{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T19:55:46Z","timestamp":1768420546998,"version":"3.49.0"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,8,13]],"date-time":"2024-08-13T00:00:00Z","timestamp":1723507200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,8,13]],"date-time":"2024-08-13T00:00:00Z","timestamp":1723507200000},"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":["12201010"],"award-info":[{"award-number":["12201010"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071400"],"award-info":[{"award-number":["12071400"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009558","name":"University Natural Science Research Project of Anhui Province","doi-asserted-by":"publisher","award":["2022AH040027"],"award-info":[{"award-number":["2022AH040027"]}],"id":[{"id":"10.13039\/501100009558","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2020YFA0713500"],"award-info":[{"award-number":["2020YFA0713500"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2025,7]]},"DOI":"10.1007\/s11075-024-01910-z","type":"journal-article","created":{"date-parts":[[2024,8,13]],"date-time":"2024-08-13T04:02:07Z","timestamp":1723521727000},"page":"1161-1202","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Unconditionally energy stable IEQ-FEMs for the Cahn-Hilliard equation and Allen-Cahn equation"],"prefix":"10.1007","volume":"99","author":[{"given":"Yaoyao","family":"Chen","sequence":"first","affiliation":[]},{"given":"Hailiang","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Nianyu","family":"Yi","sequence":"additional","affiliation":[]},{"given":"Peimeng","family":"Yin","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,8,13]]},"reference":[{"key":"1910_CR1","doi-asserted-by":"publisher","first-page":"1017","DOI":"10.1016\/0001-6160(75)90106-6","volume":"23","author":"SM Allen","year":"1975","unstructured":"Allen, S.M., Cahn, J.W.: Coherent and incoherent equilibria in iron-rich iron-aluminum alloys. Acta Metall. 23, 1017\u20131026 (1975)","journal-title":"Acta Metall."},{"key":"1910_CR2","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1017\/S0956792500000759","volume":"3","author":"JF Blowey","year":"1992","unstructured":"Blowey, J.F., Elliott, C.M.: The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy Part II: Numerical analysis. Euro. J. Appl. Math. 3, 147\u2013179 (1992)","journal-title":"Euro. J. Appl. Math."},{"key":"1910_CR3","doi-asserted-by":"crossref","unstructured":"Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. interfacial free energy. J. Chem. Phys. 28, 258\u2013267 (1958)","DOI":"10.1063\/1.1744102"},{"key":"1910_CR4","doi-asserted-by":"publisher","first-page":"6","DOI":"10.1007\/s00211-020-01112-4","volume":"145","author":"H Chen","year":"2020","unstructured":"Chen, H., Mao, J., Shen, J.: Optimal error estimates for the scalar intermediate function finite-element schemes for gradient flows. Numer. Math. 145, 6 (2020)","journal-title":"Numer. Math."},{"key":"1910_CR5","doi-asserted-by":"crossref","unstructured":"Chen, W., Wang, C., Wang, X., Wise, S.M.: Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential. J. Comput. Phys.: X. 3, 100031 (2019)","DOI":"10.1016\/j.jcpx.2019.100031"},{"key":"1910_CR6","doi-asserted-by":"publisher","first-page":"204","DOI":"10.1016\/j.camwa.2019.02.022","volume":"78","author":"Y Chen","year":"2019","unstructured":"Chen, Y., Huang, Y., Yi, N.: A SCR-based error estimation and adaptive finite element method for the Allen-Cahn equation. Comput. Math. Appl. 78, 204\u2013223 (2019)","journal-title":"Comput. Math. Appl."},{"issue":"2","key":"1910_CR7","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1007\/s10915-023-02418-9","volume":"98","author":"Y Chen","year":"2024","unstructured":"Chen, Y., Huang, Y., Yi, N., Yin, P.: Recovery type a posteriori error estimation of adaptive finite element method for Cahn-Hilliard equation. J. Sci. Comput. 98(2), 35 (2024)","journal-title":"J. Sci. Comput."},{"key":"1910_CR8","doi-asserted-by":"publisher","first-page":"574","DOI":"10.1016\/j.cam.2018.05.039","volume":"362","author":"K Cheng","year":"2019","unstructured":"Cheng, K., Feng, W., Wang, C., Wise, S.: An energy stable fourth order finite difference scheme for the Cahn-Hilliard equation. J. Comput. Appl. Math. 362, 574\u2013595 (2019)","journal-title":"J. Comput. Appl. Math."},{"key":"1910_CR9","unstructured":"Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Universit\u00e9 Pierre et Marie Curie, Paris, France, (1974)"},{"issue":"5","key":"1910_CR10","doi-asserted-by":"publisher","first-page":"1310","DOI":"10.1137\/0728069","volume":"28","author":"Q Du","year":"1991","unstructured":"Du, Q., Nicolaides, R.: Numerical analysis of a continuum model of phase transition. SIAM J. Numer. Anal. 28(5), 1310\u20131322 (1991)","journal-title":"SIAM J. Numer. Anal."},{"key":"1910_CR11","doi-asserted-by":"crossref","unstructured":"Eyre, D.J.: Unconditionally gradient stable time marching the Cahn-Hilliard equation. In Computational and mathematical models of microstructural evolution (San Francisco, CA, 1998), volume 529 of Mater. Res. Soc. Sympos. Proc., pages 39\u201346. MRS, (1998)","DOI":"10.1557\/PROC-529-39"},{"key":"1910_CR12","doi-asserted-by":"publisher","first-page":"1093","DOI":"10.1090\/S0025-5718-07-01985-0","volume":"76","author":"X Feng","year":"2007","unstructured":"Feng, X., Karakashian, O.A.: Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition. Math. Comput. 76, 1093\u20131117 (2007)","journal-title":"Math. Comput."},{"issue":"1","key":"1910_CR13","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/s00211-002-0413-1","volume":"94","author":"X Feng","year":"2003","unstructured":"Feng, X., Prohl, A.: Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94(1), 33\u201365 (2003)","journal-title":"Numer. Math."},{"issue":"1","key":"1910_CR14","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1007\/s00211-004-0546-5","volume":"99","author":"X Feng","year":"2004","unstructured":"Feng, X., Prohl, A.: Error analysis of a mixed finite element method for the Cahn-Hilliard equation. Numer. Math. 99(1), 47\u201384 (2004)","journal-title":"Numer. Math."},{"issue":"2","key":"1910_CR15","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1007\/s10915-004-4610-1","volume":"24","author":"X Feng","year":"2005","unstructured":"Feng, X., Wu, H.: A posteriori error estimates and an adaptive finite element method for the Allen-Cahn equation and the mean curvature flow. J. Sci. Comput. 24(2), 121\u2013146 (2005)","journal-title":"J. Sci. Comput."},{"issue":"6","key":"1910_CR16","first-page":"767","volume":"26","author":"X Feng","year":"2008","unstructured":"Feng, X., Wu, H.: A posteriori error estimates for finite element approximations of the Cahn-Hilliard equation and the Hele-Shaw flow. J. Comput. Math. 26(6), 767\u2013796 (2008)","journal-title":"J. Comput. Math."},{"key":"1910_CR17","doi-asserted-by":"crossref","unstructured":"Sar\u0131ayd\u0131n-Filibelio\u01e7lu, A., Karas\u00f6zen, B., Uzunca, M.: Energy stable interior penalty discontinuous Galerkin finite element method for Cahn-Hilliard equation. Int. J. Nonlinear Sci. Numer. Simul. 18(5), 303\u2013314 (2017)","DOI":"10.1515\/ijnsns-2016-0024"},{"issue":"4","key":"1910_CR18","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1007\/PL00005429","volume":"87","author":"D Furihata","year":"2001","unstructured":"Furihata, D.: A stable and conservative finite difference scheme for the Cahn-Hilliard equation. Numer. Math. 87(4), 675\u2013699 (2001)","journal-title":"Numer. Math."},{"issue":"8","key":"1910_CR19","doi-asserted-by":"publisher","first-page":"1855","DOI":"10.1016\/j.camwa.2017.02.021","volume":"73","author":"Y Li","year":"2017","unstructured":"Li, Y., Choi, Y., Kim, J.: Computationally efficient adaptive time step method for the Cahn-Hilliard equation. Comput. Math. Appl. 73(8), 1855\u20131864 (2017)","journal-title":"Comput. Math. Appl."},{"issue":"6","key":"1910_CR20","doi-asserted-by":"publisher","first-page":"1591","DOI":"10.1016\/j.camwa.2010.06.041","volume":"60","author":"Y Li","year":"2010","unstructured":"Li, Y., Lee, H.G., Jeong, D., Kim, J.: An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation. Comput. Math. Appl. 60(6), 1591\u20131606 (2010)","journal-title":"Comput. Math. Appl."},{"key":"1910_CR21","volume":"419","author":"Y Li","year":"2022","unstructured":"Li, Y., Wang, J.: Unconditional convergence analysis of stabilized FEM-SAV method for Cahn-Hilliard equation. Appl. Math. Comput. 419, 126880 (2022)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"1910_CR22","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1137\/080720255","volume":"47","author":"H Liu","year":"2009","unstructured":"Liu, H., Yan, J.: The Direct Discontinuous Galerkin (DDG) method for diffusion problems. SIAM J. Numer. Anal. 47(1), 675\u2013698 (2009)","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"1910_CR23","doi-asserted-by":"publisher","first-page":"541","DOI":"10.4208\/cicp.010909.011209a","volume":"8","author":"H Liu","year":"2010","unstructured":"Liu, H., Yan, J.: The Direct Discontinuous Galerkin (DDG) method for diffusion with interface corrections. Commun. Comput. Phys. 8(3), 541\u2013564 (2010)","journal-title":"Commun. Comput. Phys."},{"key":"1910_CR24","doi-asserted-by":"publisher","first-page":"467","DOI":"10.1007\/s10915-018-0756-0","volume":"77","author":"H Liu","year":"2018","unstructured":"Liu, H., Yin, P.: A mixed discontinuous Galerkin method without interior penalty for time-dependent fourth order problems. J. Sci. Comput. 77, 467\u2013501 (2018)","journal-title":"J. Sci. Comput."},{"key":"1910_CR25","doi-asserted-by":"publisher","first-page":"789","DOI":"10.1007\/s10915-019-01038-6","volume":"81","author":"H Liu","year":"2019","unstructured":"Liu, H., Yin, P.: Unconditionally energy stable DG schemes for the Swift-Hohenberg equation. J. Sci. Comput. 81, 789\u2013819 (2019)","journal-title":"J. Sci. Comput."},{"key":"1910_CR26","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2020.113375","volume":"390","author":"H Liu","year":"2021","unstructured":"Liu, H., Yin, P.: Unconditionally energy stable DG schemes for the Cahn-Hilliard equation. J. Comput. Appl. Math. 390, 113375 (2021)","journal-title":"J. Comput. Appl. Math."},{"key":"1910_CR27","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2021.114015","volume":"407","author":"H Liu","year":"2021","unstructured":"Liu, H., Yin, P.: High order unconditionally energy stable RKDG schemes for the Swift-Hohenberg equation. J. Comput. Appl. Math. 407, 114015 (2021)","journal-title":"J. Comput. Appl. Math."},{"issue":"2","key":"1910_CR28","doi-asserted-by":"publisher","first-page":"1185","DOI":"10.1002\/num.22929","volume":"39","author":"H Liu","year":"2023","unstructured":"Liu, H., Yin, P.: On the SAV-DG method for a class of fourth order gradient flows. Numer. Meth. Part. D. E. 39(2), 1185\u20131200 (2023)","journal-title":"Numer. Meth. Part. D. E."},{"key":"1910_CR29","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1016\/j.jcp.2017.10.021","volume":"353","author":"J Shen","year":"2018","unstructured":"Shen, J., Xu, J., Yang, X.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407\u2013416 (2018)","journal-title":"J. Comput. Phys."},{"key":"1910_CR30","doi-asserted-by":"publisher","first-page":"474","DOI":"10.1137\/17M1150153","volume":"61","author":"J Shen","year":"2019","unstructured":"Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61, 474\u2013506 (2019)","journal-title":"SIAM Rev."},{"issue":"4","key":"1910_CR31","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-Cahn and Cahn-Hilliard equations. Discrete Cont. Dyn-A 28(4), 1669\u20131691 (2010)","journal-title":"Discrete Cont. Dyn-A"},{"key":"1910_CR32","first-page":"606","volume":"244","author":"J Shin","year":"2014","unstructured":"Shin, J., Park, S., Kim, J.: A hybrid FEM for solving the Allen-Cahn equation. Appl. Math. Comput. 244, 606\u2013612 (2014)","journal-title":"Appl. Math. Comput."},{"key":"1910_CR33","first-page":"1463","volume":"64","author":"Z-Z Sun","year":"1995","unstructured":"Sun, Z.-Z.: A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation. Math. Comput. 64, 1463\u20131471 (1995)","journal-title":"Math. Comput."},{"issue":"2","key":"1910_CR34","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1007\/s10915-022-01822-x","volume":"91","author":"J Wang","year":"2022","unstructured":"Wang, J., Pan, K., Yang, X.: Convergence analysis of the fully discrete hybridizable discontinuous Galerkin method for the Allen-Cahn equation based on the invariant energy quadratization approach. J. Sci. Comput. 91(2), 49 (2022)","journal-title":"J. Sci. Comput."},{"key":"1910_CR35","doi-asserted-by":"crossref","unstructured":"Wu, X., Zwieten, G.J.v., Zee, K.G.v.d.: Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models. Int. J. Numer. Meth. Biomed. Engng. 30, 180\u2013203 (2014)","DOI":"10.1002\/cnm.2597"},{"key":"1910_CR36","doi-asserted-by":"publisher","first-page":"1759","DOI":"10.1137\/050628143","volume":"44","author":"C Xu","year":"2006","unstructured":"Xu, C., Tang, T.: Stability analysis of large time-stepping methods for epitaxial growth models. SIAM. J. Num. Anal. 44, 1759\u20131779 (2006)","journal-title":"SIAM. J. Num. Anal."},{"key":"1910_CR37","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111278","volume":"463","author":"R Yang","year":"2022","unstructured":"Yang, R., Xing, Y.: Energy conserving discontinuous Galerkin method with scalar intermediate function technique for the nonlinear Dirac equation. J. Comput. Phys. 463, 111278 (2022)","journal-title":"J. Comput. Phys."},{"key":"1910_CR38","first-page":"509","volume":"302","author":"X Yang","year":"2016","unstructured":"Yang, X.: Linear, first and second order and unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 302, 509\u2013523 (2016)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"1910_CR39","doi-asserted-by":"publisher","first-page":"703","DOI":"10.4208\/cicp.OA-2017-0259","volume":"25","author":"X Yang","year":"2019","unstructured":"Yang, X., Zhao, J.: On linear and unconditionally energy stable algorithms for variable mobility Cahn-Hilliard type equation with logarithmic Flory-Huggins potential. Commun. Comput. Phys. 25(3), 703\u2013728 (2019)","journal-title":"Commun. Comput. Phys."},{"key":"1910_CR40","volume-title":"Efficient discontinuous Galerkin (DG) methods for time-dependent fourth order problems","author":"P Yin","year":"2019","unstructured":"Yin, P.: Efficient discontinuous Galerkin (DG) methods for time-dependent fourth order problems. Iowa State University, PhD diss. (2019)"},{"issue":"10","key":"1910_CR41","doi-asserted-by":"publisher","first-page":"2449","DOI":"10.1016\/j.cpc.2014.05.017","volume":"185","author":"S Zhai","year":"2014","unstructured":"Zhai, S., Feng, X., He, Y.: Numerical simulation of the three dimensional Allen-Cahn equation by the high-order compact ADI method. Comput. Phys. Commun. 185(10), 2449\u20132455 (2014)","journal-title":"Comput. Phys. Commun."},{"issue":"4","key":"1910_CR42","doi-asserted-by":"publisher","first-page":"3042","DOI":"10.1137\/080738398","volume":"31","author":"J Zhang","year":"2009","unstructured":"Zhang, J., Du, Q.: Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit. SIAM J. Sci. Comput. 31(4), 3042\u20133063 (2009)","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"1910_CR43","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1002\/nme.5372","volume":"110","author":"J Zhao","year":"2017","unstructured":"Zhao, J., Wang, Q., Yang, X.: Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach. Int. J. Numer. Methods Eng. 110(3), 279\u2013300 (2017)","journal-title":"Int. J. Numer. Methods Eng."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01910-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-024-01910-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01910-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T08:43:27Z","timestamp":1750149807000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-024-01910-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,13]]},"references-count":43,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["1910"],"URL":"https:\/\/doi.org\/10.1007\/s11075-024-01910-z","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,13]]},"assertion":[{"value":"6 February 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 July 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 August 2024","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":"Competing interests"}},{"value":"Not Applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval"}}]}}