{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,25]],"date-time":"2026-01-25T00:03:33Z","timestamp":1769299413605,"version":"3.49.0"},"reference-count":61,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2021,9,20]],"date-time":"2021-09-20T00:00:00Z","timestamp":1632096000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,9,20]],"date-time":"2021-09-20T00:00:00Z","timestamp":1632096000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004004","name":"Universit\u00e0 degli Studi di Trento","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004004","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2021,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We present variational approximations of boundary value problems for curvature flow (curve shortening flow) and elastic flow (curve straightening flow) in two-dimensional Riemannian manifolds that are conformally flat. For the evolving open curves we propose natural boundary conditions that respect the appropriate gradient flow structure. Based on suitable weak formulations we introduce finite element approximations using piecewise linear elements. For some of the schemes a stability result can be shown. The derived schemes can be employed in very different contexts. For example, we apply the schemes to the Angenent metric in order to numerically compute rotationally symmetric self-shrinkers for the mean curvature flow. Furthermore, we utilise the schemes to compute geodesics that are relevant for optimal interface profiles in multi-component phase field models.<\/jats:p>","DOI":"10.1007\/s00211-021-01231-6","type":"journal-article","created":{"date-parts":[[2021,9,20]],"date-time":"2021-09-20T09:14:54Z","timestamp":1632129294000},"page":"375-415","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds"],"prefix":"10.1007","volume":"149","author":[{"given":"Harald","family":"Garcke","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert","family":"N\u00fcrnberg","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,9,20]]},"reference":[{"key":"1231_CR1","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1515\/crelle-2014-0121","volume":"729","author":"B Andrews","year":"2017","unstructured":"Andrews, B., Chen, X.: Curvature flow in hyperbolic spaces. J. Reine Angew. Math. 729, 29\u201349 (2017)","journal-title":"J. Reine Angew. Math."},{"key":"1231_CR2","doi-asserted-by":"crossref","unstructured":"Angenent, S.B.: Shrinking doughnuts. In: Nonlinear Diffusion Equations and their Equilibrium States, 3 (Gregynog, 1989), Progr. Nonlinear Differential Equations Appl., vol.\u00a07, pp. 21\u201338. Birkh\u00e4user Boston, Boston, MA (1992)","DOI":"10.1007\/978-1-4612-0393-3_2"},{"key":"1231_CR3","doi-asserted-by":"publisher","first-page":"1641","DOI":"10.1093\/imanum\/draa020","volume":"41","author":"JW Barrett","year":"2021","unstructured":"Barrett, J.W., Deckelnick, K., N\u00fcrnberg, R.: A finite element error analysis for axisymmetric mean curvature flow. IMA J. Numer. Anal. 41, 1641\u20131667 (2021)","journal-title":"IMA J. Numer. Anal."},{"key":"1231_CR4","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1016\/j.jcp.2006.07.026","volume":"222","author":"JW Barrett","year":"2007","unstructured":"Barrett, J.W., Garcke, H., N\u00fcrnberg, R.: A parametric finite element method for fourth order geometric evolution equations. J. Comput. Phys. 222, 441\u2013462 (2007)","journal-title":"J. Comput. Phys."},{"key":"1231_CR5","doi-asserted-by":"publisher","first-page":"1006","DOI":"10.1137\/060653974","volume":"29","author":"JW Barrett","year":"2007","unstructured":"Barrett, J.W., Garcke, H., N\u00fcrnberg, R.: On the variational approximation of combined second and fourth order geometric evolution equations. SIAM J. Sci. Comput. 29, 1006\u20131041 (2007)","journal-title":"SIAM J. Sci. Comput."},{"key":"1231_CR6","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1093\/imanum\/drp005","volume":"30","author":"JW Barrett","year":"2010","unstructured":"Barrett, J.W., Garcke, H., N\u00fcrnberg, R.: Numerical approximation of gradient flows for closed curves in $${{\\mathbb{R}}}^d$$. IMA J. Numer. Anal. 30, 4\u201360 (2010)","journal-title":"IMA J. Numer. Anal."},{"key":"1231_CR7","doi-asserted-by":"publisher","first-page":"489","DOI":"10.1007\/s00211-011-0416-x","volume":"120","author":"JW Barrett","year":"2012","unstructured":"Barrett, J.W., Garcke, H., N\u00fcrnberg, R.: Parametric approximation of isotropic and anisotropic elastic flow for closed and open curves. Numer. Math. 120, 489\u2013542 (2012)","journal-title":"Numer. Math."},{"key":"1231_CR8","doi-asserted-by":"publisher","first-page":"1250,037","DOI":"10.1142\/S0218202512500376","volume":"22","author":"JW Barrett","year":"2012","unstructured":"Barrett, J.W., Garcke, H., N\u00fcrnberg, R.: Elastic flow with junctions: variational approximation and applications to nonlinear splines. Math. Models Methods Appl. Sci. 22, 1250,037 (2012)","journal-title":"Math. Models Methods Appl. Sci."},{"key":"1231_CR9","doi-asserted-by":"publisher","first-page":"733","DOI":"10.1016\/j.jcp.2018.10.006","volume":"376","author":"JW Barrett","year":"2019","unstructured":"Barrett, J.W., Garcke, H., N\u00fcrnberg, R.: Finite element methods for fourth order axisymmetric geometric evolution equations. J. Comput. Phys. 376, 733\u2013766 (2019)","journal-title":"J. Comput. Phys."},{"key":"1231_CR10","doi-asserted-by":"publisher","first-page":"791","DOI":"10.1007\/s00211-018-1013-z","volume":"141","author":"JW Barrett","year":"2019","unstructured":"Barrett, J.W., Garcke, H., N\u00fcrnberg, R.: Variational discretization of axisymmetric curvature flows. Numer. Math. 141, 791\u2013837 (2019)","journal-title":"Numer. Math."},{"key":"1231_CR11","doi-asserted-by":"publisher","first-page":"1987","DOI":"10.1137\/18M1227111","volume":"57","author":"JW Barrett","year":"2019","unstructured":"Barrett, J.W., Garcke, H., N\u00fcrnberg, R.: Stable discretizations of elastic flow in Riemannian manifolds. SIAM J. Numer. Anal. 57, 1987\u20132018 (2019)","journal-title":"SIAM J. Numer. Anal."},{"key":"1231_CR12","doi-asserted-by":"publisher","first-page":"1601","DOI":"10.1093\/imanum\/drz012","volume":"40","author":"JW Barrett","year":"2020","unstructured":"Barrett, J.W., Garcke, H., N\u00fcrnberg, R.: Numerical approximation of curve evolutions in Riemannian manifolds. IMA J. Numer. Anal. 40, 1601\u20131651 (2020)","journal-title":"IMA J. Numer. Anal."},{"key":"1231_CR13","doi-asserted-by":"crossref","unstructured":"Barrett, J.W., Garcke, H., N\u00fcrnberg, R.: Parametric finite element approximations of curvature driven interface evolutions. In: Bonito, A., Nochetto, R.H. (eds.) Handbook of Numerical Analysis, vol.\u00a021, pp. 275\u2013423. Elsevier, Amsterdam (2020)","DOI":"10.1016\/bs.hna.2019.05.002"},{"key":"1231_CR14","doi-asserted-by":"publisher","first-page":"833","DOI":"10.1051\/m2an\/2021014","volume":"55","author":"JW Barrett","year":"2021","unstructured":"Barrett, J.W., Garcke, H., N\u00fcrnberg, R.: Stable approximations for axisymmetric Willmore flow for closed and open surfaces. Math. Model. Numer. Anal. 55, 833\u2013885 (2021)","journal-title":"Math. Model. Numer. Anal."},{"key":"1231_CR15","doi-asserted-by":"publisher","first-page":"1115","DOI":"10.1093\/imanum\/drs041","volume":"33","author":"S Bartels","year":"2013","unstructured":"Bartels, S.: A simple scheme for the approximation of the elastic flow of inextensible curves. IMA J. Numer. Anal. 33, 1115\u20131125 (2013)","journal-title":"IMA J. Numer. Anal."},{"key":"1231_CR16","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1007\/s10851-015-0616-6","volume":"55","author":"H Benninghoff","year":"2016","unstructured":"Benninghoff, H., Garcke, H.: Segmentation and restoration of images on surfaces by parametric active contours with topology changes. J. Math. Imaging Vis. 55, 105\u2013124 (2016)","journal-title":"J. Math. Imaging Vis."},{"key":"1231_CR17","doi-asserted-by":"publisher","unstructured":"Berchenko-Kogan, Y.: The entropy of the Angenent torus is approximately 1.85122. Experiment. Math. (2019). https:\/\/doi.org\/10.1080\/10586458.2019.1583616","DOI":"10.1080\/10586458.2019.1583616"},{"key":"1231_CR18","doi-asserted-by":"crossref","unstructured":"Berchenko-Kogan, Y.: Numerically computing the index of mean curvature flow self-shrinkers. arXiv:2007.06094 (2020)","DOI":"10.1007\/s00025-021-01550-y"},{"key":"1231_CR19","doi-asserted-by":"publisher","first-page":"141","DOI":"10.4171\/IFB\/379","volume":"19","author":"E Bretin","year":"2017","unstructured":"Bretin, E., Masnou, S.: A new phase field model for inhomogeneous minimal partitions, and applications to droplets dynamics. Interfaces Free Bound. 19, 141\u2013182 (2017)","journal-title":"Interfaces Free Bound."},{"key":"1231_CR20","doi-asserted-by":"publisher","first-page":"2061","DOI":"10.1512\/iumj.2007.56.3060","volume":"56","author":"E Cabezas-Rivas","year":"2007","unstructured":"Cabezas-Rivas, E., Miquel, V.: Volume preserving mean curvature flow in the hyperbolic space. Indiana Univ. Math. J. 56, 2061\u20132086 (2007)","journal-title":"Indiana Univ. Math. J."},{"key":"1231_CR21","doi-asserted-by":"publisher","first-page":"604","DOI":"10.1006\/jcph.2001.6960","volume":"175","author":"LT Cheng","year":"2002","unstructured":"Cheng, L.T., Burchard, P., Merriman, B., Osher, S.: Motion of curves constrained on surfaces using a level-set approach. J. Comput. Phys. 175, 604\u2013644 (2002)","journal-title":"J. Comput. Phys."},{"key":"1231_CR22","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1080\/10586458.1994.10504572","volume":"3","author":"DL Chopp","year":"1994","unstructured":"Chopp, D.L.: Computation of self-similar solutions for mean curvature flow. Exp. Math. 3, 1\u201315 (1994)","journal-title":"Exp. Math."},{"issue":"175","key":"1231_CR23","doi-asserted-by":"publisher","first-page":"755","DOI":"10.4007\/annals.2012.175.2.7","volume":"2","author":"TH Colding","year":"2012","unstructured":"Colding, T.H., Minicozzi, W.P., II.: Generic mean curvature flow I: generic singularities. Ann. Math. 2(175), 755\u2013833 (2012)","journal-title":"Ann. Math."},{"key":"1231_CR24","doi-asserted-by":"crossref","unstructured":"Dall\u2019Acqua, A., Laux, T., Lin, C.C., Pozzi, P., Spener, A.: The elastic flow of curves on the sphere. Geom. Flows 3, 1\u201313 (2018)","DOI":"10.1515\/geofl-2018-0001"},{"key":"1231_CR25","doi-asserted-by":"crossref","unstructured":"Dall\u2019Acqua, A., Lin, C.C., Pozzi, P.: Evolution of open elastic curves in $$\\mathbb{R}^n$$ subject to fixed length and natural boundary conditions. Analysis (Berlin) 34, 209\u2013222 (2014)","DOI":"10.1515\/anly-2014-1249"},{"key":"1231_CR26","doi-asserted-by":"crossref","unstructured":"Dall\u2019Acqua, A., Lin, C.C., Pozzi, P.: A gradient flow for open elastic curves with fixed length and clamped ends. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5(17), 1031\u20131066 (2017)","DOI":"10.2422\/2036-2145.201511_009"},{"key":"1231_CR27","doi-asserted-by":"crossref","unstructured":"Dall\u2019Acqua, A., Lin, C.C., Pozzi, P.: Elastic flow of networks: long-time existence result. Geom. Flows 4, 83\u2013136 (2019)","DOI":"10.1515\/geofl-2019-0005"},{"key":"1231_CR28","doi-asserted-by":"crossref","unstructured":"Dall\u2019Acqua, A., Pozzi, P.: A Willmore-Helfrich $$L^2$$-flow of curves with natural boundary conditions. Commun. Anal. Geom. 22, 617\u2013669 (2014)","DOI":"10.4310\/CAG.2014.v22.n4.a2"},{"key":"1231_CR29","unstructured":"Dall\u2019Acqua, A., Spener, A.: The elastic flow of curves in the hyperbolic plane. arXiv:1710.09600 (2017)"},{"key":"1231_CR30","unstructured":"Dall\u2019Acqua, A., Spener, A.: Circular solutions to the elastic flow in hyperbolic space. In: Proceedings of analysis on shapes of solutions to partial differential equations, (2017), RIMS K\u00f4ky\u00fbroku, vol. 2082. Kyoto, Japan (2018)"},{"key":"1231_CR31","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1145\/992200.992206","volume":"30","author":"TA Davis","year":"2004","unstructured":"Davis, T.A.: Algorithm 832: UMFPACK V4.3\u2014an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30, 196\u2013199 (2004)","journal-title":"ACM Trans. Math. Software"},{"key":"1231_CR32","doi-asserted-by":"publisher","first-page":"645","DOI":"10.1090\/S0025-5718-08-02176-5","volume":"78","author":"K Deckelnick","year":"2009","unstructured":"Deckelnick, K., Dziuk, G.: Error analysis for the elastic flow of parametrized curves. Math. Comput. 78, 645\u2013671 (2009)","journal-title":"Math. Comput."},{"key":"1231_CR33","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1017\/S0962492904000224","volume":"14","author":"K Deckelnick","year":"2005","unstructured":"Deckelnick, K., Dziuk, G., Elliott, C.M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14, 139\u2013232 (2005)","journal-title":"Acta Numer."},{"key":"1231_CR34","doi-asserted-by":"publisher","first-page":"7213","DOI":"10.1090\/tran\/6907","volume":"369","author":"G Drugan","year":"2017","unstructured":"Drugan, G., Kleene, S.J.: Immersed self-shrinkers. Trans. Am. Math. Soc. 369, 7213\u20137250 (2017)","journal-title":"Trans. Am. Math. Soc."},{"key":"1231_CR35","doi-asserted-by":"publisher","first-page":"3725","DOI":"10.1007\/s12220-017-9976-z","volume":"28","author":"G Drugan","year":"2018","unstructured":"Drugan, G., Nguyen, X.H.: Shrinking doughnuts via variational methods. J. Geom. Anal. 28, 3725\u20133746 (2018)","journal-title":"J. Geom. Anal."},{"key":"1231_CR36","doi-asserted-by":"crossref","unstructured":"Elliott, C.M.: Private communication (2009)","DOI":"10.1016\/S0145-4145(08)79416-4"},{"key":"1231_CR37","doi-asserted-by":"crossref","unstructured":"Epstein, C.L., Gage, M.: The curve shortening flow. In: Wave Motion: Theory, Modelling, and Computation (Berkeley, Calif., 1986), Math. Sci. Res. Inst. Publ., vol.\u00a07, pp. 15\u201359. Springer, New York (1987)","DOI":"10.1007\/978-1-4613-9583-6_2"},{"key":"1231_CR38","doi-asserted-by":"crossref","unstructured":"Garcke, H., Haas, R.: Modeling of Nonisothermal Multi-Component, Multi-Phase Systems with Convection, vol. chap. 20, pp. 325\u2013338. Wiley, London (2008)","DOI":"10.1002\/9783527624041.ch20"},{"key":"1231_CR39","doi-asserted-by":"publisher","first-page":"2019","DOI":"10.1016\/j.jde.2018.08.019","volume":"266","author":"H Garcke","year":"2019","unstructured":"Garcke, H., Menzel, J., Pluda, A.: Willmore flow of planar networks. J. Differ. Equ. 266, 2019\u20132051 (2019)","journal-title":"J. Differ. Equ."},{"key":"1231_CR40","doi-asserted-by":"publisher","first-page":"1253","DOI":"10.1080\/03605302.2020.1771364","volume":"45","author":"H Garcke","year":"2020","unstructured":"Garcke, H., Menzel, J., Pluda, A.: Long time existence of solutions to an elastic flow of networks. Commun. Partial Differ. Equ. 45, 1253\u20131305 (2020)","journal-title":"Commun. Partial Differ. Equ."},{"key":"1231_CR41","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1137\/S0036139998334895","volume":"60","author":"H Garcke","year":"1999","unstructured":"Garcke, H., Stoth, B., Nestler, B.: A multiphase field concept: numerical simulations of moving phase boundaries and multiple junctions. SIAM J. Appl. Math. 60, 295\u2013315 (1999)","journal-title":"SIAM J. Appl. Math."},{"key":"1231_CR42","unstructured":"Haas, R.: Modeling and analysis for general non-isothermal convective phase field systems. University Regensburg, Regensburg (2007) (Ph.D. thesis)"},{"key":"1231_CR43","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1007\/s00526-012-0550-z","volume":"48","author":"M Helmers","year":"2013","unstructured":"Helmers, M.: Kinks in two-phase lipid bilayer membranes. Calc. Var. Partial Differ. Equ. 48, 211\u2013242 (2013)","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"1231_CR44","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1093\/qmath\/hau027","volume":"66","author":"M Helmers","year":"2015","unstructured":"Helmers, M.: Convergence of an approximation for rotationally symmetric two-phase lipid bilayer membranes. Q. J. Math. 66, 143\u2013170 (2015)","journal-title":"Q. J. Math."},{"key":"1231_CR45","doi-asserted-by":"publisher","first-page":"285","DOI":"10.4310\/jdg\/1214444099","volume":"31","author":"G Huisken","year":"1990","unstructured":"Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285\u2013299 (1990)","journal-title":"J. Differ. Geom."},{"key":"1231_CR46","volume-title":"Riemannian Geometry and Geometric Analysis","author":"J Jost","year":"2005","unstructured":"Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, Berlin (2005)"},{"key":"1231_CR47","doi-asserted-by":"publisher","first-page":"3943","DOI":"10.1090\/S0002-9947-2014-05721-8","volume":"366","author":"S Kleene","year":"2014","unstructured":"Kleene, S., M\u00f8ller, N.M.: Self-shrinkers with a rotational symmetry. Trans. Am. Math. Soc. 366, 3943\u20133963 (2014)","journal-title":"Trans. Am. Math. Soc."},{"key":"1231_CR48","first-page":"453","volume":"52","author":"N Koiso","year":"2015","unstructured":"Koiso, N.: On motion of an elastic wire in a Riemannian manifold and singular perturbation. Osaka J. Math. 52, 453\u2013473 (2015)","journal-title":"Osaka J. Math."},{"key":"1231_CR49","unstructured":"Kraus, D., Roth, O.: Conformal metrics. In: Topics in Modern Function Theory, Ramanujan Math. Soc. Lect. Notes Ser., vol.\u00a019, pp. 41\u201383. Ramanujan Math. Soc., Mysore, India (2013), (see also https:\/\/arxiv.org\/abs\/0805.2235)"},{"key":"1231_CR50","doi-asserted-by":"crossref","unstructured":"K\u00fchnel, W.: Differential Geometry: Curves - Surfaces - Manifolds, Student Mathematical Library, vol. 77. American Mathematical Society, Providence, RI (2015)","DOI":"10.1090\/stml\/077"},{"key":"1231_CR51","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4310\/jdg\/1214438990","volume":"20","author":"J Langer","year":"1984","unstructured":"Langer, J., Singer, D.A.: The total squared curvature of closed curves. J. Differ. Geom. 20, 1\u201322 (1984)","journal-title":"J. Differ. Geom."},{"key":"1231_CR52","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1007\/BF00127856","volume":"5","author":"J Langer","year":"1987","unstructured":"Langer, J., Singer, D.A.: Curve-straightening in Riemannian manifolds. Ann. Global Anal. Geom. 5, 133\u2013150 (1987)","journal-title":"Ann. Global Anal. Geom."},{"key":"1231_CR53","doi-asserted-by":"publisher","first-page":"3743","DOI":"10.1090\/S0002-9947-98-01977-1","volume":"350","author":"A Linn\u00e9r","year":"1998","unstructured":"Linn\u00e9r, A.: Curve-straightening and the Palais\u2013Smale condition. Trans. Am. Math. Soc. 350, 3743\u20133765 (1998)","journal-title":"Trans. Am. Math. Soc."},{"key":"1231_CR54","unstructured":"Liu, Z.H.: The Morse index of mean curvature flow self-shrinkers. Massachusetts Institute of Technology, Ann Arbor, MI (2016) (Ph.D. thesis)"},{"key":"1231_CR55","doi-asserted-by":"publisher","first-page":"9596","DOI":"10.1016\/j.jcp.2008.07.011","volume":"227","author":"E Maitre","year":"2008","unstructured":"Maitre, E., Santosa, F.: Level set methods for optimization problems involving geometry and constraints. II. Optimization over a fixed surface. J. Comput. Phys. 227, 9596\u20139611 (2008)","journal-title":"J. Comput. Phys."},{"key":"1231_CR56","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1080\/00036810500333604","volume":"85","author":"K Mikula","year":"2006","unstructured":"Mikula, K., \u0160ev\u010dovi\u010d, D.: Evolution of curves on a surface driven by the geodesic curvature and external force. Appl. Anal. 85, 345\u2013362 (2006)","journal-title":"Appl. Anal."},{"key":"1231_CR57","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1515\/geofl-2020-0002","volume":"5","author":"M M\u00fcller","year":"2020","unstructured":"M\u00fcller, M., Spener, A.: On the convergence of the elastic flow in the hyperbolic plane. Geom. Flows 5, 40\u201377 (2020)","journal-title":"Geom. Flows"},{"key":"1231_CR58","first-page":"497","volume":"32","author":"E Schippers","year":"2007","unstructured":"Schippers, E.: The calculus of conformal metrics. Ann. Acad. Sci. Fenn. Math. 32, 497\u2013521 (2007)","journal-title":"Ann. Acad. Sci. Fenn. Math."},{"key":"1231_CR59","unstructured":"Schmidt, A., Siebert, K.G.: Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA. Lecture Notes in Computational Science and Engineering, vol. 42. Springer, Berlin (2005)"},{"key":"1231_CR60","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1016\/j.jcp.2006.09.008","volume":"223","author":"A Spira","year":"2007","unstructured":"Spira, A., Kimmel, R.: Geometric curve flows on parametric manifolds. J. Comput. Phys. 223, 235\u2013249 (2007)","journal-title":"J. Comput. Phys."},{"key":"1231_CR61","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1007\/BF01192093","volume":"2","author":"A Stone","year":"1994","unstructured":"Stone, A.: A density function and the structure of singularities of the mean curvature flow. Calc. Var. Partial Differ. Equ. 2, 443\u2013480 (1994)","journal-title":"Calc. Var. Partial Differ. Equ."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-021-01231-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-021-01231-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-021-01231-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,9]],"date-time":"2023-11-09T01:33:58Z","timestamp":1699493638000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-021-01231-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,20]]},"references-count":61,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,10]]}},"alternative-id":["1231"],"URL":"https:\/\/doi.org\/10.1007\/s00211-021-01231-6","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,20]]},"assertion":[{"value":"4 December 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 July 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 July 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 September 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}