{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T18:15:08Z","timestamp":1783188908867,"version":"3.54.6"},"reference-count":42,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12571391"],"award-info":[{"award-number":["12571391"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12171322"],"award-info":[{"award-number":["12171322"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12271366"],"award-info":[{"award-number":["12271366"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Journal of Computational and Applied Mathematics"],"published-print":{"date-parts":[[2026,12]]},"DOI":"10.1016\/j.cam.2026.117819","type":"journal-article","created":{"date-parts":[[2026,5,26]],"date-time":"2026-05-26T23:38:09Z","timestamp":1779838689000},"page":"117819","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["hp-version space-time Galerkin discretization for fourth-order linear parabolic equations"],"prefix":"10.1016","volume":"488","author":[{"given":"Zhe","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xinghui","family":"Yong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lijun","family":"Yi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.cam.2026.117819_bib0001","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1016\/S0022-247X(03)00474-8","article-title":"A fourth-order parabolic equation modeling epitaxial thin film growth","volume":"286","author":"King","year":"2003","journal-title":"J. Math. Anal. Appl."},{"key":"10.1016\/j.cam.2026.117819_bib0002","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1137\/S003614459529284X","article-title":"Thin films with high surface tension","volume":"40","author":"Myers","year":"1998","journal-title":"SIAM Rev."},{"key":"10.1016\/j.cam.2026.117819_bib0003","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1016\/S0022-5096(98)00102-1","article-title":"A continuum model of kinetic roughening and coarsening in thin films","volume":"47","author":"Ortiz","year":"1999","journal-title":"J. Mech. Phys. Solids"},{"key":"10.1016\/j.cam.2026.117819_bib0004","doi-asserted-by":"crossref","first-page":"1397","DOI":"10.1080\/03605309408821059","article-title":"Slow dynamics for the Cahn-Hilliard equation in higher space dimensions, part I: spectral estimates","volume":"19","author":"Alikakos","year":"1994","journal-title":"Commun. Part. Differ. Equ."},{"key":"10.1016\/j.cam.2026.117819_bib0005","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s002050050072","article-title":"Slow dynamics for the Cahn-Hilliard equation in higher space dimensions: the motion of bubbles","volume":"141","author":"Alikakos","year":"1998","journal-title":"Arch. Ration. Mech. Anal."},{"key":"10.1016\/j.cam.2026.117819_bib0006","doi-asserted-by":"crossref","first-page":"405","DOI":"10.3934\/cpaa.2005.4.405","article-title":"On two classes of generalized viscous Cahn-Hilliard equations","volume":"4","author":"Rossi","year":"2005","journal-title":"Commun. Pure Appl. Anal."},{"key":"10.1016\/j.cam.2026.117819_bib0007","first-page":"212","article-title":"Dispersal models","volume":"37","author":"Calder\u00f3n","year":"1991","journal-title":"Rev. Union Mat. Argent."},{"key":"10.1016\/j.cam.2026.117819_bib0008","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/BF00276132","article-title":"A generalized diffusion model for growth and dispersal in a population","volume":"12","author":"Cohen","year":"1981","journal-title":"J. Math. Biol."},{"key":"10.1016\/j.cam.2026.117819_bib0009","doi-asserted-by":"crossref","first-page":"574","DOI":"10.1016\/j.cam.2018.05.039","article-title":"An energy stable fourth order finite difference scheme for the Cahn-Hilliard equation","volume":"362","author":"Cheng","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2026.117819_bib0010","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/S0045-7825(96)01176-0","article-title":"Implicit time splitting for fourth-order parabolic equations","volume":"148","author":"Christov","year":"1997","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"10.1016\/j.cam.2026.117819_bib0011","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0025-5718-1967-0221785-2","article-title":"Some stable difference approximations to a fourth-order parabolic partial differential equation","volume":"21","author":"Fairweather","year":"1967","journal-title":"Math. Comp."},{"key":"10.1016\/j.cam.2026.117819_bib0012","doi-asserted-by":"crossref","first-page":"1667","DOI":"10.3934\/dcdsb.2014.19.1667","article-title":"Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations","volume":"19","author":"Ju","year":"2014","journal-title":"Discrete Contin. Dyn. Syst. Ser. B."},{"key":"10.1016\/j.cam.2026.117819_bib0013","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1007\/s002110050377","article-title":"Finite element approximation of a fourth order nonlinear degenerate parabolic equation","volume":"80","author":"Barrett","year":"1998","journal-title":"Numer. Math."},{"key":"10.1016\/j.cam.2026.117819_bib0014","doi-asserted-by":"crossref","first-page":"1745","DOI":"10.1002\/num.22373","article-title":"Weak Galerkin finite element methods for a fourth order parabolic equation","volume":"35","author":"Chai","year":"2019","journal-title":"Numer. Methods Part. Differ. Equ."},{"key":"10.1016\/j.cam.2026.117819_bib0015","doi-asserted-by":"crossref","first-page":"2163","DOI":"10.1090\/mcom\/2936","article-title":"Adaptive discontinuous Galerkin approximations to fourth order parabolic problems","volume":"84","author":"Georgoulis","year":"2015","journal-title":"Math. Comput."},{"key":"10.1016\/j.cam.2026.117819_bib0016","doi-asserted-by":"crossref","first-page":"2928","DOI":"10.3934\/dcdsb.2023207","article-title":"Differential-spectral approximation based on reduced-dimension scheme for fourth-order parabolic equation","volume":"29","author":"Pan","year":"2024","journal-title":"Discrete Contin. Dyn. Syst. Ser. B."},{"key":"10.1016\/j.cam.2026.117819_bib0017","doi-asserted-by":"crossref","first-page":"578","DOI":"10.1137\/1036141","article-title":"The p and h\u2013p versions of the finite element method, basic principles and properties","volume":"36","author":"Babu\u0161ka","year":"1994","journal-title":"SIAM Rev."},{"key":"10.1016\/j.cam.2026.117819_bib0018","series-title":"P-and hp-Finite Element Methods","author":"Schwab","year":"1998"},{"key":"10.1016\/j.cam.2026.117819_bib0019","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1090\/S0025-5718-1972-0321301-2","article-title":"One-step piecewise polynomial Galerkin methods for initial value problems","volume":"26","author":"Hulme","year":"1972","journal-title":"Math. Comp."},{"key":"10.1016\/j.cam.2026.117819_bib0020","doi-asserted-by":"crossref","first-page":"881","DOI":"10.1090\/S0025-5718-1972-0315899-8","article-title":"Discrete Galerkin and related one-step methods for ordinary differential equations","volume":"26","author":"Hulme","year":"1972","journal-title":"Math. Comp."},{"key":"10.1016\/j.cam.2026.117819_bib0021","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1090\/S0025-5718-1981-0606506-0","article-title":"Discontinuous Galerkin methods for ordinary differential equations","volume":"36","author":"Delfour","year":"1981","journal-title":"Math. Comp."},{"key":"10.1016\/j.cam.2026.117819_bib0022","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/s100920070002","article-title":"An hp a-priori error analysis of the DG time-stepping method for initial value problems","volume":"37","author":"Sch\u00f6tzau","year":"2000","journal-title":"Calcolo"},{"key":"10.1016\/j.cam.2026.117819_bib0023","doi-asserted-by":"crossref","first-page":"1455","DOI":"10.1093\/imanum\/drae036","article-title":"hp-version C1-continuous Petrov-Galerkin method for nonlinear second-order initial value problems with application to wave equations","volume":"45","author":"Wang","year":"2025","journal-title":"IMA J. Numer. Anal."},{"key":"10.1016\/j.cam.2026.117819_bib0024","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/s10444-020-09800-3","article-title":"An hp-version of the c-continuous Galerkin time-stepping method for nonlinear second-order initial value problems","volume":"46","author":"Wei","year":"2020","journal-title":"Adv. Comput. Math."},{"key":"10.1016\/j.cam.2026.117819_bib0025","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1007\/s10915-004-4796-2","article-title":"An a priori error analysis of the hp-version of the continuous Galerkin FEM for nonlinear initial value problems","volume":"25","author":"Wihler","year":"2005","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.cam.2026.117819_bib0026","doi-asserted-by":"crossref","first-page":"301","DOI":"10.4208\/eajam.310315.070815a","article-title":"An L\u221e-error estimate for the h-p version continuous Petrov-Galerkin method for nonlinear initial value problems","volume":"5","author":"Yi","year":"2015","journal-title":"East Asian J. Appl. Math."},{"key":"10.1016\/j.cam.2026.117819_bib0027","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1051\/m2an:2004013","article-title":"Galerkin time-stepping methods for nonlinear parabolic equations","volume":"38","author":"Akrivis","year":"2004","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"10.1016\/j.cam.2026.117819_bib0028","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1090\/S0025-5718-1989-0983310-2","article-title":"Continuous finite elements in space and time for the heat equation","volume":"52","author":"Aziz","year":"1989","journal-title":"Math. Comp."},{"issue":"88","key":"10.1016\/j.cam.2026.117819_bib0029","first-page":"31","article-title":"Space-time continuous Galerkin discretization of linear parabolic equations: error profile and postprocessing","volume":"104","author":"Zhang","year":"2025","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.cam.2026.117819_bib0030","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1051\/m2an\/1985190406111","article-title":"Time discretization of parabolic problems by the discontinuous Galerkin method","volume":"19","author":"Eriksson","year":"1985","journal-title":"RAIRO Mod\u00e9l. Math. Anal. Num\u00e9r."},{"key":"10.1016\/j.cam.2026.117819_bib0031","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1137\/0715059","article-title":"Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain","volume":"15","author":"Jamet","year":"1978","journal-title":"SIAM J. Numer. Anal."},{"key":"10.1016\/j.cam.2026.117819_bib0032","series-title":"Galerkin Finite Element Methods for Parabolic Problems","author":"Thom\u00e9e","year":"2006"},{"key":"10.1016\/j.cam.2026.117819_bib0033","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1137\/S0036142999352394","article-title":"Time discretization of parabolic problems by the hp-version of the discontinuous Galerkin finite element method","volume":"38","author":"Sch\u00f6tzau","year":"2000","journal-title":"SIAM J. Numer. Anal."},{"key":"10.1016\/j.cam.2026.117819_bib0034","doi-asserted-by":"crossref","first-page":"6685","DOI":"10.1016\/S0045-7825(01)00258-4","article-title":"hp-discontinuous Galerkin time stepping for parabolic problems","volume":"190","author":"Werder","year":"2001","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"10.1016\/j.cam.2026.117819_bib0035","doi-asserted-by":"crossref","first-page":"A2073","DOI":"10.1137\/23M1572787","article-title":"Efficient and parallel solution of high-order continuous time Galerkin for dissipative and wave propagation problems","volume":"46","author":"chen","year":"2024","journal-title":"SIAM J. Sci. Comput."},{"key":"10.1016\/j.cam.2026.117819_bib0036","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-75934-0","article-title":"The mathematical theory of finite element methods","author":"Brenner","year":"2008"},{"key":"10.1016\/j.cam.2026.117819_bib0037","article-title":"The finite element method for elliptic problems","volume":"vol. 4","author":"Ciarlet","year":"1978"},{"key":"10.1016\/j.cam.2026.117819_bib0038","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1007\/BF01395885","article-title":"The h-p version of the finite element method for elliptic equations of order 2m","volume":"53","author":"Guo","year":"1988","journal-title":"Numer. Math."},{"key":"10.1016\/j.cam.2026.117819_bib0039","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1051\/m2an\/1990240202651","article-title":"The p-version of the finite element method for elliptic equations of order 2l","volume":"24","author":"Suri","year":"1990","journal-title":"RAIRO Mod\u00e9l. Math. Anal. Num\u00e9r."},{"issue":"30","key":"10.1016\/j.cam.2026.117819_bib0040","first-page":"32","article-title":"hp-optimal interior penalty discontinuous Galerkin methods for the biharmonic problem","volume":"96","author":"Dong","year":"2023","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.cam.2026.117819_bib0041","doi-asserted-by":"crossref","first-page":"2677","DOI":"10.1137\/15M1006489","article-title":"An h-p version of the continuous Petrov-Galerkin finite element method for Volterra integro-differential equations with smooth and nonsmooth kernels","volume":"53","author":"Yi","year":"2015","journal-title":"SIAM J. Numer. Anal."},{"key":"10.1016\/j.cam.2026.117819_bib0042","doi-asserted-by":"crossref","first-page":"1190","DOI":"10.1093\/imrn\/rnv156","article-title":"One-dimensional interpolation inequalities, Carlson-Landau inequalities, and magnetic schr\u00f6dinger operators","author":"Ilyin","year":"2016","journal-title":"Int. Math. Res. Not."}],"container-title":["Journal of Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0377042726004619?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0377042726004619?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T17:16:02Z","timestamp":1783185362000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0377042726004619"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,12]]},"references-count":42,"alternative-id":["S0377042726004619"],"URL":"https:\/\/doi.org\/10.1016\/j.cam.2026.117819","relation":{},"ISSN":["0377-0427"],"issn-type":[{"value":"0377-0427","type":"print"}],"subject":[],"published":{"date-parts":[[2026,12]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"hp-version space-time Galerkin discretization for fourth-order linear parabolic equations","name":"articletitle","label":"Article Title"},{"value":"Journal of Computational and Applied Mathematics","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cam.2026.117819","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"117819"}}