{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T15:40:07Z","timestamp":1772293207010,"version":"3.50.1"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2015,6,13]],"date-time":"2015-06-13T00:00:00Z","timestamp":1434153600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2015,6,13]],"date-time":"2015-06-13T00:00:00Z","timestamp":1434153600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS1312701"],"award-info":[{"award-number":["DMS1312701"]}],"id":[{"id":"10.13039\/100000001","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":[[2016,3]]},"DOI":"10.1007\/s10915-015-0055-y","type":"journal-article","created":{"date-parts":[[2015,6,12]],"date-time":"2015-06-12T08:25:35Z","timestamp":1434097535000},"page":"1102-1121","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":["A Decoupled Unconditionally Stable Numerical Scheme for the Cahn\u2013Hilliard\u2013Hele-Shaw System"],"prefix":"10.1007","volume":"66","author":[{"given":"Daozhi","family":"Han","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2015,6,13]]},"reference":[{"issue":"2","key":"55_CR1","doi-asserted-by":"publisher","first-page":"492","DOI":"10.1063\/1.1425843","volume":"14","author":"H-G Lee","year":"2002","unstructured":"Lee, H.-G., Lowengrub, J.S., Goodman, J.: Modeling pinchoff and reconnection in a Hele-Shaw cell. I. The models and their calibration. Phys. Fluids 14(2), 492\u2013513 (2002). doi:\n                    10.1063\/1.1425843","journal-title":"Phys. Fluids"},{"issue":"2","key":"55_CR2","doi-asserted-by":"publisher","first-page":"514","DOI":"10.1063\/1.1425844","volume":"14","author":"H-G Lee","year":"2002","unstructured":"Lee, H.-G., Lowengrub, J.S., Goodman, J.: Modeling pinchoff and reconnection in a Hele-Shaw cell. II. Analysis and simulation in the nonlinear regime. Phys. Fluids 14(2), 514\u2013545 (2002). doi:\n                    10.1063\/1.1425844","journal-title":"Phys. Fluids"},{"issue":"1","key":"55_CR3","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1007\/s10915-010-9363-4","volume":"44","author":"SM Wise","year":"2010","unstructured":"Wise, S.M.: Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn\u2013Hilliard\u2013Hele-Shaw system of equations. J. Sci. Comput. 44(1), 38\u201368 (2010). doi:\n                    10.1007\/s10915-010-9363-4","journal-title":"J. Sci. Comput."},{"issue":"3","key":"55_CR4","doi-asserted-by":"publisher","first-page":"1320","DOI":"10.1137\/110827119","volume":"50","author":"X Feng","year":"2012","unstructured":"Feng, X., Wise, S.: Analysis of a Darcy\u2013Cahn\u2013Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation. SIAM J. Numer. Anal. 50(3), 1320\u20131343 (2012). doi:\n                    10.1137\/110827119","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"55_CR5","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1016\/j.anihpc.2012.06.003","volume":"30","author":"X Wang","year":"2013","unstructured":"Wang, X., Zhang, Z.: Well-posedness of the Hele-Shaw\u2013Cahn\u2013Hilliard system. Ann. Inst. H. Poincar\u00e9 Anal. Non Lin\u00e9aire 30(3), 367\u2013384 (2013). doi:\n                    10.1016\/j.anihpc.2012.06.003","journal-title":"Ann. Inst. H. Poincar\u00e9 Anal. Non Lin\u00e9aire"},{"issue":"2","key":"55_CR6","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1007\/s00205-008-0160-2","volume":"194","author":"H Abels","year":"2009","unstructured":"Abels, H.: On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Ration. Mech. Anal. 194(2), 463\u2013506 (2009). doi:\n                    10.1007\/s00205-008-0160-2","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"1978","key":"55_CR7","doi-asserted-by":"publisher","first-page":"2617","DOI":"10.1098\/rspa.1998.0273","volume":"454","author":"J Lowengrub","year":"1998","unstructured":"Lowengrub, J., Truskinovsky, L.: Quasi-incompressible Cahn\u2013Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454(1978), 2617\u20132654 (1998). doi:\n                    10.1098\/rspa.1998.0273","journal-title":"R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci."},{"issue":"3\u20134","key":"55_CR8","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1016\/S0167-2789(03)00030-7","volume":"179","author":"C Liu","year":"2003","unstructured":"Liu, C., Shen, J.: A phase field model for the mixture of two incompressible fluids and its approximation by a fourier-spectral method. Phys. D Nonlinear Phenom. 179(3\u20134), 211\u2013228 (2003). doi:\n                    10.1016\/S0167-2789(03)00030-7","journal-title":"Phys. D Nonlinear Phenom."},{"issue":"2","key":"55_CR9","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1016\/j.jcp.2003.07.035","volume":"193","author":"J Kim","year":"2004","unstructured":"Kim, J., Kang, K., Lowengrub, J.: Conservative multigrid methods for Cahn\u2013Hilliard fluids. J. Comput. Phys. 193(2), 511\u2013543 (2004). doi:\n                    10.1016\/j.jcp.2003.07.035","journal-title":"J. Comput. Phys."},{"issue":"3","key":"55_CR10","doi-asserted-by":"publisher","first-page":"1049","DOI":"10.1137\/050638333","volume":"44","author":"X Feng","year":"2006","unstructured":"Feng, X.: Fully discrete finite element approximations of the Navier-Stokes\u2013Cahn\u2013Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44(3), 1049\u20131072 (2006). doi:\n                    10.1137\/050638333","journal-title":"SIAM J. Numer. Anal."},{"issue":"6","key":"55_CR11","doi-asserted-by":"publisher","first-page":"2241","DOI":"10.1137\/050648110","volume":"29","author":"D Kay","year":"2007","unstructured":"Kay, D., Welford, R.: Efficient numerical solution of Cahn\u2013Hilliard\u2013Navier\u2013Stokes fluids in 2d. SIAM J. Sci. Comput. 29(6), 2241\u20132257 (2007). doi:\n                    10.1137\/050648110","journal-title":"SIAM J. Sci. Comput."},{"issue":"5","key":"55_CR12","doi-asserted-by":"publisher","first-page":"743","DOI":"10.1007\/s11401-010-0599-y","volume":"31","author":"J Shen","year":"2010","unstructured":"Shen, J., Yang, X.: Energy stable schemes for Cahn\u2013Hilliard phase-field model of two-phase incompressible flows. Chin. Ann. Math. Ser. B 31(5), 743\u2013758 (2010). doi:\n                    10.1007\/s11401-010-0599-y","journal-title":"Chin. Ann. Math. Ser. B"},{"issue":"4","key":"55_CR13","doi-asserted-by":"publisher","first-page":"697","DOI":"10.1051\/m2an\/2010072","volume":"45","author":"F Boyer","year":"2011","unstructured":"Boyer, F., Minjeaud, S.: Numerical schemes for a three component Cahn\u2013Hilliard model. ESAIM Math. Model. Numer. Anal. 45(4), 697\u2013738 (2011). doi:\n                    10.1051\/m2an\/2010072","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"55_CR14","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/j.jcp.2015.02.046","volume":"290","author":"D Han","year":"2015","unstructured":"Han, D., Wang, X.: A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn\u2013Hilliard\u2013Navier\u2013Stokes equation. J. Comput. Phys. 290, 139\u2013156 (2015). doi:\n                    10.1016\/j.jcp.2015.02.046","journal-title":"J. Comput. Phys."},{"key":"55_CR15","doi-asserted-by":"publisher","first-page":"2161","DOI":"10.1103\/PhysRevA.45.R2161","volume":"45","author":"A Shinozaki","year":"1992","unstructured":"Shinozaki, A., Oono, Y.: Spinodal decomposition in a Hele-Shaw cell. Phys. Rev. A 45, 2161\u20132164 (1992). doi:\n                    10.1103\/PhysRevA.45.R2161","journal-title":"Phys. Rev. A"},{"key":"55_CR16","doi-asserted-by":"publisher","first-page":"053006","DOI":"10.1103\/PhysRevE.89.053006","volume":"89","author":"C-Y Chen","year":"2014","unstructured":"Chen, C.-Y., Huang, Y.-S., Miranda, J.A.: Radial Hele-Shaw flow with suction: fully nonlinear pattern formation. Phys. Rev. E 89, 053006 (2014). doi:\n                    10.1103\/PhysRevE.89.053006","journal-title":"Phys. Rev. E"},{"issue":"1\u20132","key":"55_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.mcm.2010.07.007","volume":"53","author":"SM Wise","year":"2011","unstructured":"Wise, S.M., Lowengrub, J.S., Cristini, V.: An adaptive multigrid algorithm for simulating solid tumor growth using mixture models. Math. Comput. Model. 53(1\u20132), 1\u201320 (2011). doi:\n                    10.1016\/j.mcm.2010.07.007","journal-title":"Math. Comput. Model."},{"issue":"18","key":"55_CR18","doi-asserted-by":"publisher","first-page":"3048","DOI":"10.1002\/mma.3043","volume":"37","author":"D Han","year":"2014","unstructured":"Han, D., Sun, D., Wang, X.: Two-phase flows in karstic geometry. Math. Methods Appl. Sci. 37(18), 3048\u20133063 (2014). doi:\n                    10.1002\/mma.3043","journal-title":"Math. Methods Appl. Sci."},{"key":"55_CR19","doi-asserted-by":"crossref","unstructured":"Eyre, D.J.: Unconditionally gradient stable time marching the Cahn\u2013Hilliard equation. In: Computational and Mathematical Models of Microstructural Evolution (San Francisco, CA, 1998). Mater. Res. Soc. Sympos. Proc., vol. 529, pp. 39\u201346. MRS, Warrendale, PA (1998)","DOI":"10.1557\/PROC-529-39"},{"key":"55_CR20","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1016\/j.jcp.2014.01.037","volume":"264","author":"R Guo","year":"2014","unstructured":"Guo, R., Xia, Y., Xu, Y.: An efficient fully-discrete local discontinuous Galerkin method for the Cahn\u2013Hilliard\u2013Hele-Shaw system. J. Comput. Phys. 264, 23\u201340 (2014). doi:\n                    10.1016\/j.jcp.2014.01.037","journal-title":"J. Comput. Phys."},{"issue":"2","key":"55_CR21","doi-asserted-by":"publisher","first-page":"584","DOI":"10.1002\/num.21721","volume":"29","author":"S Minjeaud","year":"2013","unstructured":"Minjeaud, S.: An unconditionally stable uncoupled scheme for a triphasic Cahn\u2013Hilliard\/Navier\u2013Stokes model. Numer. Methods Partial Differential Equ. 29(2), 584\u2013618 (2013). doi:\n                    10.1002\/num.21721","journal-title":"Numer. Methods Partial Differential Equ."},{"issue":"1","key":"55_CR22","doi-asserted-by":"publisher","first-page":"122","DOI":"10.1137\/130921593","volume":"36","author":"J Shen","year":"2014","unstructured":"Shen, J., Yang, X.: Decoupled energy stable schemes for phase-field models of two-phase complex fluids. SIAM J. Sci. Comput. 36(1), 122\u2013145 (2014). doi:\n                    10.1137\/130921593","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"55_CR23","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1137\/140971154","volume":"53","author":"J Shen","year":"2015","unstructured":"Shen, J., Yang, X.: Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows. SIAM J. Numer. Anal. 53(1), 279\u2013296 (2015). doi:\n                    10.1137\/140971154","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"55_CR24","doi-asserted-by":"crossref","first-page":"929","DOI":"10.4208\/cicp.171211.130412a","volume":"13","author":"C Collins","year":"2013","unstructured":"Collins, C., Shen, J., Wise, S.M.: An efficient, energy stable scheme for the Cahn\u2013Hilliard\u2013Brinkman system. Commun. Comput. Phys. 13(4), 929\u2013957 (2013)","journal-title":"Commun. Comput. Phys."},{"issue":"44\u201347","key":"55_CR25","doi-asserted-by":"publisher","first-page":"6011","DOI":"10.1016\/j.cma.2005.10.010","volume":"195","author":"JL Guermond","year":"2006","unstructured":"Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg. 195(44\u201347), 6011\u20136045 (2006). doi:\n                    10.1016\/j.cma.2005.10.010","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"issue":"8","key":"55_CR26","doi-asserted-by":"publisher","first-page":"2834","DOI":"10.1016\/j.jcp.2008.12.036","volume":"228","author":"J-L Guermond","year":"2009","unstructured":"Guermond, J.-L., Salgado, A.: A splitting method for incompressible flows with variable density based on a pressure Poisson equation. J. Comput. Phys. 228(8), 2834\u20132846 (2009). doi:\n                    10.1016\/j.jcp.2008.12.036","journal-title":"J. Comput. Phys."},{"key":"55_CR27","doi-asserted-by":"publisher","unstructured":"Rannacher, R.: On chorin\u2019s projection method for the incompressible Navier\u2013Stokes equations. In: Heywood, J., Masuda, K., Rautmann, R., Solonnikov, V. (eds.) The Navier\u2013Stokes Equations II Theory and Numerical Methods. Lecture Notes in Mathematics, vol. 1530, pp. 167\u2013183. Springer, (1992). doi:\n                    10.1007\/BFb0090341","DOI":"10.1007\/BFb0090341"},{"issue":"1","key":"55_CR28","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/0168-9274(95)00132-8","volume":"21","author":"J Shen","year":"1996","unstructured":"Shen, J.: On a new pseudocompressibility method for the incompressible Navier\u2013Stokes equations. Appl. Numer. Math. 21(1), 71\u201390 (1996). doi:\n                    10.1016\/0168-9274(95)00132-8","journal-title":"Appl. Numer. Math."},{"issue":"10","key":"55_CR29","doi-asserted-by":"publisher","first-page":"3887","DOI":"10.1016\/j.jde.2014.07.013","volume":"257","author":"D Han","year":"2014","unstructured":"Han, D., Wang, X., Wu, H.: Existence and uniqueness of global weak solutions to a Cahn\u2013Hilliard\u2013Stokes\u2013Darcy system for two phase incompressible flows in karstic geometry. J. Differential Equations 257(10), 3887\u20133933 (2014). doi:\n                    10.1016\/j.jde.2014.07.013","journal-title":"J. Differential Equations"},{"key":"55_CR30","doi-asserted-by":"publisher","unstructured":"Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15, p. 350. Springer, (1991). doi:\n                    10.1007\/978-1-4612-3172-1","DOI":"10.1007\/978-1-4612-3172-1"},{"key":"55_CR31","doi-asserted-by":"crossref","unstructured":"Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, vol. 40, p. 530. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002)","DOI":"10.1137\/1.9780898719208"},{"issue":"1","key":"55_CR32","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1137\/130950628","volume":"53","author":"AE Diegel","year":"2015","unstructured":"Diegel, A.E., Feng, X.H., Wise, S.M.: Analysis of a mixed finite element method for a Cahn\u2013Hilliard\u2013Darcy\u2013Stokes system. SIAM J. Numer. Anal. 53(1), 127\u2013152 (2015). doi:\n                    10.1137\/130950628","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"55_CR33","doi-asserted-by":"publisher","first-page":"15","DOI":"10.4171\/IFB\/178","volume":"10","author":"D Kay","year":"2008","unstructured":"Kay, D., Styles, V., Welford, R.: Finite element approximation of a Cahn\u2013Hilliard\u2013Navier\u2013Stokes system. Interfaces Free Bound. 10(1), 15\u201343 (2008). doi:\n                    10.4171\/IFB\/178","journal-title":"Interfaces Free Bound."},{"issue":"3","key":"55_CR34","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1007\/s00220-002-0693-4","volume":"229","author":"RV Kohn","year":"2002","unstructured":"Kohn, R.V., Otto, F.: Upper bounds on coarsening rates. Commun. Math. Phys. 229(3), 375\u2013395 (2002). doi:\n                    10.1007\/s00220-002-0693-4","journal-title":"Commun. Math. Phys."},{"issue":"4","key":"55_CR35","doi-asserted-by":"publisher","first-page":"1669","DOI":"10.3934\/dcds.2010.28.1669","volume":"28","author":"J Shen","year":"2010","unstructured":"Shen, J., Yang, X.: Numerical approximations of Allen\u2013Cahn and Cahn\u2013Hilliard equations. Discrete Contin. Dyn. Syst. 28(4), 1669\u20131691 (2010). doi:\n                    10.3934\/dcds.2010.28.1669","journal-title":"Discrete Contin. Dyn. Syst."},{"issue":"3\u20134","key":"55_CR36","first-page":"251","volume":"20","author":"F Hecht","year":"2012","unstructured":"Hecht, F.: New development in freefem++. J. Numer. Math. 20(3\u20134), 251\u2013265 (2012)","journal-title":"J. Numer. Math."},{"key":"55_CR37","unstructured":"Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II\/B, pp. 469\u20131202. Springer, (1990). Nonlinear monotone operators. Translated from the German by the author and Leo F, Boron"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-015-0055-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-015-0055-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-015-0055-y","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-015-0055-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,17]],"date-time":"2020-05-17T09:14:25Z","timestamp":1589706865000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-015-0055-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,6,13]]},"references-count":37,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2016,3]]}},"alternative-id":["55"],"URL":"https:\/\/doi.org\/10.1007\/s10915-015-0055-y","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,6,13]]},"assertion":[{"value":"28 October 2014","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 May 2015","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 June 2015","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 June 2015","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}