{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,12,24]],"date-time":"2022-12-24T23:15:19Z","timestamp":1671923719523},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2021,4,5]],"date-time":"2021-04-05T00:00:00Z","timestamp":1617580800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,4,5]],"date-time":"2021-04-05T00:00:00Z","timestamp":1617580800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2021,6]]},"DOI":"10.1007\/s40314-021-01494-7","type":"journal-article","created":{"date-parts":[[2021,4,5]],"date-time":"2021-04-05T21:02:36Z","timestamp":1617656556000},"update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Some algorithms for the mean curvature flow under topological changes"],"prefix":"10.1007","volume":"40","author":[{"given":"Arthur","family":"Bousquet","sequence":"first","affiliation":[]},{"given":"Yukun","family":"Li","sequence":"additional","affiliation":[]},{"given":"Guanqian","family":"Wang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,4,5]]},"reference":[{"key":"1494_CR1","unstructured":"Bousquet A, Li Y, Wang G (2019) Some algorithms for the mean curvature flow under topological changes. arXiv:1908.09690"},{"key":"1494_CR2","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1016\/j.jcp.2006.07.026","volume":"222","author":"J Barrett","year":"2007","unstructured":"Barrett J, Garcke H, N\u00fcrnberg R (2007) A parametric finite element method for fourth order geometric evolution equations. J Comput Phys 222:441\u2013467","journal-title":"J Comput Phys"},{"key":"1494_CR3","doi-asserted-by":"publisher","first-page":"761","DOI":"10.1090\/S0025-5718-2010-02444-5","volume":"80","author":"S Bartels","year":"2011","unstructured":"Bartels S, M\u00fcller R (2011) Quasi-optimal and robust a posteriori error estimates in $$L^{\\infty }(L^2)$$ for the approximation of Allen\u2013Cahn equations past singularities. Math Comput 80:761\u2013780","journal-title":"Math Comput"},{"key":"1494_CR4","doi-asserted-by":"publisher","first-page":"110","DOI":"10.1137\/090751530","volume":"49","author":"S Bartels","year":"2011","unstructured":"Bartels S, M\u00fcller R, Ortner C (2011) Robust a priori and a posteriori error analysis for the approximation of Allen\u2013Cahn and Ginzburg\u2013Landau equations past topological changes. SIAM J Numer Anal 49:110\u2013134","journal-title":"SIAM J Numer Anal"},{"key":"1494_CR5","unstructured":"Bartels S (2010) A lower bound for the spectrum of the linearized Allen\u2013Cahn operator near a singularity. Preprint"},{"key":"1494_CR6","doi-asserted-by":"crossref","first-page":"749","DOI":"10.4310\/jdg\/1214446564","volume":"33","author":"Y Chen","year":"1991","unstructured":"Chen Y, Giga Y, Goto S et al (1991) Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J Differ Geom 33:749\u2013786","journal-title":"J Differ Geom"},{"key":"1494_CR7","doi-asserted-by":"crossref","unstructured":"Church M, Guo Z, Jimack P, Madzvamuse A, Promislow K, Wetton B, Wise S,Yang F (2019) High accuracy benchmark problems for Allen\u2013Cahn and Cahn\u2013Hilliard dynamics. Commun Comput Phys 26","DOI":"10.4208\/cicp.OA-2019-0006"},{"key":"1494_CR8","doi-asserted-by":"crossref","unstructured":"Cesaroni A, Dipierro S, Novaga M, Valdinoci E (2018) Fattening and nonfattening phenomena for planar nonlocal curvature flows. Math Ann 375:687\u2013736","DOI":"10.1007\/s00208-018-1793-6"},{"key":"1494_CR9","doi-asserted-by":"crossref","unstructured":"Deckelnick K, Dziuk G (2003) Numerical approximation of mean curvature flow of graphs and level sets. In: Mathematical aspects of evolving interfaces, pp 53\u201387","DOI":"10.1007\/978-3-540-39189-0_2"},{"key":"1494_CR10","doi-asserted-by":"publisher","first-page":"1913","DOI":"10.1137\/040606417","volume":"65","author":"Q Du","year":"2005","unstructured":"Du Q, Liu C, Wang X (2005) Retrieving topological information for phase field models. SIAM J Appl Math 65:1913\u20131932","journal-title":"SIAM J Appl Math"},{"key":"1494_CR11","first-page":"543","volume":"37","author":"C Elliott","year":"2017","unstructured":"Elliott C, Fritz H (2017) On approximations of the curve shortening flow and of the mean curvature flow based on the DeTurck trick. IMA J Numer Anal 37:543\u2013603","journal-title":"IMA J Numer Anal"},{"key":"1494_CR12","doi-asserted-by":"publisher","first-page":"1097","DOI":"10.1002\/cpa.3160450903","volume":"45","author":"L Evans","year":"1992","unstructured":"Evans L, Soner H, Souganidis P (1992) Phase transitions and generalized motion by mean curvature. Commun Pure Appl Math 45:1097\u20131123","journal-title":"Commun Pure Appl Math"},{"key":"1494_CR13","doi-asserted-by":"crossref","first-page":"635","DOI":"10.4310\/jdg\/1214446559","volume":"33","author":"L Evans","year":"1991","unstructured":"Evans L, Spruck J (1991) Motion of level sets by mean curvature I. J Differ Geom 33:635\u2013681","journal-title":"J Differ Geom"},{"key":"1494_CR14","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 (2003) Numerical analysis of the Allen\u2013Cahn equation and approximation for mean curvature flows. Numer Math. 94:33\u201365","journal-title":"Numer Math."},{"key":"1494_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 (2005) A posteriori error estimates and an adaptive finite element algorithm for the Allen\u2013Cahn equation and the mean curvature flow. J Sci Comput 24:121\u2013146","journal-title":"J Sci Comput"},{"key":"1494_CR16","first-page":"54","volume":"2","author":"X Feng","year":"2014","unstructured":"Feng X, Li Y, Prohl A (2014) Finite element approximations of the stochastic mean curvature flow of planar curves of graphs. Stoch Partial Differ Equ Anal Comput 2:54\u201383","journal-title":"Stoch Partial Differ Equ Anal Comput"},{"key":"1494_CR17","doi-asserted-by":"publisher","first-page":"1622","DOI":"10.1093\/imanum\/dru058","volume":"35","author":"X Feng","year":"2014","unstructured":"Feng X, Li Y (2014) Analysis of symmetric interior penalty discontinuous Galerkin methods for the Allen\u2013Cahn equation and the mean curvature flow. IMA J Numer Anal 35:1622\u20131651","journal-title":"IMA J Numer Anal"},{"key":"1494_CR18","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1137\/15M1009962","volume":"54","author":"X Feng","year":"2016","unstructured":"Feng X, Li Y, Xing Y (2016) Analysis of mixed interior penalty discontinuous Galerkin methods for the Cahn\u2013Hilliard equation and the Hele\u2013Shaw flow. SIAM J Numer Anal 54:825\u2013847","journal-title":"SIAM J Numer Anal"},{"key":"1494_CR19","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 (2017) Finite element methods for the stochastic Allen\u2013Cahn equation with gradient-type multiplicative noise. SIAM J Numer Anal 55:194\u2013216","journal-title":"SIAM J Numer Anal"},{"key":"1494_CR20","unstructured":"Feng X, Li Y, Zhang Y (2018) Strong convergence of a fully discrete finite element method for a class of semilinear stochastic partial differential equations with multiplicative noise. Comput Math. arXiv:1811.05028"},{"key":"1494_CR21","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1016\/0021-9991(81)90145-5","volume":"39","author":"C Hirt","year":"1981","unstructured":"Hirt C, Nichols B (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39:201\u2013225","journal-title":"J Comput Phys"},{"key":"1494_CR22","doi-asserted-by":"crossref","first-page":"417","DOI":"10.4310\/jdg\/1214454300","volume":"38","author":"T Ilmanen","year":"1993","unstructured":"Ilmanen T et al (1993) Convergence of the Allen\u2013Cahn equation to Brakke\u2019s motion by mean curvature. J Differ Geom 38:417\u2013461","journal-title":"J Differ Geom"},{"key":"1494_CR23","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1051\/m2an:2004006","volume":"38","author":"D Kessler","year":"2004","unstructured":"Kessler D, Nochetto R, Schmidt A (2004) A posteriori error control for the Allen\u2013Cahn problem: circumventing Gronwall\u2019s inequality. ESAIM Math Model Numer Anal 38:129\u2013142","journal-title":"ESAIM Math Model Numer Anal"},{"key":"1494_CR24","doi-asserted-by":"crossref","unstructured":"Kov\u00e1cs B, Li B, Lubich C (2018) A convergent evolving finite element algorithm for mean curvature flow of closed surfaces. arXiv:1805.06667","DOI":"10.1007\/s00211-019-01074-2"},{"key":"1494_CR25","unstructured":"Li Y (2015) Numerical methods for deterministic and stochastic phase field models of phase transition and related geometric flows. Ph.D. thesis, The University of Tennessee"},{"key":"1494_CR26","doi-asserted-by":"publisher","first-page":"1862","DOI":"10.1007\/s10915-018-0834-3","volume":"78","author":"Y Li","year":"2019","unstructured":"Li Y (2019) Error analysis of a fully discrete Morley finite element approximation for the Cahn\u2013Hilliard equation. J Sci Comput 78:1862\u20131892","journal-title":"J Sci Comput"},{"key":"1494_CR27","doi-asserted-by":"publisher","first-page":"1019","DOI":"10.1137\/0731054","volume":"31","author":"R Leveque","year":"1994","unstructured":"Leveque R, Li Z (1994) The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J Numer Anal 31:1019\u20131044","journal-title":"SIAM J Numer Anal"},{"key":"1494_CR28","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 (2018) Optimal strong rates of convergence for a space-time discretization of the stochastic Allen\u2013Cahn equation with multiplicative noise. Comput Methods Appl Math 18:297\u2013311","journal-title":"Comput Methods Appl Math"},{"issue":"151","key":"1494_CR29","doi-asserted-by":"publisher","first-page":"773","DOI":"10.1090\/S0025-5718-1980-0572855-7","volume":"35","author":"J Nocedal","year":"1980","unstructured":"Nocedal J (1980) Updating quasi-Newton matrices with limited storage. Math Comput 35(151):773\u2013782","journal-title":"Math Comput"},{"key":"1494_CR30","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1016\/0021-9991(88)90002-2","volume":"79","author":"S Osher","year":"1998","unstructured":"Osher S, Sethian J (1998) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton\u2013Jacobi formulations. J Comput Phys 79:12\u201349","journal-title":"J Comput Phys"},{"key":"1494_CR31","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1017\/S0962492902000077","volume":"11","author":"C Peskin","year":"2002","unstructured":"Peskin C (2002) The immersed boundary method. Acta Numer 11:479\u2013517","journal-title":"Acta Numer"},{"key":"1494_CR32","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1080\/14786449208621456","volume":"33","author":"L Rayleigh","year":"1892","unstructured":"Rayleigh L (1892) On the theory of surface forces. II. Compressible fluids. Lond Edinb Dublin Philos Mag J Sci 33:209\u2013220","journal-title":"Lond Edinb Dublin Philos Mag J Sci"},{"key":"1494_CR33","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-71041-7","volume-title":"Spectral methods: algorithms, analysis and applications","author":"J Shen","year":"2011","unstructured":"Shen J, Tang T, Wang L (2011) Spectral methods: algorithms, analysis and applications, vol 41. Springer Science & Business Media, Berlin"},{"key":"1494_CR34","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1016\/0021-9991(92)90307-K","volume":"100","author":"S Unverdi","year":"1992","unstructured":"Unverdi S, Tryggvason G (1992) A front-tracking method for viscous, incompressible, multi-fluid flows. J Comput Phys 100:25\u201337","journal-title":"J Comput Phys"},{"key":"1494_CR35","doi-asserted-by":"publisher","first-page":"2215","DOI":"10.1137\/S0036142994262068","volume":"33","author":"N Walkington","year":"1996","unstructured":"Walkington N (1996) Algorithms for computing motion by mean curvature. SIAM J Numer Anal 33:2215\u20132238","journal-title":"SIAM J Numer Anal"},{"key":"1494_CR36","doi-asserted-by":"crossref","unstructured":"Wu S, Li Y (2019) Analysis of the Morley element for the Cahn\u2013Hilliard equation and the Hele\u2013Shaw flow. ESAIM Math Model Numer Anal. arXiv:1808.08581","DOI":"10.1051\/m2an\/2019085"},{"key":"1494_CR37","doi-asserted-by":"publisher","first-page":"826","DOI":"10.1016\/j.cma.2018.09.017","volume":"345","author":"J Xu","year":"2019","unstructured":"Xu J, Li Y, Wu S, Bousquet A (2019) On the stability and accuracy of partially and fully implicit schemes for phase field modeling. Comput Methods Appl Mech Eng 345:826\u2013853","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1494_CR38","first-page":"3042","volume":"31","author":"J Zhang","year":"2009","unstructured":"Zhang J, Du Q (2009) Numerical studies of discrete approximations to the Allen\u2013Cahn equation in the sharp interface limit. SIAM J Numer Anal 31:3042\u20133063","journal-title":"SIAM J Numer Anal"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01494-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-021-01494-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01494-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,23]],"date-time":"2022-12-23T12:13:15Z","timestamp":1671797595000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-021-01494-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,5]]},"references-count":38,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,6]]}},"alternative-id":["1494"],"URL":"https:\/\/doi.org\/10.1007\/s40314-021-01494-7","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,5]]},"assertion":[{"value":"22 October 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 March 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 March 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 April 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"104"}}