{"id":"https://openalex.org/W4377088181","doi":"https://doi.org/10.1137/22m1486844","title":"Optimal \\(\\boldsymbol{{L^2}}\\) Error Estimates of Unconditionally Stable Finite Element Schemes for the Cahn\u2013Hilliard\u2013Navier\u2013Stokes System","display_name":"Optimal \\(\\boldsymbol{{L^2}}\\) Error Estimates of Unconditionally Stable Finite Element Schemes for the Cahn\u2013Hilliard\u2013Navier\u2013Stokes System","publication_year":2023,"publication_date":"2023-05-19","ids":{"openalex":"https://openalex.org/W4377088181","doi":"https://doi.org/10.1137/22m1486844"},"language":"en","primary_location":{"id":"doi:10.1137/22m1486844","is_oa":false,"landing_page_url":"https://doi.org/10.1137/22m1486844","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5088867146","display_name":"Wentao Cai","orcid":"https://orcid.org/0000-0003-2393-8213"},"institutions":[{"id":"https://openalex.org/I4210139958","display_name":"Beijing Computational Science Research Center","ror":"https://ror.org/04tavf782","country_code":"CN","type":"facility","lineage":["https://openalex.org/I4210139958"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Wentao Cai","raw_affiliation_strings":["Beijing Computational Science Research Center, Beijing 100193, China"],"affiliations":[{"raw_affiliation_string":"Beijing Computational Science Research Center, Beijing 100193, China","institution_ids":["https://openalex.org/I4210139958"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014226512","display_name":"Weiwei Sun","orcid":"https://orcid.org/0000-0002-2079-4480"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Weiwei Sun","raw_affiliation_strings":["Research Center for Mathematics, Advanced Institute of Natural Sciences, Beijing Normal University, Zhuhai, 519087, China and Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science (IRADS), BNU-HKBU United International College, Zhuhai, 519087, China"],"affiliations":[{"raw_affiliation_string":"Research Center for Mathematics, Advanced Institute of Natural Sciences, Beijing Normal University, Zhuhai, 519087, China and Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science (IRADS), BNU-HKBU United International College, Zhuhai, 519087, China","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5024180528","display_name":"Jilu Wang","orcid":"https://orcid.org/0009-0001-9468-249X"},"institutions":[{"id":"https://openalex.org/I204983213","display_name":"Harbin Institute of Technology","ror":"https://ror.org/01yqg2h08","country_code":"CN","type":"education","lineage":["https://openalex.org/I204983213"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jilu Wang","raw_affiliation_strings":["School of Science, Harbin Institute of Technology, Shenzhen 518055, China"],"affiliations":[{"raw_affiliation_string":"School of Science, Harbin Institute of Technology, Shenzhen 518055, China","institution_ids":["https://openalex.org/I204983213"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5058611661","display_name":"Zongze Yang","orcid":"https://orcid.org/0000-0002-2125-6932"},"institutions":[{"id":"https://openalex.org/I14243506","display_name":"Hong Kong Polytechnic University","ror":"https://ror.org/0030zas98","country_code":"HK","type":"education","lineage":["https://openalex.org/I14243506"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Zongze Yang","raw_affiliation_strings":["Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong","institution_ids":["https://openalex.org/I14243506"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5088867146"],"corresponding_institution_ids":["https://openalex.org/I4210139958"],"apc_list":null,"apc_paid":null,"fwci":2.6203,"has_fulltext":false,"cited_by_count":28,"citation_normalized_percentile":{"value":0.90586469,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"61","issue":"3","first_page":"1218","last_page":"1245"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9954000115394592,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10670","display_name":"Aluminum Alloy Microstructure Properties","score":0.9855999946594238,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7412063479423523},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.7247943878173828},{"id":"https://openalex.org/keywords/superconvergence","display_name":"Superconvergence","score":0.6953477263450623},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.6712067127227783},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5638217329978943},{"id":"https://openalex.org/keywords/cahn\u2013hilliard-equation","display_name":"Cahn\u2013Hilliard equation","score":0.5354955196380615},{"id":"https://openalex.org/keywords/error-analysis","display_name":"Error analysis","score":0.501288890838623},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.46953046321868896},{"id":"https://openalex.org/keywords/approximation-error","display_name":"Approximation error","score":0.4636897146701813},{"id":"https://openalex.org/keywords/interpolation","display_name":"Interpolation (computer graphics)","score":0.4531533718109131},{"id":"https://openalex.org/keywords/projection","display_name":"Projection (relational algebra)","score":0.4504304528236389},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.4250618815422058},{"id":"https://openalex.org/keywords/biharmonic-equation","display_name":"Biharmonic equation","score":0.4240800440311432},{"id":"https://openalex.org/keywords/navier\u2013stokes-equations","display_name":"Navier\u2013Stokes equations","score":0.4116281569004059},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.31625792384147644},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.1765459179878235},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.14228367805480957},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.13642346858978271},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.12293678522109985},{"id":"https://openalex.org/keywords/image","display_name":"Image (mathematics)","score":0.11494404077529907}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7412063479423523},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.7247943878173828},{"id":"https://openalex.org/C83295009","wikidata":"https://www.wikidata.org/wiki/Q7643190","display_name":"Superconvergence","level":3,"score":0.6953477263450623},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.6712067127227783},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5638217329978943},{"id":"https://openalex.org/C24822716","wikidata":"https://www.wikidata.org/wiki/Q5017440","display_name":"Cahn\u2013Hilliard equation","level":3,"score":0.5354955196380615},{"id":"https://openalex.org/C3018824978","wikidata":"https://www.wikidata.org/wiki/Q2894891","display_name":"Error analysis","level":2,"score":0.501288890838623},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.46953046321868896},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.4636897146701813},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.4531533718109131},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.4504304528236389},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.4250618815422058},{"id":"https://openalex.org/C145784156","wikidata":"https://www.wikidata.org/wiki/Q859808","display_name":"Biharmonic equation","level":3,"score":0.4240800440311432},{"id":"https://openalex.org/C2781278361","wikidata":"https://www.wikidata.org/wiki/Q201321","display_name":"Navier\u2013Stokes equations","level":3,"score":0.4116281569004059},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.31625792384147644},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.1765459179878235},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.14228367805480957},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.13642346858978271},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.12293678522109985},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.11494404077529907},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C84655787","wikidata":"https://www.wikidata.org/wiki/Q8067817","display_name":"Compressibility","level":2,"score":0.0},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/22m1486844","is_oa":false,"landing_page_url":"https://doi.org/10.1137/22m1486844","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1822535219","display_name":null,"funder_award_id":"11901142","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2797088476","display_name":null,"funder_award_id":"LY22A010019","funder_id":"https://openalex.org/F4320338464","funder_display_name":"Natural Science Foundation of Zhejiang Province"},{"id":"https://openalex.org/G6296395921","display_name":null,"funder_award_id":"12071040","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6945854197","display_name":null,"funder_award_id":"2021M700356","funder_id":"https://openalex.org/F4320321543","funder_display_name":"China Postdoctoral Science Foundation"},{"id":"https://openalex.org/G8476700986","display_name":null,"funder_award_id":"12071020","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321543","display_name":"China Postdoctoral Science Foundation","ror":"https://ror.org/0426zh255"},{"id":"https://openalex.org/F4320338464","display_name":"Natural Science Foundation of Zhejiang Province","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":38,"referenced_works":["https://openalex.org/W1566292686","https://openalex.org/W1976156829","https://openalex.org/W2010502651","https://openalex.org/W2011723842","https://openalex.org/W2017385247","https://openalex.org/W2025043986","https://openalex.org/W2029966914","https://openalex.org/W2042669961","https://openalex.org/W2046127134","https://openalex.org/W2047378168","https://openalex.org/W2048343906","https://openalex.org/W2049914863","https://openalex.org/W2051862019","https://openalex.org/W2053384289","https://openalex.org/W2066359652","https://openalex.org/W2077488436","https://openalex.org/W2078273688","https://openalex.org/W2083988343","https://openalex.org/W2084494000","https://openalex.org/W2128204029","https://openalex.org/W2151085908","https://openalex.org/W2206898513","https://openalex.org/W2340417499","https://openalex.org/W2497754766","https://openalex.org/W2500585602","https://openalex.org/W2594425100","https://openalex.org/W2607475371","https://openalex.org/W2766149883","https://openalex.org/W2962738836","https://openalex.org/W2962769912","https://openalex.org/W2963167146","https://openalex.org/W2963577202","https://openalex.org/W2964351848","https://openalex.org/W2991443752","https://openalex.org/W3047289271","https://openalex.org/W3118262253","https://openalex.org/W3134167459","https://openalex.org/W3204708751"],"related_works":["https://openalex.org/W2503543076","https://openalex.org/W2738351852","https://openalex.org/W2094577950","https://openalex.org/W2031157121","https://openalex.org/W1565666028","https://openalex.org/W2891538916","https://openalex.org/W2001758285","https://openalex.org/W2737775198","https://openalex.org/W2070923604","https://openalex.org/W2389506962"],"abstract_inverted_index":{".The":[0],"paper":[1],"is":[2,29,63,135,167],"concerned":[3],"with":[4],"the":[5,16,27,40,42,47,52,79,82,97,108,120,125,128,144,151],"analysis":[6,58,73,134],"of":[7,46,81,127,146],"a":[8,32],"popular":[9],"convex-splitting":[10,98],"finite":[11],"element":[12,190],"method":[13,28],"(FEM)":[14],"for":[15,54,59,96,116,150],"Cahn\u2013Hilliard\u2013Navier\u2013Stokes":[17],"system,":[18,41],"which":[19],"has":[20],"been":[21],"widely":[22],"used":[23],"in":[24,39,74,93,107,119],"practice.":[25],"Since":[26],"based":[30,136],"on":[31,137],"combined":[33,61],"approximation":[34,43],"to":[35,44,69,78,124,160,174,185],"multiple":[36],"variables":[37,48],"involved":[38],"one":[45],"may":[49,112],"seriously":[50],"affect":[51],"accuracy":[53],"others.":[55],"Optimal-order":[56],"error":[57,72,91,105,188,193],"such":[60],"approximations":[62],"challenging.":[64],"The":[65],"previous":[66],"works":[67],"failed":[68],"present":[70,88],"optimal":[71,90,104,187],"\\(L^2\\)":[75,94],"-norm":[76,95],"due":[77,123],"weakness":[80],"traditional":[83,109],"approach.":[84],"Here":[85],"we":[86],"first":[87],"an":[89],"estimate":[92],"FEMs.":[99],"We":[100],"also":[101,158],"show":[102],"that":[103,168],"estimates":[106,149],"(interpolation)":[110],"sense":[111],"not":[113],"always":[114],"hold":[115],"all":[117],"components":[118],"coupled":[121,181],"system":[122],"nature":[126],"pollution/influence":[129],"from":[130],"lower-order":[131],"approximations.":[132],"Our":[133],"two":[138],"newly":[139],"introduced":[140],"elliptic":[141],"quasi-projections":[142],"and":[143,178],"superconvergence":[145],"negative":[147],"norm":[148],"corresponding":[152],"projection":[153],"errors.":[154],"Numerical":[155],"examples":[156],"are":[157],"presented":[159],"illustrate":[161],"our":[162,169],"theoretical":[163],"results.":[164],"More":[165],"important":[166],"approach":[170],"can":[171],"be":[172],"extended":[173],"many":[175],"other":[176,179],"FEMs":[177],"strongly":[180],"phase":[182],"field":[183],"models":[184],"obtain":[186],"estimates.KeywordsCahn\u2013Hilliard\u2013Navier\u2013Stokesfinite":[189],"methodsRitz":[191],"quasi-projectionoptimal":[192],"estimatesunconditional":[194],"stabilityMSC":[195],"codes65M1265N3065M6035K55":[196]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":19},{"year":2024,"cited_by_count":7},{"year":2023,"cited_by_count":1}],"updated_date":"2026-04-22T08:38:42.863108","created_date":"2025-10-10T00:00:00"}
