{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,17]],"date-time":"2025-12-17T13:00:57Z","timestamp":1765976457758},"reference-count":33,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100018618","name":"National Numerical Wind Tunnel Project of China","doi-asserted-by":"publisher","award":["NNW2019ZT4-B08"],"award-info":[{"award-number":["NNW2019ZT4-B08"]}],"id":[{"id":"10.13039\/501100018618","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11722112","12071455"],"award-info":[{"award-number":["11722112","12071455"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, we mainly study the error analysis of an unconditionally energy stable local discontinuous Galerkin (LDG) scheme for the Cahn\u2013Hilliard equation with concentration-dependent mobility.\nThe time discretization is based on the invariant energy quadratization (IEQ) method.\nThe fully discrete scheme leads to a linear algebraic system to solve at each time step.\nThe main difficulty in the error estimates is the lack of control on some jump terms at cell boundaries in the LDG discretization.\nSpecial treatments are needed for the initial condition and the non-constant mobility term of the Cahn\u2013Hilliard equation.\nFor the analysis of the non-constant mobility term, we take full advantage of the semi-implicit time-discrete method and bound some numerical variables in <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>L<\/m:mi>\n                              <m:mi mathvariant=\"normal\">\u221e<\/m:mi>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2020-0066_ineq_0001.png\" \/>\n                        <jats:tex-math>L^{\\infty}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-norm by the mathematical induction method.\nThe optimal error results are obtained for the fully discrete scheme.<\/jats:p>","DOI":"10.1515\/cmam-2020-0066","type":"journal-article","created":{"date-parts":[[2021,4,22]],"date-time":"2021-04-22T18:50:49Z","timestamp":1619117449000},"page":"729-751","source":"Crossref","is-referenced-by-count":4,"title":["Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn\u2013Hilliard Equation with Concentration-Dependent Mobility"],"prefix":"10.1515","volume":"21","author":[{"given":"Fengna","family":"Yan","sequence":"first","affiliation":[{"name":"School of Mathematics , Hefei University of Technology , Hefei , Anhui 230601 , P. R. China"}]},{"given":"Yan","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences , University of Science and Technology of China , Hefei , Anhui 230026 , P. R. China"}]}],"member":"374","published-online":{"date-parts":[[2021,4,20]]},"reference":[{"key":"2023033111373257946_j_cmam-2020-0066_ref_001","doi-asserted-by":"crossref","unstructured":"M. Baccouch,\nOptimal energy-conserving local discontinuous Galerkin method for the one-dimensional sine-Gordon equation,\nInt. J. Comput. Math. 94 (2017), no. 2, 316\u2013344.","DOI":"10.1080\/00207160.2015.1105364"},{"key":"2023033111373257946_j_cmam-2020-0066_ref_002","doi-asserted-by":"crossref","unstructured":"J. W. Barrett and J. F. Blowey,\nFinite element approximation of a model for phase separation of a multi-component alloy with a concentration-dependent mobility matrix,\nIMA J. Numer. Anal. 18 (1998), no. 2, 287\u2013328.","DOI":"10.1093\/imanum\/18.2.287"},{"key":"2023033111373257946_j_cmam-2020-0066_ref_003","doi-asserted-by":"crossref","unstructured":"J. W. Barrett and J. F. Blowey,\nFinite element approximation of the Cahn\u2013Hilliard equation with concentration dependent mobility,\nMath. Comp. 68 (1999), no. 226, 487\u2013517.","DOI":"10.1090\/S0025-5718-99-01015-7"},{"key":"2023033111373257946_j_cmam-2020-0066_ref_004","doi-asserted-by":"crossref","unstructured":"J. W. Barrett, J. F. Blowey and H. Garcke,\nFinite element approximation of the Cahn\u2013Hilliard equation with degenerate mobility,\nSIAM J. Numer. Anal. 37 (1999), no. 1, 286\u2013318.","DOI":"10.1137\/S0036142997331669"},{"key":"2023033111373257946_j_cmam-2020-0066_ref_005","doi-asserted-by":"crossref","unstructured":"J. W. Cahn and J. E. Hilliard,\nFree energy of a nonuniform system I. Interfacial free energy,\nJ. Chem. Phys. 28 (1958), 258\u2013267.","DOI":"10.1063\/1.1744102"},{"key":"2023033111373257946_j_cmam-2020-0066_ref_006","doi-asserted-by":"crossref","unstructured":"W. Chen, C. Wang, X. Wang and S. M. Wise,\nPositivity-preserving, energy stable numerical schemes for the Cahn\u2013Hilliard equation with logarithmic potential,\nJ. Comput. Phys. X 3 (2019), Article ID 100031.","DOI":"10.1016\/j.jcpx.2019.100031"},{"key":"2023033111373257946_j_cmam-2020-0066_ref_007","doi-asserted-by":"crossref","unstructured":"S. M. Choo and Y. J. Lee,\nA discontinuous Galerkin method for the Cahn\u2013Hilliard equation,\nJ. Appl. Math. Comput. 18 (2005), no. 1\u20132, 113\u2013126.","DOI":"10.1007\/BF02936559"},{"key":"2023033111373257946_j_cmam-2020-0066_ref_008","doi-asserted-by":"crossref","unstructured":"B. Cockburn, G. Kanschat, I. Perugia and D. Sch\u00f6tzau,\nSuperconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids,\nSIAM J. Numer. 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