{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T19:31:32Z","timestamp":1769974292116,"version":"3.49.0"},"reference-count":50,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>A Cahn\u2013Hilliard\u2013Allen\u2013Cahn phase-field model coupled with a heat transfer\nequation, particularly with full non-diagonal mobility matrices, is studied. After reformulating the problem with respect to the inverse of temperature, we proposed and analysed a structure-preserving approximation for the semi-discretisation in space and then a fully discrete approximation using conforming finite elements and time-stepping methods. We prove structure-preserving property and discrete stability using relative entropy methods for the semi-discrete and fully discrete case. The theoretical results are illustrated by numerical experiments.<\/jats:p>","DOI":"10.1515\/cmam-2023-0274","type":"journal-article","created":{"date-parts":[[2024,9,2]],"date-time":"2024-09-02T14:52:49Z","timestamp":1725288769000},"page":"373-396","source":"Crossref","is-referenced-by-count":5,"title":["Variational Approximation for a Non-Isothermal Coupled Phase-Field System: Structure-Preservation &amp; Nonlinear Stability"],"prefix":"10.1515","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4987-2398","authenticated-orcid":false,"given":"Aaron","family":"Brunk","sequence":"first","affiliation":[{"name":"Institute of Mathematics , 9182 Johannes Gutenberg-University , Mainz , Germany"}]},{"given":"Oliver","family":"Habrich","sequence":"additional","affiliation":[{"name":"Institute for Numerical Mathematics , 27266 Johannes Kepler University , Linz , Austria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1049-7463","authenticated-orcid":false,"given":"Timileyin David","family":"Oyedeji","sequence":"additional","affiliation":[{"name":"Mechanics of Functional Materials Division , 26536 Technical University Darmstadt , Darmstadt , Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5505-7117","authenticated-orcid":false,"given":"Yangyiwei","family":"Yang","sequence":"additional","affiliation":[{"name":"Mechanics of Functional Materials Division , 26536 Technical University Darmstadt , Darmstadt , Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5906-5341","authenticated-orcid":false,"given":"Bai-Xiang","family":"Xu","sequence":"additional","affiliation":[{"name":"Mechanics of Functional Materials Division , 26536 Technical University Darmstadt , Darmstadt , Germany"}]}],"member":"374","published-online":{"date-parts":[[2024,9,3]]},"reference":[{"key":"2025032910202860823_j_cmam-2023-0274_ref_001","doi-asserted-by":"crossref","unstructured":"G.  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Math. 99,\nBirkh\u00e4user, Basel (1991), 1\u201332.","DOI":"10.1007\/978-3-0348-5715-4_1"},{"key":"2025032910202860823_j_cmam-2023-0274_ref_004","doi-asserted-by":"crossref","unstructured":"H. W.  Alt and I.  Paw\u0142ow,\nA mathematical model of dynamics of nonisothermal phase separation,\nPhys. D 59 (1992), no. 4, 389\u2013416.","DOI":"10.1016\/0167-2789(92)90078-2"},{"key":"2025032910202860823_j_cmam-2023-0274_ref_005","doi-asserted-by":"crossref","unstructured":"H. W.  Alt and I.  Paw\u0142ow,\nExistence of solutions for non-isothermal phase separation,\nAdv. Math. Sci. Appl. 1 (1992), no. 2, 319\u2013409.","DOI":"10.1007\/978-3-0348-5715-4_1"},{"key":"2025032910202860823_j_cmam-2023-0274_ref_006","doi-asserted-by":"crossref","unstructured":"E.  Bonetti, P.  Colli and M.  Fremond,\nA phase field model with thermal memory governed by the entropy balance,\nMath. Models Methods Appl. 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Anal. 92 (1986), no. 3, 205\u2013245.","DOI":"10.1007\/BF00254827"},{"key":"2025032910202860823_j_cmam-2023-0274_ref_012","doi-asserted-by":"crossref","unstructured":"C.  Charach and P. C.  Fife,\nOn thermodynamically consistent schemes for phase field equations,\nOpen Syst. Inf. Dyn. 5 (1998), no. 2, 99\u2013123.","DOI":"10.1023\/A:1009652531731"},{"key":"2025032910202860823_j_cmam-2023-0274_ref_013","doi-asserted-by":"crossref","unstructured":"C.  Chen and X.  Yang,\nFast, provably unconditionally energy stable, and second-order accurate algorithms for the anisotropic Cahn\u2013Hilliard model,\nComput. Methods Appl. Mech. Engrg. 351 (2019), 35\u201359.","DOI":"10.1016\/j.cma.2019.03.030"},{"key":"2025032910202860823_j_cmam-2023-0274_ref_014","doi-asserted-by":"crossref","unstructured":"R.  Chen and S.  Gu,\nOn novel linear schemes for the Cahn\u2013Hilliard equation based on an improved invariant energy quadratization approach,\nJ. Comput. Appl. 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Niezg\u00f3dka,\nEvolution systems of nonlinear variational inequalities arising from phase change problems,\nNonlinear Anal. 22 (1994), no. 9, 1163\u20131180.","DOI":"10.1016\/0362-546X(94)90235-6"},{"key":"2025032910202860823_j_cmam-2023-0274_ref_036","doi-asserted-by":"crossref","unstructured":"Y.  Li and J.  Yang,\nConsistency-enhanced SAV BDF2 time-marching method with relaxation for the incompressible Cahn\u2013Hilliard\u2013Navier\u2013Stokes binary fluid model,\nCommun. Nonlinear Sci. Numer. Simul. 118 (2023), Article ID 107055.","DOI":"10.1016\/j.cnsns.2022.107055"},{"key":"2025032910202860823_j_cmam-2023-0274_ref_037","doi-asserted-by":"crossref","unstructured":"A.  Marveggio and G.  Schimperna,\nOn a non-isothermal Cahn\u2013Hilliard model based on a microforce balance,\nJ. Differential Equations 274 (2021), 924\u2013970.","DOI":"10.1016\/j.jde.2020.10.030"},{"key":"2025032910202860823_j_cmam-2023-0274_ref_038","doi-asserted-by":"crossref","unstructured":"R. I.  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Zheng,\nGlobal existence for a thermodynamically consistent model of phase field type,\nDifferential Integral Equations 5 (1992), no. 2, 241\u2013253.","DOI":"10.57262\/die\/1371043970"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0274\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0274\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,29]],"date-time":"2025-03-29T10:25:19Z","timestamp":1743243919000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0274\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,3]]},"references-count":50,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,11,14]]},"published-print":{"date-parts":[[2025,4,1]]}},"alternative-id":["10.1515\/cmam-2023-0274"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2023-0274","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,9,3]]}}}