{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T00:46:50Z","timestamp":1771030010704,"version":"3.50.1"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T00:00:00Z","timestamp":1760572800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T00:00:00Z","timestamp":1760572800000},"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":["12301525"],"award-info":[{"award-number":["12301525"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100018542","name":"Natural Science Foundation of Sichuan Province","doi-asserted-by":"publisher","award":["2023NSFSC1324"],"award-info":[{"award-number":["2023NSFSC1324"]}],"id":[{"id":"10.13039\/501100018542","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2025,12]]},"DOI":"10.1007\/s10444-025-10261-9","type":"journal-article","created":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T08:00:41Z","timestamp":1760601641000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Pathwise uniform convergence of a full discretization for a three-dimensional stochastic Allen-Cahn equation with multiplicative noise"],"prefix":"10.1007","volume":"51","author":[{"given":"Binjie","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qin","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,10,16]]},"reference":[{"key":"10261_CR1","doi-asserted-by":"publisher","first-page":"1084","DOI":"10.1016\/0001-6160(79)90196-2","volume":"27","author":"SM Allen","year":"1979","unstructured":"Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1084\u20131095 (1979)","journal-title":"Acta Metall."},{"key":"10261_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-8518-8","volume-title":"Well-Posedness of Parabolic Difference Equations","author":"A Ashyralyev","year":"1994","unstructured":"Ashyralyev, A., Sobolerskii, P.E.: Well-Posedness of Parabolic Difference Equations. Birkh\u00e4user, Basel (1994)"},{"key":"10261_CR3","doi-asserted-by":"publisher","first-page":"1597","DOI":"10.1090\/S0025-5718-02-01488-6","volume":"72","author":"NY Bakaev","year":"2002","unstructured":"Bakaev, N.Y., Thom\u00e9e, V., Wahlbin, L.B.: Maximum-norm estimates for resolvents of elliptic finite element operators. Math. Comput. 72, 1597\u20131610 (2002)","journal-title":"Math. Comput."},{"key":"10261_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-62226-2","volume-title":"Stochastic Calculus","author":"P Baldi","year":"2017","unstructured":"Baldi, P.: Stochastic Calculus. Springer, Cham (2017)"},{"key":"10261_CR5","first-page":"211","volume":"11","author":"S Becker","year":"2023","unstructured":"Becker, S., Gess, B., Jentzen, A., Kloeden, P.E.: Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations. Stoch PDE: Anal. Comp. 11, 211\u2013268 (2023)","journal-title":"Stoch PDE: Anal. Comp."},{"key":"10261_CR6","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/j.spa.2018.02.008","volume":"129","author":"S Becker","year":"2019","unstructured":"Becker, S., Jentzen, A.: Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equations. Stoch. Process. Appl. 129, 28\u201369 (2019)","journal-title":"Stoch. Process. Appl."},{"key":"10261_CR7","doi-asserted-by":"publisher","first-page":"2096","DOI":"10.1093\/imanum\/dry052","volume":"39","author":"C-E Br\u00e9hier","year":"2019","unstructured":"Br\u00e9hier, C.-E., Cui, J., Hong, J.: Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen-Cahn equation. IMA J. Numer. Anal. 39, 2096\u20132134 (2019)","journal-title":"IMA J. Numer. Anal."},{"key":"10261_CR8","first-page":"2181","volume":"12","author":"D Breit","year":"2024","unstructured":"Breit, D., Prohl, A.: Weak error analysis for the stochastic Allen-Cahn equation. Stoch PDE: Anal. Comp. 12, 2181\u20132245 (2024)","journal-title":"Stoch PDE: Anal. Comp."},{"key":"10261_CR9","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0","volume-title":"The Mathematical Theory of Finite Element Methods","author":"SC Brenner","year":"2008","unstructured":"Brenner, S.C., Scott, R.: The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (2008)"},{"key":"10261_CR10","doi-asserted-by":"publisher","first-page":"428","DOI":"10.1137\/090761835","volume":"48","author":"S Cox","year":"2010","unstructured":"Cox, S., Neerven, J.: Convergence rates of the splitting scheme for parabolic linear stochastic Cauchy problems. SIAM J. Numer. Anal. 48, 428\u2013451 (2010)","journal-title":"SIAM J. Numer. Anal."},{"key":"10261_CR11","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/s00211-013-0538-4","volume":"125","author":"S Cox","year":"2013","unstructured":"Cox, S., Neerven, J.: Pathwise H\u00f6lder convergence of the implicit-linear Euler scheme for semi-linear SPDEs with multiplicative noise. Numer. Math. 125, 259\u2013345 (2013)","journal-title":"Numer. Math."},{"key":"10261_CR12","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1007\/BF01400302","volume":"23","author":"J Douglas Jr","year":"1975","unstructured":"Douglas, J., Jr., Dupont, T., Wahlbin, L.: The stability in $$l^q$$ of the $$ l^2$$-projection into finite element function spaces. Numer. Math. 23, 193\u2013197 (1975)","journal-title":"Numer. Math."},{"key":"10261_CR13","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-0387-8","volume-title":"Methods of Nonlinear Analysis","author":"P Dr\u00e1bek","year":"2013","unstructured":"Dr\u00e1bek, P., Milota, J.: Methods of Nonlinear Analysis. Birkh\u00e4user, Basel (2013)"},{"key":"10261_CR14","doi-asserted-by":"publisher","first-page":"1097","DOI":"10.1002\/cpa.3160450903","volume":"45","author":"LC Evans","year":"1992","unstructured":"Evans, L.C., Soner, H.M., Souganidis, P.E.: Phase transitions and generalized motion by mean curvature. Commm. Pure Appl. Math. 45, 1097\u20131123 (1992)","journal-title":"Commm. Pure Appl. Math."},{"key":"10261_CR15","doi-asserted-by":"publisher","first-page":"194","DOI":"10.1137\/15M1022124","volume":"55","author":"X Feng","year":"2017","unstructured":"Feng, X., Li, Y., Zhang, Y.: Finite element methods for the stochastic Allen-Cahn equation with gradient-type multiplicative noises. SIAM J. Numer. Anal. 55, 194\u2013216 (2017)","journal-title":"SIAM J. Numer. Anal."},{"key":"10261_CR16","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1007\/BF01213390","volume":"102","author":"T Funaki","year":"1995","unstructured":"Funaki, T.: The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Relat. Fields 102, 221\u2013288 (1995)","journal-title":"Probab. Theory Relat. Fields"},{"key":"10261_CR17","doi-asserted-by":"crossref","unstructured":"Gy\u00f6ngy, I., Millet, A.: Rate of convergence of implicit approximations for stochastic evolution equations. In: Stochastic Differential Equations: Theory and Applications. Interdisciplinary Mathematical Sciences, vol. 2, pp. 281\u2013310. World Scientific Publishing, Hackensack, NJ (2007)","DOI":"10.1142\/9789812770639_0011"},{"key":"10261_CR18","first-page":"225","volume":"4","author":"I Gy\u00f6ngy","year":"2016","unstructured":"Gy\u00f6ngy, I., Sabanis, S., \u0160i\u0161ka, D.: Convergence of tamed Euler scheme for a class of stochastic evolution equations. Stoch. Partial Differ. Equ. Anal. Comput. 4, 225\u2013245 (2016)","journal-title":"Stoch. Partial Differ. Equ. Anal. Comput."},{"key":"10261_CR19","doi-asserted-by":"publisher","first-page":"1489","DOI":"10.4310\/CMS.2018.v16.n6.a2","volume":"16","author":"M Hutzenthaler","year":"2018","unstructured":"Hutzenthaler, M., Jentzen, A., Salimova, D.: Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto-Sivashinsky equations. Commun. Math. Sci. 16, 1489\u20131529 (2018)","journal-title":"Commun. Math. Sci."},{"key":"10261_CR20","doi-asserted-by":"crossref","unstructured":"T.\u00a0Hyt\u00f6nen, J.\u00a0van Neerven, M.\u00a0Veraar, and L.\u00a0Weis. Analysis in Banach Spaces, Volume II: Probabilistic methods and operator theory. Springer, Cham, 2017","DOI":"10.1007\/978-3-319-69808-3"},{"key":"10261_CR21","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1007\/s00032-009-0100-0","volume":"77","author":"A Jentzen","year":"2009","unstructured":"Jentzen, A., Kloeden, P.E.: The numerical approximation of stochastic partial differential equations. Milan J. Math. 77, 205\u2013244 (2009)","journal-title":"Milan J. Math."},{"key":"10261_CR22","first-page":"565","volume":"6","author":"A Jentzen","year":"2018","unstructured":"Jentzen, A., Pu\u0161nik, P.: Exponential moments for numerical approximations of stochastic partial differential equations. Stoch. Partial Differ. Equ. Anal. Comput. 6, 565\u2013617 (2018)","journal-title":"Stoch. Partial Differ. Equ. Anal. Comput."},{"key":"10261_CR23","doi-asserted-by":"publisher","first-page":"1005","DOI":"10.1093\/imanum\/drz009","volume":"40","author":"A Jentzen","year":"2020","unstructured":"Jentzen, A., Pu\u0161nik, P.: Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. IMA J. Numer. Anal. 40, 1005\u20131050 (2020)","journal-title":"IMA J. Numer. Anal."},{"key":"10261_CR24","first-page":"241","volume":"234","author":"T Kemmochi","year":"2016","unstructured":"Kemmochi, T.: Discrete maximal regularity for abstract Cauchy problems. Studia Math. 234, 241\u2013263 (2016)","journal-title":"Studia Math."},{"key":"10261_CR25","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/s00028-024-00975-6","volume":"24","author":"K Klioba","year":"2024","unstructured":"Klioba, K., Veraar, M.: Temporal approximation of stochastic evolution equations with irregular nonlinearities. J. Evol. Equ. 24, 43 (2024)","journal-title":"J. Evol. Equ."},{"key":"10261_CR26","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1002\/cpa.20144","volume":"60","author":"R Kohn","year":"2007","unstructured":"Kohn, R., Otto, F., Reznikoff, M.G., Vanden-Eijnden, E.: Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation. Comm. Pure Appl. Math. 60, 393\u2013438 (2007)","journal-title":"Comm. Pure Appl. Math."},{"key":"10261_CR27","doi-asserted-by":"publisher","first-page":"708","DOI":"10.1137\/17M1121627","volume":"56","author":"M Kov\u00e1cs","year":"2018","unstructured":"Kov\u00e1cs, M., Furihata, D., Larsson, S., Lindgren, F.: Strong convergence of a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation. SIAM J. Numer. 56, 708\u2013731 (2018)","journal-title":"SIAM J. Numer."},{"key":"10261_CR28","doi-asserted-by":"publisher","first-page":"966","DOI":"10.1002\/mana.201600283","volume":"291","author":"M Kov\u00e1cs","year":"2018","unstructured":"Kov\u00e1cs, M., Larsson, S., Lindgren, F.: On the discretisation in time of the stochastic Allen-Cahn equation. Math. Nachr. 291, 966\u2013995 (2018)","journal-title":"Math. Nachr."},{"key":"10261_CR29","doi-asserted-by":"publisher","first-page":"2407","DOI":"10.1137\/110828150","volume":"49","author":"M Kov\u00e1cs","year":"2011","unstructured":"Kov\u00e1cs, M., Larsson, S., Mesforush, A.: Finite element approximation of the Cahn-Hilliard-cook equations. SIAM J. Numer. 49, 2407\u20132429 (2011)","journal-title":"SIAM J. Numer."},{"key":"10261_CR30","doi-asserted-by":"crossref","unstructured":"Li, B.: Maximum-norm stability and maximal $$L^p$$ regularity of FEMs for parabolic equations with Lipschitz continuous coefficients. Numer. Math. 131, 489\u2013516 (2015)","DOI":"10.1007\/s00211-015-0698-5"},{"key":"10261_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/s40072-025-00375-y","author":"B Li","year":"2025","unstructured":"Li, B., Xie, X.: Stability and convergence of the Euler scheme for stochastic evolution equations in Banach spaces. Stoch PDE: Anal Comp (2025). https:\/\/doi.org\/10.1007\/s40072-025-00375-y","journal-title":"Stoch PDE: Anal Comp"},{"key":"10261_CR32","doi-asserted-by":"crossref","unstructured":"Li, B., Zhou, Q.: Convergence of a spatial semidiscretization for a three-dimensional stochastic Allen-Cahn equation with multiplicative noise (2025). arXiv:2401.09834","DOI":"10.2139\/ssrn.4782199"},{"key":"10261_CR33","doi-asserted-by":"crossref","unstructured":"Li, B., Zhou, Z.: Discrete stochastic maximal $$L^p$$-regularity and convergence of a spatial semidiscretization for a linear stochastic heat equation (2025). arXiv:2311.04615","DOI":"10.2139\/ssrn.4782199"},{"key":"10261_CR34","doi-asserted-by":"publisher","first-page":"572","DOI":"10.1016\/j.jde.2013.04.021","volume":"255","author":"W Liu","year":"2013","unstructured":"Liu, W.: Well-posedness of stochastic partial differential equations with Lyapunov condition. J. Differ. Equ. 255, 572\u2013592 (2013)","journal-title":"J. Differ. Equ."},{"key":"10261_CR35","doi-asserted-by":"publisher","first-page":"2902","DOI":"10.1016\/j.jfa.2010.05.012","volume":"259","author":"W Liu","year":"2010","unstructured":"Liu, W., R\u00f6ckner, M.: SPDE in Hilbert space with locally monotone coefficients. J. Funct. Anal. 259, 2902\u20132922 (2010)","journal-title":"J. Funct. Anal."},{"key":"10261_CR36","first-page":"559","volume":"9","author":"Z Liu","year":"2021","unstructured":"Liu, Z., Qiao, Z.: Strong approximation of monotone stochastic partial differential equations driven by multiplicative noise. Stoch PDE: Anal. Comp. 9, 559\u2013602 (2021)","journal-title":"Stoch PDE: Anal. Comp."},{"issue":"2","key":"10261_CR37","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1515\/cmam-2017-0023","volume":"18","author":"A Majee","year":"2018","unstructured":"Majee, A., Prohl, A.: Optimal strong rates of convergence for a space-time discretization of the stochastic Allen-Cahn equation with multiplicative noise. Comput. Methods Appl. Math. 18(2), 297\u2013311 (2018)","journal-title":"Comput. Methods Appl. Math."},{"key":"10261_CR38","first-page":"516","volume":"10","author":"J van Neerven","year":"2022","unstructured":"van Neerven, J., Verrar, M.: Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretization schemes. Stoch PDE: Anal. Comp. 10, 516\u2013581 (2022)","journal-title":"Stoch PDE: Anal. Comp."},{"key":"10261_CR39","doi-asserted-by":"crossref","unstructured":"van Neerven, J., Verrar, M., Weis, L.: Stochastic integration in UMD Banach spaces. Ann. Probab. 35, 1438\u20131478 (2007)","DOI":"10.1214\/009117906000001006"},{"key":"10261_CR40","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1214\/10-AOP626","volume":"40","author":"J van Neerven","year":"2012","unstructured":"van Neerven, J., Veraar, M., Weis, L.: Stochastic maximal $$ l^p $$-regularity. Ann. Probab. 40, 788\u2013812 (2012)","journal-title":"Ann. Probab."},{"key":"10261_CR41","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-05714-9","volume-title":"Stochastic Differential Equations, Backward SDEs, Partial Differential Equations","author":"E Pardoux","year":"2014","unstructured":"Pardoux, E., Rascanu, A.: Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. Springer, Cham (2014)"},{"key":"10261_CR42","doi-asserted-by":"crossref","unstructured":"Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, 2 edition (2014)","DOI":"10.1017\/CBO9781107295513"},{"key":"10261_CR43","doi-asserted-by":"publisher","first-page":"1171","DOI":"10.1007\/s10915-019-00973-8","volume":"80","author":"R Qi","year":"2019","unstructured":"Qi, R., Wang, X.: Optimal error estimates of Galerkin finite element methods for stochastic Allen-Cahn equation with additive noise. J. Sci. Comput. 80, 1171\u20131194 (2019)","journal-title":"J. Sci. Comput."},{"key":"10261_CR44","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1007\/s10444-023-10072-w","volume":"49","author":"X Qi","year":"2023","unstructured":"Qi, X., Zhang, Y., Xu, C.: An efficient approximation to the stochastic Allen-Cahn equation with random diffusion coefficient field and multiplicative noise. Adv. Comput. Math. 49, 73 (2023)","journal-title":"Adv. Comput. Math."},{"key":"10261_CR45","first-page":"175","volume":"1","author":"M R\u00f6ger","year":"2013","unstructured":"R\u00f6ger, M., Weber, H.: Tightness for a stochastic Allen-Cahn equation. Stoch PDE: Anal. Comp. 1, 175\u2013203 (2013)","journal-title":"Stoch PDE: Anal. Comp."},{"key":"10261_CR46","doi-asserted-by":"publisher","first-page":"743","DOI":"10.1090\/S0025-5718-2014-02873-1","volume":"84","author":"M Sauer","year":"2015","unstructured":"Sauer, M., Stannat, W.: Lattice approximation for stochastic reaction diffusion equations with one-sided Lipschitz condition. Math. Comp. 84, 743\u2013766 (2015)","journal-title":"Math. Comp."},{"key":"10261_CR47","volume-title":"An Introduction to Sobolev Spaces and Interpolation Spaces","author":"L Tartar","year":"2007","unstructured":"Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces. Springer, Berlin (2007)"},{"key":"10261_CR48","doi-asserted-by":"publisher","first-page":"6271","DOI":"10.1016\/j.spa.2020.05.011","volume":"130","author":"X Wang","year":"2020","unstructured":"Wang, X.: An efficient explicit full-discrete scheme for strong approximation of stochastic Allen-Cahn equation. Stoch. Process. Appl. 130, 6271\u20136299 (2020)","journal-title":"Stoch. Process. Appl."},{"key":"10261_CR49","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-04631-5","volume-title":"Abstract Parabolic Evolution Equations and Their Applications","author":"A Yagi","year":"2010","unstructured":"Yagi, A.: Abstract Parabolic Evolution Equations and Their Applications. Springer, Berlin (2010)"},{"key":"10261_CR50","first-page":"541","volume":"21","author":"S Zhang","year":"1995","unstructured":"Zhang, S.: Successive subdivisions of tetrahedra and multigrid methods on tetrahedral meshes. Houston J. Math. 21, 541\u2013556 (1995)","journal-title":"Houston J. Math."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-025-10261-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-025-10261-9","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-025-10261-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T13:02:17Z","timestamp":1767963737000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-025-10261-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,16]]},"references-count":50,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["10261"],"URL":"https:\/\/doi.org\/10.1007\/s10444-025-10261-9","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,16]]},"assertion":[{"value":"26 November 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 September 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 October 2025","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":"Conflict of interest"}}],"article-number":"46"}}