref_043","DOI":"10.1137\/0732016"},{"doi-asserted-by":"crossref","unstructured":"J. Shen,\nOn error estimates of the projection methods for the Navier\u2013Stokes equations: Second-order schemes,\nMath. Comp. 65 (1996), no. 215, 1039\u20131065.","key":"2025032910202776849_j_cmam-2023-0261_ref_044","DOI":"10.1090\/S0025-5718-96-00750-8"},{"doi-asserted-by":"crossref","unstructured":"J. Shen and X. Yang,\nError estimates for finite element approximations of consistent splitting schemes for incompressible flows,\nDiscrete Contin. Dyn. Syst. Ser. B 8 (2007), no. 3, 663\u2013676.","key":"2025032910202776849_j_cmam-2023-0261_ref_045","DOI":"10.3934\/dcdsb.2007.8.663"},{"doi-asserted-by":"crossref","unstructured":"R. T\u00e9mam,\nSur l\u2019approximation de la solution des \u00e9quations de Navier\u2013Stokes par la m\u00e9thode des pas fractionnaires. II,\nArch. Ration. Mech. Anal. 33 (1969), 377\u2013385.","key":"2025032910202776849_j_cmam-2023-0261_ref_046","DOI":"10.1007\/BF00247696"},{"doi-asserted-by":"crossref","unstructured":"L. J. P. Timmermans, P. D. Minev and F. N. van de Vosse,\nAn approximate projection scheme for incompressible flow using spectral elements,\nInternat. J. Numer. Methods Fluids 22 (1996), no. 7, 673\u2013688.","key":"2025032910202776849_j_cmam-2023-0261_ref_047","DOI":"10.1002\/(SICI)1097-0363(19960415)22:7<673::AID-FLD373>3.0.CO;2-O"},{"doi-asserted-by":"crossref","unstructured":"J. van Kan,\nA second-order accurate pressure-correction scheme for viscous incompressible flow,\nSIAM J. Sci. Statist. Comput. 7 (1986), no. 3, 870\u2013891.","key":"2025032910202776849_j_cmam-2023-0261_ref_048","DOI":"10.1137\/0907059"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0261\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0261\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,29]],"date-time":"2025-03-29T10:23:52Z","timestamp":1743243832000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0261\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,14]]},"references-count":48,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,11,14]]},"published-print":{"date-parts":[[2025,4,1]]}},"alternative-id":["10.1515\/cmam-2023-0261"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2023-0261","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2024,11,14]]}}}