{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,26]],"date-time":"2026-04-26T07:45:40Z","timestamp":1777189540258,"version":"3.51.4"},"reference-count":53,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,3,18]],"date-time":"2025-03-18T00:00:00Z","timestamp":1742256000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,3,18]],"date-time":"2025-03-18T00:00:00Z","timestamp":1742256000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004377","name":"The Hong Kong Polytechnic University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004377","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2025,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We propose a high-order multistep projection method for the harmonic map heat flow from a bounded domain <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\varOmega \\subset \\mathbb {R}^d$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03a9<\/mml:mi>\n                    <mml:mo>\u2282<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mi>R<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mi>d<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> into a given <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\mathcal {N}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>N<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-dimensional smooth surface <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Gamma \\subset \\mathbb {R}^{{{\\mathcal {N}}}+1}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u0393<\/mml:mi>\n                    <mml:mo>\u2282<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mi>R<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mi>N<\/mml:mi>\n                        <mml:mo>+<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. At every time level, an auxiliary numerical solution is solved by a multistep backward difference formula with a mass-lumping finite element method in space, and then projected onto the surface <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Gamma $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0393<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. The projected numerical solution is used in the backward difference formula and the extrapolation of nonlinearities in the following time levels. Such projection algorithms are convenient in computation while still preserving the pointwise geometric constraint of the solution to stay on the target surface <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Gamma $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0393<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. The convergence of some low-order single-step projection algorithms based on the backward Euler and Crank\u2013Nicolson schemes have been studied in many articles for harmonic map heat flow and related models into the unit sphere, while the convergence of high-order multistep projection methods still remains open. In this article, we propose a high-order multistep projection method for harmonic map heat flows into a general smooth surface (not necessarily the unit sphere) and prove its optimal-order convergence by combining four techniques, i.e., decomposition of the Nevanlinna\u2013Odeh multiplier technique into approximately normal and tangential components separately, an almost orthogonal relation between the error functions associated to the auxiliary and projected numerical solutions, pointwise <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$L^\\infty $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mi>\u221e<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> error estimates, the use of orthogonal projection onto the target surface <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Gamma $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0393<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. Numerical results are provided to support the theoretical analysis on the convergence of the high-order multistep projection methods.<\/jats:p>","DOI":"10.1007\/s00211-025-01464-9","type":"journal-article","created":{"date-parts":[[2025,3,18]],"date-time":"2025-03-18T02:34:26Z","timestamp":1742265266000},"page":"629-661","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Convergence of multistep projection methods for harmonic map heat flows into general surfaces"],"prefix":"10.1007","volume":"157","author":[{"given":"Genming","family":"Bai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinping","family":"Gui","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Buyang","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,3,18]]},"reference":[{"key":"1464_CR1","doi-asserted-by":"publisher","first-page":"464","DOI":"10.1137\/140962619","volume":"53","author":"G Akrivis","year":"2015","unstructured":"Akrivis, G.: Stability of implicit-explicit backward difference formulas for nonlinear parabolic equations. SIAM J. Numer. Anal. 53, 464\u2013484 (2015)","journal-title":"SIAM J. Numer. Anal."},{"issue":"329","key":"1464_CR2","doi-asserted-by":"publisher","first-page":"995","DOI":"10.1090\/mcom\/3597","volume":"90","author":"G Akrivis","year":"2021","unstructured":"Akrivis, G., Feischl, M., Kov\u00e1cs, B., Lubich, C.: Higher-order linearly implicit full discretization of the Landau\u2013Lifshitz\u2013Gilbert equation. Math. Comput. 90(329), 995\u20131038 (2021)","journal-title":"Math. Comput."},{"key":"1464_CR3","doi-asserted-by":"publisher","first-page":"2195","DOI":"10.1090\/mcom3055","volume":"85","author":"G Akrivis","year":"2016","unstructured":"Akrivis, G., Katsoprinakis, E.: Backward difference formulae: new multipliers and stability properties for parabolic equations. Math. Comput. 85, 2195\u20132216 (2016)","journal-title":"Math. Comput."},{"issue":"4","key":"1464_CR4","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1007\/s00211-015-0702-0","volume":"131","author":"G Akrivis","year":"2015","unstructured":"Akrivis, G., Lubich, C.: Fully implicit, linearly implicit and implicit-explicit backward difference formulae for quasi-linear parabolic equations. Numer. Math. 131(4), 713\u2013735 (2015)","journal-title":"Numer. Math."},{"issue":"2","key":"1464_CR5","first-page":"187","volume":"1","author":"F Alouges","year":"2008","unstructured":"Alouges, F.: A new finite element scheme for Landau\u2013Lifchitz equations. Discrete Contin. Dyn. Syst. Ser. S 1(2), 187\u2013196 (2008)","journal-title":"Discrete Contin. Dyn. Syst. Ser. S"},{"issue":"2","key":"1464_CR6","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1142\/S0218202506001169","volume":"16","author":"F Alouges","year":"2006","unstructured":"Alouges, F., Jaisson, P.: Convergence of a finite element discretization for the Landau\u2013Lifshitz equations in micromagnetism. Math. Models Methods Appl. Sci. 16(2), 299\u2013316 (2006)","journal-title":"Math. Models Methods Appl. Sci."},{"key":"1464_CR7","doi-asserted-by":"publisher","first-page":"1071","DOI":"10.1016\/0362-546X(92)90196-L","volume":"18","author":"F Alouges","year":"1992","unstructured":"Alouges, F., Soyeur, A.: On global weak solutions for Landau\u2013Lifshitz equations: existence and nonuniqueness. Nonlinear Anal. 18, 1071\u20131084 (1992)","journal-title":"Nonlinear Anal."},{"key":"1464_CR8","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1007\/s00211-014-0615-3","volume":"128","author":"F Alouges","year":"2014","unstructured":"Alouges, F., Kritsikis, E., Steiner, J., Toussaint, J.-C.: A convergent and precise finite element scheme for Landau-Lifshitz-Gilbert equation. Numer. Math. 128, 407\u2013430 (2014)","journal-title":"Numer. Math."},{"issue":"3","key":"1464_CR9","doi-asserted-by":"publisher","first-page":"1639","DOI":"10.1137\/20M1335431","volume":"59","author":"R An","year":"2021","unstructured":"An, R., Gao, H., Sun, W.: Optimal error analysis of Euler and Crank-Nicolson projection finite difference schemes for Landau\u2013Lifshitz equation. SIAM J. Numer. Anal. 59(3), 1639\u20131662 (2021)","journal-title":"SIAM J. Numer. Anal."},{"key":"1464_CR10","doi-asserted-by":"publisher","first-page":"979","DOI":"10.1007\/s10915-017-0479-7","volume":"74","author":"R An","year":"2018","unstructured":"An, R., Su, J.: Optimal error estimates of semi-implicit Galerkin method for time-dependent nematic liquid crystal flows. J. Sci. Comput. 74, 979\u20131008 (2018)","journal-title":"J. Sci. Comput."},{"key":"1464_CR11","doi-asserted-by":"crossref","unstructured":"An, R., Sun, W.: Analysis of backward Euler projection FEM for the Landau-Lifshitz equation. IMA J. Numer. Anal. (2021)","DOI":"10.1093\/imanum\/drab038"},{"issue":"271","key":"1464_CR12","doi-asserted-by":"publisher","first-page":"1263","DOI":"10.1090\/S0025-5718-09-02300-X","volume":"79","author":"S Bartels","year":"2010","unstructured":"Bartels, S.: Numerical analysis of a finite element scheme for the approximation of harmonic maps into surfaces. Math. comput. 79(271), 1263\u20131301 (2010)","journal-title":"Math. comput."},{"key":"1464_CR13","doi-asserted-by":"publisher","first-page":"1269","DOI":"10.1090\/S0025-5718-09-02221-2","volume":"78","author":"S Bartels","year":"2009","unstructured":"Bartels, S., Lubich, C., Prohl, A.: Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers. Math. Comput. 78, 1269\u20131292 (2009)","journal-title":"Math. Comput."},{"key":"1464_CR14","doi-asserted-by":"publisher","first-page":"1847","DOI":"10.1090\/S0025-5718-07-02026-1","volume":"76","author":"S Bartels","year":"2007","unstructured":"Bartels, S., Prohl, A.: Constraint preserving implicit finite element discretization of harmonic map flow into spheres. Math. Comput. 76, 1847\u20131859 (2007)","journal-title":"Math. Comput."},{"key":"1464_CR15","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1007\/s00211-009-0282-y","volume":"115","author":"L Ba\u0148as","year":"2010","unstructured":"Ba\u0148as, L., Prohl, A., Sch\u00e4tzle, R.: Finite element approximations of harmonic map heat flows and wave maps into spheres of nonconstant radii. Numer. Math. 115, 395\u2013432 (2010)","journal-title":"Numer. Math."},{"key":"1464_CR16","doi-asserted-by":"publisher","first-page":"1704","DOI":"10.1137\/07068254X","volume":"46","author":"R Becker","year":"2008","unstructured":"Becker, R., Feng, X., Prohl, A.: Finite element approximations of the Ericksen-Leslie model for nematic liquid crystal flow. SIAM J. Numer. Anal. 46, 1704\u20131731 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"1464_CR17","series-title":"Texts in Applied Mathematics","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, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008)","edition":"3"},{"key":"1464_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110831","volume":"451","author":"Y Cai","year":"2022","unstructured":"Cai, Y., Chen, J., Wang, C., Xie, C.: A second-order numerical method for Landau\u2013Lifshitz\u2013Gilbert equation with large damping parameters. J. Comput. Phys. 451, 110831 (2022)","journal-title":"J. Comput. Phys."},{"key":"1464_CR19","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/j.apnum.2021.05.027","volume":"168","author":"J Chen","year":"2021","unstructured":"Chen, J., Wang, C., Xie, C.: Convergence analysis of a second-order semi-implicit projection method for Landau\u2013Lifshitz equation. Appl. Numer. Math. 168, 55\u201374 (2021)","journal-title":"Appl. Numer. Math."},{"issue":"2","key":"1464_CR20","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1007\/BF02921640","volume":"8","author":"Y Chen","year":"1998","unstructured":"Chen, Y., Lin, F.H.: Evolution equations with a free boundary condition. J. Geom. Anal. 8(2), 179\u2013197 (1998)","journal-title":"J. Geom. Anal."},{"issue":"4","key":"1464_CR21","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1023\/A:1004420829610","volume":"35","author":"M Chin-Joe-Kong","year":"1999","unstructured":"Chin-Joe-Kong, M., Mulder, W.A., Van Veldhuizen, M.: Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation. J. Eng. Math. 35(4), 405\u2013426 (1999)","journal-title":"J. Eng. Math."},{"issue":"2","key":"1464_CR22","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1016\/0045-7825(72)90006-0","volume":"1","author":"PG Ciarlet","year":"1972","unstructured":"Ciarlet, P.G., Raviart, P.A.: Interpolation theory over curved elements, with applications to finite element methods. Comput. Methods Appl. Mech. Eng. 1(2), 217\u2013249 (1972)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1464_CR23","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1093\/imanum\/dri011","volume":"25","author":"I Cimr\u00e1k","year":"2005","unstructured":"Cimr\u00e1k, I.: Error estimates for a semi-implicit numerical scheme solving the Landau\u2013Lifshitz equation with an exchange field. IMA. J. Numer. Anal. 25, 611\u2013634 (2005)","journal-title":"IMA. J. Numer. Anal."},{"issue":"6","key":"1464_CR24","doi-asserted-by":"publisher","first-page":"2047","DOI":"10.1137\/S0036142997329554","volume":"38","author":"G Cohen","year":"2001","unstructured":"Cohen, G., Joly, P., Roberts, J.E., Tordjman, N.: Higher order triangular finite elements with mass lumping for the wave equation. SIAM J. Numer. Anal. 38(6), 2047\u20132078 (2001)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"1464_CR25","doi-asserted-by":"publisher","first-page":"805","DOI":"10.1137\/070708135","volume":"47","author":"A Demlow","year":"2009","unstructured":"Demlow, A.: Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces. SIAM J. Numer. Anal. 47(2), 805\u2013827 (2009)","journal-title":"SIAM J. Numer. Anal."},{"key":"1464_CR26","doi-asserted-by":"publisher","first-page":"109","DOI":"10.2307\/2373037","volume":"86","author":"J Eells Jr","year":"1964","unstructured":"Eells, J., Jr., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109\u2013160 (1964)","journal-title":"Am. J. Math."},{"key":"1464_CR27","series-title":"Graduate Studies in Mathematics","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/019","volume-title":"Partial Differential Equations","author":"LC Evans","year":"2010","unstructured":"Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence (2010)","edition":"2"},{"key":"1464_CR28","doi-asserted-by":"publisher","first-page":"1786","DOI":"10.1137\/16M1065161","volume":"55","author":"M Feischl","year":"2017","unstructured":"Feischl, M., Tran, T.: The eddy current-LLG equations: FEM-BEM coupling and a priori error estimates. SIAM J. Numer. Anal. 55, 1786\u20131819 (2017)","journal-title":"SIAM J. Numer. Anal."},{"key":"1464_CR29","doi-asserted-by":"publisher","first-page":"2574","DOI":"10.1137\/130936476","volume":"52","author":"H Gao","year":"2014","unstructured":"Gao, H.: Optimal error estimates of a linearized backward Euler FEM for the Landau\u2013Lifshitz equation. SIAM J. Numer. Anal. 52, 2574\u20132593 (2014)","journal-title":"SIAM J. Numer. Anal."},{"key":"1464_CR30","doi-asserted-by":"publisher","first-page":"781","DOI":"10.1090\/S0025-5718-2010-02429-9","volume":"80","author":"V Girault","year":"2011","unstructured":"Girault, V., Guillen-Gonzalez, F.: Mixed formulation, approximation and decoupling algorithm for a penalized nematic liquid crystals model. Math. Comput. 80, 781\u2013819 (2011)","journal-title":"Math. Comput."},{"issue":"3","key":"1464_CR31","doi-asserted-by":"publisher","first-page":"849","DOI":"10.1093\/imanum\/drs026","volume":"33","author":"P Grohs","year":"2013","unstructured":"Grohs, P.: Quasi-interpolation in Riemannian manifolds. IMA J. Numer. Anal. 33(3), 849\u2013874 (2013)","journal-title":"IMA J. Numer. Anal."},{"issue":"6","key":"1464_CR32","doi-asserted-by":"publisher","first-page":"1357","DOI":"10.1007\/s10208-014-9230-z","volume":"15","author":"P Grohs","year":"2015","unstructured":"Grohs, P., Hardering, H., Sander, O.: Optimal a priori discretization error bounds for geodesic finite elements. Found. Comput. Math. 15(6), 1357\u20131411 (2015)","journal-title":"Found. Comput. Math."},{"issue":"1","key":"1464_CR33","doi-asserted-by":"publisher","first-page":"404","DOI":"10.1137\/18M1176798","volume":"57","author":"P Grohs","year":"2019","unstructured":"Grohs, P., Hardering, H., Sander, O., Sprecher, M.: Projection-based finite elements for nonlinear function spaces. SIAM J. Numer. Anal. 57(1), 404\u2013428 (2019)","journal-title":"SIAM J. Numer. Anal."},{"key":"1464_CR34","unstructured":"Grohs, P., Sprecher, M.: Projection-based quasiinterpolation in manifolds. SAM Rep. 23 (2013)"},{"issue":"1","key":"1464_CR35","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1137\/21M1402212","volume":"60","author":"X Gui","year":"2022","unstructured":"Gui, X., Li, B., Wang, J.: Convergence of renormalized finite element methods for heat flow of harmonic maps. SIAM J. Numer. Anal. 60(1), 312\u2013338 (2022)","journal-title":"SIAM J. Numer. Anal."},{"key":"1464_CR36","doi-asserted-by":"publisher","first-page":"2565","DOI":"10.1137\/17M1116799","volume":"55","author":"JV Guti\u00e9rrez-Santacreu","year":"2017","unstructured":"Guti\u00e9rrez-Santacreu, J.V., Restelli, M.: Inf-sup stable finite element methods for the Landau\u2013Lifshitz\u2013Gilbert and harmonic map heat flow equations. SIAM J. Numer. Anal. 55, 2565\u20132591 (2017)","journal-title":"SIAM J. Numer. Anal."},{"key":"1464_CR37","unstructured":"Hairer, E., Wanner, G.: Solving ordinary differential equations. II, volume\u00a014 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2010. Stiff and differential-algebraic problems, Second revised edition, paperback"},{"key":"1464_CR38","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/s00211-017-0941-3","volume":"139","author":"H Hardering","year":"2018","unstructured":"Hardering, H.: $$L^2$$-discretization error bounds for maps into Riemannian manifolds. Numerische Mathematik 139, 381\u2013410 (2018)","journal-title":"Numerische Mathematik"},{"key":"1464_CR39","doi-asserted-by":"publisher","first-page":"797","DOI":"10.1007\/s00211-019-01074-2","volume":"143","author":"B Kov\u00e1cs","year":"2019","unstructured":"Kov\u00e1cs, B., Li, B., Lubich, C.: A convergent evolving finite element algorithm for mean curvature flow of closed surfaces. Numer. Math. 143, 797\u2013853 (2019)","journal-title":"Numer. Math."},{"issue":"4","key":"1464_CR40","doi-asserted-by":"publisher","first-page":"443","DOI":"10.4171\/ifb\/446","volume":"22","author":"B Kov\u00e1cs","year":"2020","unstructured":"Kov\u00e1cs, B., Li, B., Lubich, C.: A convergent algorithm for forced mean curvature flow driven by diffusion on the surface. Interfaces Free Bound. 22(4), 443\u2013464 (2020)","journal-title":"Interfaces Free Bound."},{"key":"1464_CR41","unstructured":"Lang, U.: Lecture notes in Riemannian and metric geometry. ETH Z\u00fcrich (2022)"},{"issue":"6","key":"1464_CR42","doi-asserted-by":"publisher","first-page":"A3957","DOI":"10.1137\/20M1333456","volume":"42","author":"B Li","year":"2020","unstructured":"Li, B., Yang, J., Zhou, Z.: Arbitrarily high-order exponential cut-off methods for preserving maximum principle of parabolic equations. SIAM J. Sci. Comput. 42(6), A3957\u2013A3978 (2020)","journal-title":"SIAM J. Sci. Comput."},{"issue":"6","key":"1464_CR43","doi-asserted-by":"publisher","first-page":"789","DOI":"10.1002\/cpa.3160420605","volume":"42","author":"F-H Lin","year":"1989","unstructured":"Lin, F.-H.: Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. Commun. Pure Appl. Math. 42(6), 789\u2013814 (1989)","journal-title":"Commun. Pure Appl. Math."},{"key":"1464_CR44","doi-asserted-by":"publisher","first-page":"1365","DOI":"10.1093\/imanum\/drs044","volume":"33","author":"C Lubich","year":"2013","unstructured":"Lubich, C., Mansour, D., Venkataraman, C.: Backward difference time discretization of parabolic differential equations on evolving surfaces. IMA J. Numer. Anal. 33, 1365\u20131385 (2013)","journal-title":"IMA J. Numer. Anal."},{"issue":"4","key":"1464_CR45","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1080\/01630568108816097","volume":"3","author":"O Nevanlinna","year":"1981","unstructured":"Nevanlinna, O., Odeh, F.M.: Multiplier techniques for linear multistep methods. Numer. Funct. Anal. Optim. 3(4), 377\u2013423 (1981)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"1464_CR46","doi-asserted-by":"crossref","unstructured":"Prohl, A.: Computational micromagnetism. Advances in Numerical Mathematics. B. G. Teubner, Stuttgart (2001)","DOI":"10.1007\/978-3-663-09498-2"},{"issue":"3","key":"1464_CR47","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1145\/2998441","volume":"43","author":"F Rathgeber","year":"2017","unstructured":"Rathgeber, F., Ham, D.A., Mitchell, L., Lange, M., Luporini, F., McRae, A.T., Bercea, G.T., Markall, G.R., Kelly, P.H.: Firedrake: automating the finite element method by composing abstractions. ACM Trans. Math. Softw. 43(3), 24 (2017)","journal-title":"ACM Trans. Math. Softw."},{"issue":"4","key":"1464_CR48","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1002\/cpa.20205","volume":"61","author":"T Riviere","year":"2008","unstructured":"Riviere, T., Struwe, M.: Partial regularity for harmonic maps and related problems. Commun. Pure Appl. Math. J. Issued Courant Inst. Math. Sci. 61(4), 451\u2013463 (2008)","journal-title":"Commun. Pure Appl. Math. J. Issued Courant Inst. Math. Sci."},{"key":"1464_CR49","volume-title":"Geometric Wave Equations","author":"JMI Shatah","year":"2000","unstructured":"Shatah, J.M.I., Struwe, M.: Geometric Wave Equations, vol. 2. American Mathematical Society (2000)"},{"key":"1464_CR50","volume-title":"Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems","author":"M Struwe","year":"2008","unstructured":"Struwe, M.: Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, vol. 34. Springer (2008)"},{"issue":"5","key":"1464_CR51","doi-asserted-by":"publisher","first-page":"701","DOI":"10.1109\/83.918563","volume":"10","author":"B Tang","year":"2001","unstructured":"Tang, B., Sapiro, G., Caselles, V.: Color image enhancement via chromaticity diffusion. IEEE Trans. Image Process. 10(5), 701\u2013707 (2001)","journal-title":"IEEE Trans. Image Process."},{"issue":"1","key":"1464_CR52","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1007\/BF01947068","volume":"83","author":"KK Uhlenbeck","year":"1982","unstructured":"Uhlenbeck, K.K.: Removable singularities in Yang-Mills fields. Commun. Math. Phys. 83(1), 11\u201329 (1982)","journal-title":"Commun. Math. Phys."},{"issue":"6","key":"1464_CR53","doi-asserted-by":"publisher","first-page":"2085","DOI":"10.1137\/S0036142901396715","volume":"40","author":"LA Vese","year":"2002","unstructured":"Vese, L.A., Osher, S.J.: Numerical methods for $$p$$-harmonic flows and applications to image processing. SIAM J. Numer. Anal. 40(6), 2085\u20132104 (2002). https:\/\/doi.org\/10.1137\/S0036142901396715","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-025-01464-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-025-01464-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-025-01464-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,2]],"date-time":"2025-06-02T10:06:02Z","timestamp":1748858762000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-025-01464-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,18]]},"references-count":53,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,4]]}},"alternative-id":["1464"],"URL":"https:\/\/doi.org\/10.1007\/s00211-025-01464-9","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,18]]},"assertion":[{"value":"8 June 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 May 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 December 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 March 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}