{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,11]],"date-time":"2026-08-11T02:29:17Z","timestamp":1786415357701,"version":"3.56.0"},"reference-count":55,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T00:00:00Z","timestamp":1784592000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T00:00:00Z","timestamp":1784592000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12471369, 12241101."],"award-info":[{"award-number":["12471369, 12241101."]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Science Foundation grant DMS","award":["2310340"],"award-info":[{"award-number":["2310340"]}]},{"name":"National Science Foundation grant DMS","award":["123B2015, 12601002"],"award-info":[{"award-number":["123B2015, 12601002"]}]},{"name":"National Science Foundation grant DMS","award":["12271237"],"award-info":[{"award-number":["12271237"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2026,9]]},"DOI":"10.1007\/s10915-026-03400-x","type":"journal-article","created":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T06:50:20Z","timestamp":1784616620000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Convergence Analysis of a Bounds-Preserving Numerical Scheme for the Quasi-Incompressible Cahn-Hilliard-Darcy System"],"prefix":"10.1007","volume":"108","author":[{"given":"Wenbin","family":"Chen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daozhi","family":"Han","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-1171-6251","authenticated-orcid":false,"given":"Qianqian","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoming","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,21]]},"reference":[{"key":"3400_CR1","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1016\/j.apnum.2024.11.005","volume":"209","author":"D Acosta-Soba","year":"2025","unstructured":"Acosta-Soba, D., Guill\u00e9n-Gonz\u00e1lez, F., Rodr\u00edguez-Galv\u00e1n, J.R., Wang, J.: Property-preserving numerical approximation of a Cahn-Hilliard-Navier-Stokes model with variable density and degenerate mobility. Appl. Numer. Math. 209, 68\u201383 (2025)","journal-title":"Appl. Numer. Math."},{"issue":"1","key":"3400_CR2","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1137\/22M1488934","volume":"66","author":"GR Barrenechea","year":"2024","unstructured":"Barrenechea, G.R., John, V., Knobloch, P.: Finite element methods respecting the discrete maximum principle for convection-diffusion equations. SIAM Rev. 66(1), 3\u201388 (2024). https:\/\/doi.org\/10.1137\/22M1488934","journal-title":"SIAM Rev."},{"key":"3400_CR3","doi-asserted-by":"publisher","unstructured":"Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd ed. Springer, New York (2008). https:\/\/doi.org\/10.1007\/978-0-387-75934-0","DOI":"10.1007\/978-0-387-75934-0"},{"issue":"313","key":"3400_CR4","doi-asserted-by":"publisher","first-page":"2057","DOI":"10.1090\/mcom\/3280","volume":"87","author":"Y Cai","year":"2018","unstructured":"Cai, Y., Shen, J.: Error estimates for a fully discretized scheme to a Cahn-Hilliard phase-field model for two-phase incompressible flows. Math. Comput. 87(313), 2057\u20132090 (2018)","journal-title":"Math. Comput."},{"issue":"1","key":"3400_CR5","doi-asserted-by":"publisher","first-page":"376","DOI":"10.1137\/23M1562068","volume":"62","author":"JA Carrillo","year":"2024","unstructured":"Carrillo, J.A., Wang, L., Wei, C.: Structure preserving primal dual methods for gradient flows with nonlinear mobility transport distances. SIAM J. Numer. Anal. 62(1), 376\u2013399 (2024). https:\/\/doi.org\/10.1137\/23M1562068","journal-title":"SIAM J. Numer. Anal."},{"key":"3400_CR6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109782","volume":"423","author":"L Chen","year":"2020","unstructured":"Chen, L., Zhao, J.: A novel second-order linear scheme for the Cahn-Hilliard-Navier-Stokes equations. J. Comput. Phys. 423, 109782 (2020)","journal-title":"J. Comput. Phys."},{"key":"3400_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-020-01341-7","volume":"85","author":"W Chen","year":"2020","unstructured":"Chen, W., Han, D., Wang, X., Zhang, Y.: Uniquely solvable and energy stable decoupled numerical schemes for the Cahn-Hilliard-Navier-Stokes-Darcy-Boussinesq system. J. Sci. Comput. 85, 1\u201328 (2020)","journal-title":"J. Sci. Comput."},{"issue":"6","key":"3400_CR8","doi-asserted-by":"publisher","first-page":"1823","DOI":"10.1002\/num.22841","volume":"38","author":"W Chen","year":"2022","unstructured":"Chen, W., Han, D., Wang, X., Zhang, Y.: Conservative unconditionally stable decoupled numerical schemes for the Cahn-Hilliard-Navier-Stokes-Darcy-Boussinesq system. Numer. Methods Partial Differential Equations 38(6), 1823\u20131842 (2022)","journal-title":"Numer. Methods Partial Differential Equations"},{"key":"3400_CR9","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2024.115981","volume":"450","author":"W Chen","year":"2024","unstructured":"Chen, W., Jing, J., Liu, Q., Wang, C., Wang, X.: Convergence analysis of a second order numerical scheme for the Flory-Huggins-Cahn-Hilliard-Navier-Stokes system. J. Comput. Appl. Math. 450, 115981 (2024)","journal-title":"J. Comput. Appl. Math."},{"issue":"3","key":"3400_CR10","doi-asserted-by":"publisher","first-page":"633","DOI":"10.4208\/cicp.OA-2023-0038","volume":"35","author":"W Chen","year":"2024","unstructured":"Chen, W., Jing, J., Liu, Q., Wang, C., Wang, X.: A second order numerical scheme of the Cahn-Hilliard-Navier-Stokes system with Flory-Huggins potential. Commun. Comput. Phys. 35(3), 633\u2013661 (2024)","journal-title":"Commun. Comput. Phys."},{"issue":"2","key":"3400_CR11","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1007\/s10915-022-01872-1","volume":"92","author":"W Chen","year":"2022","unstructured":"Chen, W., Jing, J., Wang, C., Wang, X.: A positivity preserving, energy stable finite difference scheme for the Flory-Huggins-Cahn-Hilliard-Navier-Stokes system. J. Sci. Comput. 92(2), 31 (2022)","journal-title":"J. Sci. Comput."},{"key":"3400_CR12","doi-asserted-by":"publisher","unstructured":"Chen, W., Jing, J.,Wang, C.,Wang, X., Wise, S.M.: A modified Crank-Nicolson numerical scheme for the Flory-Huggins Cahn-Hilliard model. Commun. Comput. Phys. 31(1), 60\u201393 (2022). https:\/\/doi.org\/10.4208\/cicp.OA-2021-0074","DOI":"10.4208\/cicp.OA-2021-0074"},{"issue":"3","key":"3400_CR13","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1007\/s10915-023-02296-1","volume":"96","author":"W Chen","year":"2023","unstructured":"Chen, W., Jing, J., Wu, H.: A uniquely solvable, positivity-preserving and unconditionally energy stable numerical scheme for the functionalized Cahn-Hilliard equation with logarithmic potential. J. Sci. Comput. 96(3), 75 (2023)","journal-title":"J. Sci. Comput."},{"issue":"2\u20133","key":"3400_CR14","first-page":"275","volume":"19","author":"W Chen","year":"2022","unstructured":"Chen, W., Liu, Q., Shen, J.: Error estimates and blow-up analysis of a finite-element approximation for the parabolic-elliptic Keller-Segel system. Int. J. Numer. Anal. Model. 19(2\u20133), 275\u2013298 (2022)","journal-title":"Int. J. Numer. Anal. Model."},{"issue":"301","key":"3400_CR15","doi-asserted-by":"publisher","first-page":"2231","DOI":"10.1090\/mcom3052","volume":"85","author":"W Chen","year":"2016","unstructured":"Chen, W., Liu, Y., Wang, C., Wise, S.M.: Convergence analysis of a fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equation. Math. Comput. 85(301), 2231\u20132257 (2016)","journal-title":"Math. Comput."},{"issue":"3","key":"3400_CR16","doi-asserted-by":"publisher","first-page":"2621","DOI":"10.1093\/imanum\/drab046","volume":"42","author":"W Chen","year":"2022","unstructured":"Chen, W., Wang, S., Zhang, Y., Han, D., Wang, C., Wang, X.: Error estimate of a decoupled numerical scheme for the Cahn-Hilliard-Stokes-Darcy system. IMA J. Numer. Anal. 42(3), 2621\u20132655 (2022)","journal-title":"IMA J. Numer. Anal."},{"key":"3400_CR17","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1016\/j.jcp.2015.12.006","volume":"308","author":"Y Chen","year":"2016","unstructured":"Chen, Y., Shen, J.: Efficient, adaptive energy stable schemes for the incompressible Cahn-Hilliard-Navier-Stokes phase-field models. J. Comput. Phys. 308, 40\u201356 (2016)","journal-title":"J. Comput. Phys."},{"key":"3400_CR18","doi-asserted-by":"publisher","unstructured":"Cheng, Q., Shen, J.: A new Lagrange multiplier approach for constructing structure preserving schemes, II. Bound preserving. SIAM Journal on Numerical Analysis 60(3), 970\u2013998 (2022). https:\/\/doi.org\/10.1137\/21M144877X","DOI":"10.1137\/21M144877X"},{"key":"3400_CR19","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1007\/BF01385847","volume":"63","author":"MIM Copetti","year":"1992","unstructured":"Copetti, M.I.M., Elliott, C.M.: Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy. Numer. Math. 63, 39\u201365 (1992)","journal-title":"Numer. Math."},{"key":"3400_CR20","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1007\/s00211-017-0887-5","volume":"137","author":"AE Diegel","year":"2017","unstructured":"Diegel, A.E., Wang, C., Wang, X., Wise, S.M.: Convergence analysis and error estimates for a second order accurate finite element method for the Cahn-Hilliard-Navier-Stokes system. Numer. Math. 137, 495\u2013534 (2017)","journal-title":"Numer. Math."},{"issue":"2","key":"3400_CR21","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1137\/19M1243750","volume":"63","author":"Q Du","year":"2021","unstructured":"Du, Q., Ju, L., Li, X., Qiao, Z.: Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes. SIAM Rev. 63(2), 317\u2013359 (2021). https:\/\/doi.org\/10.1137\/19M1243750","journal-title":"SIAM Rev."},{"issue":"7","key":"3400_CR22","doi-asserted-by":"publisher","first-page":"3764","DOI":"10.1002\/mma.8015","volume":"45","author":"C Duan","year":"2022","unstructured":"Duan, C., Chen, W., Liu, C., Wang, C., Zhou, S.: Convergence analysis of structure-preserving numerical methods for nonlinear Fokker-Planck equations with nonlocal interactions. Math. Methods Appl. Sci. 45(7), 3764\u20133781 (2022)","journal-title":"Math. Methods Appl. Sci."},{"key":"3400_CR23","doi-asserted-by":"publisher","first-page":"A26","DOI":"10.1017\/jfm.2025.10186","volume":"1013","author":"MF ten Eikelder","year":"2025","unstructured":"ten Eikelder, M.F.: A unified framework for N-phase Navier-Stokes Cahn-Hilliard Allen-Cahn mixture models with non-matching densities. J. Fluid Mech. 1013, A26 (2025). https:\/\/doi.org\/10.1017\/jfm.2025.10186","journal-title":"J. Fluid Mech."},{"key":"3400_CR24","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.114186","volume":"387","author":"G Fu","year":"2021","unstructured":"Fu, G., Han, D.: A linear second-order in time unconditionally energy stable finite element scheme for a cahn-hilliard phase-field model for two-phase incompressible flow of variable densities. Comput. Methods Appl. Mech. Eng. 387, 114186 (2021)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"3400_CR25","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2023.112375","volume":"491","author":"G Fu","year":"2023","unstructured":"Fu, G., Osher, S., Li, W.: High order spatial discretization for variational time implicit schemes: Wasserstein gradient flows and reaction-diffusion systems. J. Comput. Phys. 491, 112375 (2023). https:\/\/doi.org\/10.1016\/j.jcp.2023.112375","journal-title":"J. Comput. Phys."},{"issue":"4","key":"3400_CR26","doi-asserted-by":"publisher","first-page":"23087","DOI":"10.1002\/num.23087","volume":"40","author":"Y Gao","year":"2024","unstructured":"Gao, Y., Han, D.: Fully decoupled unconditionally stable Crank-Nicolson leapfrog numerical methods for the Cahn-Hilliard-Darcy system. Numer. Methods Partial Differ. Equ 40(4), 23087 (2024)","journal-title":"Numer. Methods Partial Differ. Equ"},{"key":"3400_CR27","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2024.113340","volume":"518","author":"Y Gao","year":"2024","unstructured":"Gao, Y., Han, D., Wang, X.: A second-order, mass-conservative, unconditionally stable and bound-preserving finite element method for the quasi-incompressible Cahn-Hilliard-Darcy system. J. Comput. Phys. 518, 113340 (2024)","journal-title":"J. Comput. Phys."},{"key":"3400_CR28","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/j.apnum.2015.09.002","volume":"99","author":"H Garcke","year":"2016","unstructured":"Garcke, H., Hinze, M., Kahle, C.: A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow. Appl. Numer. Math. 99, 151\u2013171 (2016)","journal-title":"Appl. Numer. Math."},{"key":"3400_CR29","doi-asserted-by":"crossref","unstructured":"Giesselmann, J., Pryer, T.: Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow model. ESAIM: Mathematical Modelling and Numerical Analysis 49(1), 275\u2013301 (2015)","DOI":"10.1051\/m2an\/2014033"},{"key":"3400_CR30","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1016\/j.cpc.2017.05.002","volume":"219","author":"Y Gong","year":"2017","unstructured":"Gong, Y., Zhao, J., Wang, Q.: An energy stable algorithm for a quasi-incompressible hydrodynamic phase-field model of viscous fluid mixtures with variable densities and viscosities. Comput. Phys. Commun. 219, 20\u201334 (2017)","journal-title":"Comput. Phys. Commun."},{"issue":"1","key":"3400_CR31","doi-asserted-by":"publisher","first-page":"B138","DOI":"10.1137\/17M1111759","volume":"40","author":"Y Gong","year":"2018","unstructured":"Gong, Y., Zhao, J., Yang, X., Wang, Q.: Fully discrete second-order linear schemes for hydrodynamic phase field models of binary viscous fluid flows with variable densities. SIAM J. Sci. Comput. 40(1), B138\u2013B167 (2018)","journal-title":"SIAM J. Sci. Comput."},{"issue":"5","key":"3400_CR32","doi-asserted-by":"publisher","first-page":"1473","DOI":"10.4208\/cicp.scpde14.39s","volume":"19","author":"G Gr\u00fcn","year":"2016","unstructured":"Gr\u00fcn, G., Guill\u00e9n-Gonz\u00e1lez, F., Metzger, S.: On fully decoupled, convergent schemes for diffuse interface models for two-phase flow with general mass densities. Commun. Comput. Phys. 19(5), 1473\u20131502 (2016)","journal-title":"Commun. Comput. Phys."},{"key":"3400_CR33","doi-asserted-by":"publisher","first-page":"708","DOI":"10.1016\/j.jcp.2013.10.028","volume":"257","author":"G Gr\u00fcn","year":"2014","unstructured":"Gr\u00fcn, G., Klingbeil, F.: Two-phase flow with mass density contrast: stable schemes for a thermodynamic consistent and frame-indifferent diffuse-interface model. J. Comput. Phys. 257, 708\u2013725 (2014)","journal-title":"J. Comput. Phys."},{"issue":"6","key":"3400_CR34","doi-asserted-by":"publisher","first-page":"643","DOI":"10.4208\/jcm.1405-m4410","volume":"32","author":"F Guill\u00e9n-Gonz\u00e1lez","year":"2014","unstructured":"Guill\u00e9n-Gonz\u00e1lez, F., Tierra, G.: Splitting schemes for a Navier-Stokes-Cahn-Hilliard model for two fluids with different densities. J. Comput. Math. 32(6), 643\u2013664 (2014)","journal-title":"J. Comput. Math."},{"key":"3400_CR35","doi-asserted-by":"publisher","unstructured":"Guill\u00e9n-Gonz\u00e1lez, F., Tierra, G.: Energy-stable and boundedness preserving numerical schemes for the Cahn-Hilliard equation with degenerate mobility. Appl. Numer. Math 196, 62\u201382 (2024). https:\/\/doi.org\/10.1016\/j.apnum.2023.10.006","DOI":"10.1016\/j.apnum.2023.10.006"},{"issue":"349","key":"3400_CR36","doi-asserted-by":"publisher","first-page":"2185","DOI":"10.1090\/mcom\/3916","volume":"93","author":"Y Guo","year":"2024","unstructured":"Guo, Y., Wang, C., Wise, S.M., Zhang, Z.: Convergence analysis of a positivity-preserving numerical scheme for the Cahn-Hilliard-Stokes system with Flory-Huggins energy potential. Math. Comput. 93(349), 2185\u20132214 (2024)","journal-title":"Math. Comput."},{"key":"3400_CR37","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110727","volume":"448","author":"Z Guo","year":"2022","unstructured":"Guo, Z., Cheng, Q., Lin, P., Liu, C., Lowengrub, J.: Second order approximation for a quasi-incompressible Navier-Stokes-Cahn-Hilliard system of two-phase flows with variable density. J. Comput. Phys. 448, 110727 (2022)","journal-title":"J. Comput. Phys."},{"key":"3400_CR38","doi-asserted-by":"publisher","first-page":"144","DOI":"10.1016\/j.cma.2017.08.011","volume":"326","author":"Z Guo","year":"2017","unstructured":"Guo, Z., Lin, P., Lowengrub, J., Wise, S.M.: Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier-Stokes-Cahn-Hilliard system: Primitive variable and projection-type schemes. Comput. Methods Appl. Mech. Eng. 326, 144\u2013174 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"3400_CR39","doi-asserted-by":"publisher","first-page":"486","DOI":"10.1016\/j.jcp.2014.07.038","volume":"276","author":"Z Guo","year":"2014","unstructured":"Guo, Z., Lin, P., Lowengrub, J.S.: A numerical method for the quasi-incompressible Cahn-Hilliard-Navier-Stokes equations for variable density flows with a discrete energy law. J. Comput. Phys. 276, 486\u2013507 (2014)","journal-title":"J. Comput. Phys."},{"key":"3400_CR40","doi-asserted-by":"publisher","first-page":"A38","DOI":"10.1017\/jfm.2020.790","volume":"907","author":"Z Guo","year":"2021","unstructured":"Guo, Z., Yu, F., Lin, P., Wise, S., Lowengrub, J.: A diffuse domain method for two-phase flows with large density ratio in complex geometries. J. Fluid Mech. 907, A38 (2021)","journal-title":"J. Fluid Mech."},{"issue":"3","key":"3400_CR41","doi-asserted-by":"publisher","first-page":"1102","DOI":"10.1007\/s10915-015-0055-y","volume":"66","author":"D Han","year":"2016","unstructured":"Han, D.: A decoupled unconditionally stable numerical scheme for the Cahn-Hilliard-Hele-Shaw system. J. Sci. Comput. 66(3), 1102\u20131121 (2016). https:\/\/doi.org\/10.1007\/s10915-015-0055-y","journal-title":"J. Sci. Comput."},{"issue":"18","key":"3400_CR42","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)","journal-title":"Math. Methods Appl. Sci."},{"key":"3400_CR43","doi-asserted-by":"publisher","first-page":"1210","DOI":"10.1007\/s10915-018-0748-0","volume":"77","author":"D Han","year":"2018","unstructured":"Han, D., Wang, X.: A second order in time, decoupled, unconditionally stable numerical scheme for the Cahn-Hilliard-Darcy system. J. Sci. Comput. 77, 1210\u20131233 (2018)","journal-title":"J. Sci. Comput."},{"key":"3400_CR44","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111177","volume":"460","author":"F Huang","year":"2022","unstructured":"Huang, F., Shen, J., Wu, K.: Bound\/positivity preserving and unconditionally stable schemes for a class of fourth order nonlinear equations. J. Comput. Phys. 460, 111177 (2022). https:\/\/doi.org\/10.1016\/j.jcp.2022.111177","journal-title":"J. Comput. Phys."},{"key":"3400_CR45","doi-asserted-by":"publisher","unstructured":"Lee, H.G., Lowengrub, J.S., Goodman, J.: Modeling pinchoff and reconnection in a Hele-Shaw cell. I. The models and their calibration. Physics of Fluids 14(2), 492\u2013513 (2002). https:\/\/doi.org\/10.1063\/1.1425843","DOI":"10.1063\/1.1425843"},{"issue":"2","key":"3400_CR46","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1007\/s10915-023-02163-z","volume":"95","author":"Q Liu","year":"2023","unstructured":"Liu, Q., Jing, J., Yuan, M., Chen, W.: A positivity-preserving, energy stable BDF2 scheme with variable steps for the Cahn-Hilliard equation with logarithmic potential. J. Sci. Comput. 95(2), 37 (2023)","journal-title":"J. Sci. Comput."},{"key":"3400_CR47","doi-asserted-by":"publisher","first-page":"679","DOI":"10.1007\/s00211-016-0813-2","volume":"135","author":"Y Liu","year":"2017","unstructured":"Liu, Y., Chen, W., Wang, C., Wise, S.M.: Error analysis of a mixed finite element method for a Cahn-Hilliard-Hele-Shaw system. Numer. Math. 135, 679\u2013709 (2017)","journal-title":"Numer. Math."},{"key":"3400_CR48","doi-asserted-by":"crossref","unstructured":"Lowengrub, J., Truskinovsky, L.: Quasi\u2013incompressible Cahn\u2013Hilliard fluids and topological transitions. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454(1978), 2617\u20132654 (1998)","DOI":"10.1098\/rspa.1998.0273"},{"key":"3400_CR49","doi-asserted-by":"crossref","unstructured":"Miranville, A.: The Cahn-Hilliard equation: recent advances and applications, SIAM (2019)","DOI":"10.1137\/1.9781611975925"},{"issue":"1","key":"3400_CR50","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)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"3400_CR51","doi-asserted-by":"publisher","first-page":"1045","DOI":"10.4208\/cicp.300711.160212a","volume":"13","author":"J Shen","year":"2013","unstructured":"Shen, J., Yang, X., Wang, Q.: Mass and volume conservation in phase field models for binary fluids. Commun. Comput. Phys 13(4), 1045\u20131065 (2013)","journal-title":"Commun. Comput. Phys"},{"key":"3400_CR52","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2019.109179","volume":"405","author":"L Shen","year":"2020","unstructured":"Shen, L., Huang, H., Lin, P., Song, Z., Xu, S.: An energy stable C0 finite element scheme for a quasi-incompressible phase-field model of moving contact line with variable density. J. Comput. Phys. 405, 109179 (2020)","journal-title":"J. Comput. Phys."},{"key":"3400_CR53","unstructured":"Thom\u00e9e, V.: Galerkin finite element methods for parabolic problems, vol.\u00a025. Springer Science & Business Media (2007)"},{"key":"3400_CR54","doi-asserted-by":"publisher","unstructured":"Wang, X., Zhang, Z.: Well-posedness of the Hele-Shaw-Cahn-Hilliard system. Annales de l\u2019Institut Henri Poincar\u00e9. Analyse Non Lin\u00e9aire 30(3), 367\u2013384 (2013). https:\/\/doi.org\/10.1016\/j.anihpc.2012.06.003","DOI":"10.1016\/j.anihpc.2012.06.003"},{"issue":"1","key":"3400_CR55","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-Hilliard-Hele-Shaw system of equations. J. Sci. Comput. 44(1), 38\u201368 (2010). https:\/\/doi.org\/10.1007\/s10915-010-9363-4","journal-title":"J. Sci. Comput."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03400-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-026-03400-x","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03400-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,11]],"date-time":"2026-08-11T01:50:26Z","timestamp":1786413026000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-026-03400-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,21]]},"references-count":55,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,9]]}},"alternative-id":["3400"],"URL":"https:\/\/doi.org\/10.1007\/s10915-026-03400-x","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,21]]},"assertion":[{"value":"31 March 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 June 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 June 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 July 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no conflict of interest.","order":1,"name":"Ethics","label":"Conflicts of Interest","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"81"}}