{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T10:22:12Z","timestamp":1775816532648,"version":"3.50.1"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,8,1]],"date-time":"2023-08-01T00:00:00Z","timestamp":1690848000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,8,2]],"date-time":"2023-08-02T00:00:00Z","timestamp":1690934400000},"content-version":"vor","delay-in-days":1,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100005682","name":"Universidad del Cauca","doi-asserted-by":"publisher","award":["VRI ID 5869"],"award-info":[{"award-number":["VRI ID 5869"]}],"id":[{"id":"10.13039\/501100005682","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005682","name":"Universidad del Cauca","doi-asserted-by":"publisher","award":["VRI ID 5869"],"award-info":[{"award-number":["VRI ID 5869"]}],"id":[{"id":"10.13039\/501100005682","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Ministerio de Ciencias - Minciencias","award":["The Bicentennial PhD Scholarship Program"],"award-info":[{"award-number":["The Bicentennial PhD Scholarship Program"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2023,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper deals with the numerical analysis for a family of nonlinear degenerate parabolic problems. The model is spatially discretized using a finite element method; an implicit Euler scheme is employed for time discretization. We deduce sufficient conditions to ensure that the fully discrete problem has a unique solution and to prove quasi-optimal error estimates for the approximation. Finally, we propose a nonlinear degenerate parabolic problem that arises from electromagnetic applications in conductive nonlinear magnetic media and deduce its solubility and convergence by using the developed abstract theory, including some numerical results to confirm the obtained theoretical results.<\/jats:p>","DOI":"10.1007\/s10444-023-10052-0","type":"journal-article","created":{"date-parts":[[2023,8,2]],"date-time":"2023-08-02T06:01:50Z","timestamp":1690956110000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Numerical analysis of nonlinear degenerate parabolic problems with application to eddy current models"],"prefix":"10.1007","volume":"49","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6844-5992","authenticated-orcid":false,"given":"Ramiro","family":"Acevedo","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7311-8478","authenticated-orcid":false,"given":"Christian","family":"G\u00f3mez","sequence":"additional","affiliation":[]},{"given":"Paulo","family":"Navia","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,8,2]]},"reference":[{"issue":"1","key":"10052_CR1","first-page":"21","volume":"37","author":"F Paronetto","year":"2004","unstructured":"Paronetto, F.: Homogenization of degenerate elliptic-parabolic equations. Asymptot. Anal. 37(1), 21\u201356 (2004)","journal-title":"Asymptot. Anal."},{"key":"10052_CR2","volume-title":"Monotone operators in Banach space and nonlinear partial differential equations","author":"R Showalter","year":"1997","unstructured":"Showalter, R.: Monotone operators in Banach space and nonlinear partial differential equations, vol. 49. American Mathematical Society, Providence (1997)"},{"key":"10052_CR3","doi-asserted-by":"crossref","unstructured":"Zlamal, M.: Finite element solution of quasistationary nonlinear magnetic field. ESAIM: Mathematical Modelling and Numerical Analysis -Mod\u00e9lisation Math\u00e9matique et Analyse Num\u00e9rique 16(2), 161\u2013191 (1982)","DOI":"10.1051\/m2an\/1982160201611"},{"issue":"4","key":"10052_CR4","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1090\/qam\/872820","volume":"44","author":"RC MacCamy","year":"1987","unstructured":"MacCamy, R.C., Suri, M.: A time-dependent interface problem for twodimensional eddy currents. Quart. Appl. Math. 44(4), 675\u2013690 (1987). https:\/\/doi.org\/10.1090\/qam\/872820","journal-title":"Quart. Appl. Math."},{"issue":"3","key":"10052_CR5","doi-asserted-by":"publisher","first-page":"984","DOI":"10.1002\/num.20670","volume":"28","author":"A Berm\u00fadez","year":"2012","unstructured":"Berm\u00fadez, A., Reales, C., Rodr\u00edguez, R., Salgado, P.: Numerical analysis of a transient eddy current axisymmetric problem involving velocity terms. Numer. Methods Partial Differential Equations 28(3), 984\u20131012 (2012). https:\/\/doi.org\/10.1002\/num.20670","journal-title":"Numer. Methods Partial Differential Equations"},{"key":"10052_CR6","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/j.camwa.2021.05.010","volume":"96","author":"R Acevedo","year":"2021","unstructured":"Acevedo, R., G\u00f3mez, C., L\u00f3pez-Rodr\u00edguez, B.: Fully discrete finite element approximation for a family of degenerate parabolic mixed equations. Computers & Mathematics with Applications 96, 155\u2013177 (2021)","journal-title":"Computers & Mathematics with Applications"},{"key":"10052_CR7","doi-asserted-by":"publisher","first-page":"134","DOI":"10.3846\/mma.2022.12846","volume":"27","author":"R Acevedo","year":"2022","unstructured":"Acevedo, R., G\u00f3mez, C., L\u00f3pez-Rodr\u00edguez, B.: Fully-discrete finite element approximation for a family of degenerate parabolic problems. Math. Model. Anal. 27, 134\u2013160 (2022). https:\/\/doi.org\/10.3846\/mma.2022.12846","journal-title":"Math. Model. Anal."},{"key":"10052_CR8","doi-asserted-by":"publisher","unstructured":"Acevedo, R., G\u00f3mez, C.: Finite element error estimates for a mixed degenerate parabolic model. C. R. Math. 360. (2022) https:\/\/doi.org\/10.5802\/crmath.308","DOI":"10.5802\/crmath.308"},{"key":"10052_CR9","doi-asserted-by":"publisher","unstructured":"Berm\u00fadez, A., Mu\u00f1oz-Sola, R., Reales, C., Rodriguez, R., Salgado, P.: A transient eddy current problem on a moving domain. numerical analysis. Adv. Comput. Math. 42(4),757\u2013789, (2016). https:\/\/doi.org\/10.1007\/s10444-015-9441-0","DOI":"10.1007\/s10444-015-9441-0"},{"issue":"4","key":"10052_CR10","doi-asserted-by":"publisher","first-page":"593","DOI":"10.1007\/s00211-005-0597-2","volume":"100","author":"F Bachinger","year":"2005","unstructured":"Bachinger, F., Langer, U., Sch\u00f6berl, J.: Numerical analysis of nonlinear multiharmonic eddy current problems. Numer. Math. 100(4), 593\u2013616 (2005). https:\/\/doi.org\/10.1007\/s00211-005-0597-2","journal-title":"Numer. Math."},{"key":"10052_CR11","volume-title":"Computational electromagnetism","author":"A Bossavit","year":"1998","unstructured":"Bossavit, A.: Computational electromagnetism. Academic Press Inc, San Diego, CA, San Diego (1998)"},{"key":"10052_CR12","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1553\/etna_vol52s230","volume":"52","author":"R Acevedo","year":"2020","unstructured":"Acevedo, R., Alvarez, E., Navia, P.: A boundary and finite element coupling for a magnetically nonlinear eddy current problem. Electronic Transaction on Numerical Analysis 52, 230\u2013248 (2020)","journal-title":"Electronic Transaction on Numerical Analysis"},{"key":"10052_CR13","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1016\/j.apnum.2013.04.007","volume":"79","author":"O B\u00edr\u00f3","year":"2014","unstructured":"B\u00edr\u00f3, O., Koczka, G., Preis, K.: Finite element solution of nonlinear eddy current problems with periodic excitation and its industrial applications. Appl. Numer. Math. 79, 3\u201317 (2014)","journal-title":"Appl. Numer. Math."},{"issue":"4","key":"10052_CR14","first-page":"636","volume":"12","author":"T Kang","year":"2015","unstructured":"Kang, T., Chen, T.: A T-$$\\psi $$ finite element method for a nonlinear degenerate eddy current model with ferromagnetic materials. Int. J. Numer. Anal. Model. 12(4), 636\u2013663 (2015)","journal-title":"Int. J. Numer. Anal. Model."},{"key":"10052_CR15","doi-asserted-by":"publisher","unstructured":"Berm\u00fadez, A., Pi\u00f1eiro, M., Salgado, P.: Mathematical and numerical analysis of a transient magnetic model with voltage drop excitations. Comput. Math. Appl. 76 (2018). https:\/\/doi.org\/10.1016\/j.camwa.2018.08.054","DOI":"10.1016\/j.camwa.2018.08.054"},{"key":"10052_CR16","doi-asserted-by":"publisher","first-page":"1045","DOI":"10.1016\/j.cma.2017.03.045","volume":"321","author":"J Chovan","year":"2017","unstructured":"Chovan, J., Geuzaine, C., Slodi\u010dka, M.: $${A- \\phi }$$ formulation of a mathematical model for the induction hardening process with a nonlinear law for the magnetic field. Computer Methods in Applied Mechanics and Engineering 321, 1045\u20131061 (2017). https:\/\/doi.org\/10.1016\/j.cma.2017.03.045","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"10052_CR17","doi-asserted-by":"publisher","unstructured":"Slodicka, Marian, Vr\u00e1bel, Vladim\u00edr: Existence and uniqueness of a solution for a field\/circuit coupled problem. ESAIM: M2AN. 51(3),1045\u20131061 (2017). https:\/\/doi.org\/10.1051\/m2an\/2016052","DOI":"10.1051\/m2an\/2016052"},{"issue":"268","key":"10052_CR18","doi-asserted-by":"publisher","first-page":"1929","DOI":"10.1090\/S0025-5718-09-02254-6","volume":"78","author":"R Acevedo","year":"2009","unstructured":"Acevedo, R., Meddahi, S., Rodr\u00edguez, R.: An $$E$$-based mixed formulation for a time-dependent eddy current problem. Math. Comp. 78(268), 1929\u20131949 (2009). https:\/\/doi.org\/10.1090\/S0025-5718-09-02254-6","journal-title":"Math. Comp."},{"issue":"2","key":"10052_CR19","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1093\/imanum\/drp049","volume":"31","author":"R Acevedo","year":"2011","unstructured":"Acevedo, R., Meddahi, S.: An E-based mixed FEM and BEM coupling for a time-dependent eddy current problem. IMA J. Numer. Anal. 31(2), 667\u2013697 (2011). https:\/\/doi.org\/10.1093\/imanum\/drp049","journal-title":"IMA J. Numer. Anal."},{"key":"10052_CR20","doi-asserted-by":"publisher","unstructured":"Berm\u00fadez, A., L\u00f3pez-Rodr\u00edguez, B., Rodr\u00edguez, R., Salgado, P.: An eddy current problem in terms of a time-primitive of the electric field with non-local source conditions. ESAIM: M2AN. 47(3),875\u2013902 (2013). https:\/\/doi.org\/10.1051\/m2an\/2013065","DOI":"10.1051\/m2an\/2013065"},{"key":"10052_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0981-2","volume-title":"Nonlinear functional analysis and its applications","author":"E Zeidler","year":"1990","unstructured":"Zeidler, E.: Nonlinear functional analysis and its applications. II\/B. Springer, Berlin, Heidelberg (1990)"},{"key":"10052_CR22","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719208","volume-title":"Finite element method for elliptic problems","author":"PG Ciarlet","year":"2002","unstructured":"Ciarlet, P.G.: Finite element method for elliptic problems. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2002)"},{"key":"10052_CR23","doi-asserted-by":"crossref","unstructured":"Quarteroni, A.M., Valli, A.: Numerical approximation of partial differential equations, 1st, (eds.): 1994. 2nd printing Springer, Germany (2008)","DOI":"10.1007\/978-3-540-85268-1"},{"key":"10052_CR24","unstructured":"Pechstein, C.: Multigrid-Newton-methods for nonlinear magnetostatic problems. Master\u2019s Thesis, Institute for Computational Mathematics, Johannes Kepler Universit\u00e4t Linz, Austria (2004)"},{"key":"10052_CR25","volume-title":"Modelling","author":"T Petzold","year":"2014","unstructured":"Petzold, T.: Modelling. Analysis and Simulation of Multifrequency Induction Hardening. Technische Universit\u00e4t Berlin, Germany (2014)"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-023-10052-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-023-10052-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-023-10052-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T20:41:37Z","timestamp":1698266497000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-023-10052-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8]]},"references-count":25,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,8]]}},"alternative-id":["10052"],"URL":"https:\/\/doi.org\/10.1007\/s10444-023-10052-0","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8]]},"assertion":[{"value":"15 July 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 May 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 August 2023","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":"64"}}