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Caccioppoli\u201d , Universit\u00e0 degli Studi di Napoli \u201cFederico II\u201d , Via Cintia, Complesso Monte S. Angelo, 80126 Napoli , Italy"}]},{"given":"Carl-Martin","family":"Pfeiler","sequence":"additional","affiliation":[{"name":"Institute of Analysis and Scientific Computing , TU Wien , Wiedner Hauptstrasse 8\u201310, 1040 , Vienna , Austria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1977-9830","authenticated-orcid":false,"given":"Dirk","family":"Praetorius","sequence":"additional","affiliation":[{"name":"Institute of Analysis and Scientific Computing , TU Wien , Wiedner Hauptstrasse 8\u201310, 1040 , Vienna , Austria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6213-1602","authenticated-orcid":false,"given":"Michele","family":"Ruggeri","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics , University of Strathclyde , 26 Richmond Street , Glasgow G1 1XH , United Kingdom"}]}],"member":"374","published-online":{"date-parts":[[2022,6,9]]},"reference":[{"key":"2023033112515587408_j_cmam-2022-0060_ref_001","doi-asserted-by":"crossref","unstructured":"C. Abert, G. Hrkac, M. Page, D. Praetorius, M. Ruggeri and D. Suess,\nSpin-polarized transport in ferromagnetic multilayers: An unconditionally convergent FEM integrator,\nComput. Math. Appl. 68 (2014), no. 6, 639\u2013654.","DOI":"10.1016\/j.camwa.2014.07.010"},{"key":"2023033112515587408_j_cmam-2022-0060_ref_002","doi-asserted-by":"crossref","unstructured":"E. Acerbi, I. Fonseca and G. Mingione,\nExistence and regularity for mixtures of micromagnetic materials,\nProc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006), no. 2072, 2225\u20132243.","DOI":"10.1098\/rspa.2006.1655"},{"key":"2023033112515587408_j_cmam-2022-0060_ref_003","doi-asserted-by":"crossref","unstructured":"G. Akrivis, M. Feischl, B. Kov\u00e1cs and C. Lubich,\nHigher-order linearly implicit full discretization of the Landau\u2013Lifshitz\u2013Gilbert equation,\nMath. Comp. 90 (2021), no. 329, 995\u20131038.","DOI":"10.1090\/mcom\/3597"},{"key":"2023033112515587408_j_cmam-2022-0060_ref_004","doi-asserted-by":"crossref","unstructured":"F. Alouges,\nA new finite element scheme for Landau\u2013Lifchitz equations,\nDiscrete Contin. Dyn. Syst. Ser. S 1 (2008), no. 2, 187\u2013196.","DOI":"10.3934\/dcdss.2008.1.187"},{"key":"2023033112515587408_j_cmam-2022-0060_ref_005","doi-asserted-by":"crossref","unstructured":"F. Alouges, A. de Bouard, B. Merlet and L. Nicolas,\nStochastic homogenization of the Landau\u2013Lifshitz\u2013Gilbert equation,\nStoch. Partial Differ. Equ. Anal. Comput. 9 (2021), no. 4, 789\u2013818.","DOI":"10.1007\/s40072-020-00185-4"},{"key":"2023033112515587408_j_cmam-2022-0060_ref_006","doi-asserted-by":"crossref","unstructured":"F. Alouges and G. Di Fratta,\nHomogenization of composite ferromagnetic materials,\nProc. A. 471 (2015), no. 2182, Article ID 20150365.","DOI":"10.1098\/rspa.2015.0365"},{"key":"2023033112515587408_j_cmam-2022-0060_ref_007","doi-asserted-by":"crossref","unstructured":"F. Alouges and P. Jaisson,\nConvergence of a finite element discretization for the Landau\u2013Lifshitz equations in micromagnetism,\nMath. Models Methods Appl. Sci. 16 (2006), no. 2, 299\u2013316.","DOI":"10.1142\/S0218202506001169"},{"key":"2023033112515587408_j_cmam-2022-0060_ref_008","doi-asserted-by":"crossref","unstructured":"F. Alouges, E. Kritsikis, J. Steiner and J.-C. Toussaint,\nA convergent and precise finite element scheme for Landau\u2013Lifschitz\u2013Gilbert equation,\nNumer. Math. 128 (2014), no. 3, 407\u2013430.","DOI":"10.1007\/s00211-014-0615-3"},{"key":"2023033112515587408_j_cmam-2022-0060_ref_009","doi-asserted-by":"crossref","unstructured":"F. Alouges, E. Kritsikis and J.-C. Toussaint,\nA convergent finite element approximation for Landau\u2013Lifschitz\u2013Gilbert equation,\nPhys. 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Rev. 100 (1955), Article ID 1243."},{"key":"2023033112515587408_j_cmam-2022-0060_ref_029","doi-asserted-by":"crossref","unstructured":"V. Girault and P.-A. Raviart,\nFinite Element Methods for Navier\u2013Stokes Equations: Theory and Algorithms,\nSpringer Ser. Comput. Math. 5,\nSpringer, Berlin, 1986.","DOI":"10.1007\/978-3-642-61623-5"},{"key":"2023033112515587408_j_cmam-2022-0060_ref_030","doi-asserted-by":"crossref","unstructured":"G. Hrkac, C.-M. Pfeiler, D. Praetorius, M. Ruggeri, A. Segatti and B. Stiftner,\nConvergent tangent plane integrators for the simulation of chiral magnetic skyrmion dynamics,\nAdv. Comput. Math. 45 (2019), no. 3, 1329\u20131368.","DOI":"10.1007\/s10444-019-09667-z"},{"key":"2023033112515587408_j_cmam-2022-0060_ref_031","unstructured":"A. Hubert and R. Sch\u00e4fer,\nMagnetic Domains: The Analysis of Magnetic Microstructures,\nSpringer, Berlin, 1998."},{"key":"2023033112515587408_j_cmam-2022-0060_ref_032","doi-asserted-by":"crossref","unstructured":"E. Kim and J. Wilkening,\nConvergence of a mass-lumped finite element method for the Landau\u2013Lifshitz equation,\nQuart. Appl. Math. 76 (2018), no. 2, 383\u2013405.","DOI":"10.1090\/qam\/1485"},{"key":"2023033112515587408_j_cmam-2022-0060_ref_033","doi-asserted-by":"crossref","unstructured":"M. Kru\u017e\u00edk and A. Prohl,\nRecent developments in the modeling, analysis, and numerics of ferromagnetism,\nSIAM Rev. 48 (2006), no. 3, 439\u2013483.","DOI":"10.1137\/S0036144504446187"},{"key":"2023033112515587408_j_cmam-2022-0060_ref_034","unstructured":"L. Landau and E. Lifshitz,\nOn the theory of the dispersion of magnetic permeability in ferromagnetic bodies,\nPhys. Z. Sowjetunion 8 (1935), 153\u2013168."},{"key":"2023033112515587408_j_cmam-2022-0060_ref_035","doi-asserted-by":"crossref","unstructured":"T. Moriya,\nAnisotropic superexchange interaction and weak ferromagnetism,\nPhys. 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