{"id":"https://openalex.org/W2046660062","doi":"https://doi.org/10.1137/080726057","title":"A Legendre\u2013Galerkin Spectral Method for Optimal Control Problems Governed by Stokes Equations","display_name":"A Legendre\u2013Galerkin Spectral Method for Optimal Control Problems Governed by Stokes Equations","publication_year":2011,"publication_date":"2011-01-01","ids":{"openalex":"https://openalex.org/W2046660062","doi":"https://doi.org/10.1137/080726057","mag":"2046660062"},"language":"en","primary_location":{"id":"doi:10.1137/080726057","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080726057","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/A5038297858","display_name":"Yanping Chen","orcid":"https://orcid.org/0000-0002-7765-8201"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yanping Chen","raw_affiliation_strings":["yanpingchen@scnu.edu.cn#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"yanpingchen@scnu.edu.cn#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5047823800","display_name":"Fenglin Huang","orcid":"https://orcid.org/0000-0001-8108-8822"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Fenglin Huang","raw_affiliation_strings":["hfl_937@sina.com and yinianyu365109@126.com#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"hfl_937@sina.com and yinianyu365109@126.com#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5020948241","display_name":"Nianyu Yi","orcid":"https://orcid.org/0000-0002-2106-8237"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Nianyu Yi","raw_affiliation_strings":["yinianyu365109@126.com#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"yinianyu365109@126.com#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100373244","display_name":"Wenbin Liu","orcid":"https://orcid.org/0000-0001-6237-5447"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wenbin Liu","raw_affiliation_strings":["W.B.Liu@ukc.ac.uk#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"W.B.Liu@ukc.ac.uk#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":8.456,"has_fulltext":false,"cited_by_count":95,"citation_normalized_percentile":{"value":0.98139472,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"49","issue":"4","first_page":"1625","last_page":"1648"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9998999834060669,"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"}},"topics":[{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9998999834060669,"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/T10173","display_name":"Computational Fluid Dynamics and Aerodynamics","score":0.9988999962806702,"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9987000226974487,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.8604533672332764},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8570550084114075},{"id":"https://openalex.org/keywords/legendre-polynomials","display_name":"Legendre polynomials","score":0.7954856157302856},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.7370107173919678},{"id":"https://openalex.org/keywords/a-priori-and-a-posteriori","display_name":"A priori and a posteriori","score":0.6782851219177246},{"id":"https://openalex.org/keywords/optimal-control","display_name":"Optimal control","score":0.6248583793640137},{"id":"https://openalex.org/keywords/discontinuous-galerkin-method","display_name":"Discontinuous Galerkin method","score":0.5065871477127075},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.49611029028892517},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4367285668849945},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.43539202213287354},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.2967153787612915},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.2840706706047058}],"concepts":[{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.8604533672332764},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8570550084114075},{"id":"https://openalex.org/C111458787","wikidata":"https://www.wikidata.org/wiki/Q215405","display_name":"Legendre polynomials","level":2,"score":0.7954856157302856},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.7370107173919678},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.6782851219177246},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.6248583793640137},{"id":"https://openalex.org/C92244383","wikidata":"https://www.wikidata.org/wiki/Q428273","display_name":"Discontinuous Galerkin method","level":3,"score":0.5065871477127075},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.49611029028892517},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4367285668849945},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.43539202213287354},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.2967153787612915},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2840706706047058},{"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/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/080726057","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080726057","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"},{"id":"pmh:oai:kar.kent.ac.uk:28219","is_oa":false,"landing_page_url":"https://kar.kent.ac.uk/28219/","pdf_url":null,"source":{"id":"https://openalex.org/S4377196264","display_name":"Kent Academic Repository (University of Kent)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I20581793","host_organization_name":"University of Kent","host_organization_lineage":["https://openalex.org/I20581793"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":41,"referenced_works":["https://openalex.org/W94458352","https://openalex.org/W201429281","https://openalex.org/W1543766063","https://openalex.org/W1549220217","https://openalex.org/W1912930600","https://openalex.org/W1965880205","https://openalex.org/W1968702788","https://openalex.org/W1974032925","https://openalex.org/W1978282183","https://openalex.org/W1984815757","https://openalex.org/W1987244070","https://openalex.org/W1995404822","https://openalex.org/W1997701847","https://openalex.org/W1997999015","https://openalex.org/W1998697817","https://openalex.org/W2006850990","https://openalex.org/W2009985306","https://openalex.org/W2014303698","https://openalex.org/W2018383942","https://openalex.org/W2022977680","https://openalex.org/W2029123910","https://openalex.org/W2030028670","https://openalex.org/W2039146619","https://openalex.org/W2041523977","https://openalex.org/W2046628455","https://openalex.org/W2047352485","https://openalex.org/W2053632598","https://openalex.org/W2072625513","https://openalex.org/W2077931396","https://openalex.org/W2082849992","https://openalex.org/W2089984340","https://openalex.org/W2093581986","https://openalex.org/W2094464929","https://openalex.org/W2131778920","https://openalex.org/W2135572939","https://openalex.org/W2146598639","https://openalex.org/W2199169705","https://openalex.org/W2410898212","https://openalex.org/W3032164133","https://openalex.org/W3084022122","https://openalex.org/W3150423993"],"related_works":["https://openalex.org/W2736737644","https://openalex.org/W3106446297","https://openalex.org/W2385225694","https://openalex.org/W2120874885","https://openalex.org/W2790191500","https://openalex.org/W2162026849","https://openalex.org/W4229335327","https://openalex.org/W3084022122","https://openalex.org/W2394687667","https://openalex.org/W2040975642"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3],"study":[4],"the":[5,21,26,39,42,49,71,75,94,102,106,118],"Legendre\u2013Galerkin":[6,72],"spectral":[7],"approximation":[8,73],"of":[9,35,48,74,96,105],"distributed":[10],"optimal":[11,107],"control":[12,23,77,91],"problems":[13,24,92],"governed":[14],"by":[15,30],"Stokes":[16],"equations.":[17],"We":[18,57],"show":[19],"that":[20],"discretized":[22],"satisfy":[25],"well-known":[27],"Babu\u0161ka\u2013Brezzi":[28],"conditions":[29],"choosing":[31],"an":[32,97],"appropriate":[33],"pair":[34],"discretization":[36,50],"spaces":[37,51],"for":[38,70,90],"velocity":[40],"and":[41,67,83],"pressure.":[43],"Constructing":[44],"suitable":[45],"base":[46],"functions":[47],"leads":[52],"to":[53,101,116],"sparse":[54],"coefficient":[55],"matrices.":[56],"first":[58],"derive":[59],"a":[60,81,84],"priori":[61,82],"error":[62,86,119],"estimates":[63,87],"in":[64],"both":[65,80],"$H^1$":[66],"$L^2$":[68],"norms":[69],"unconstrained":[76],"problems.":[78],"Then":[79],"posteriori":[85],"are":[88,114],"obtained":[89],"with":[93],"constraints":[95],"integral":[98],"type,":[99],"thanks":[100],"higher":[103],"regularity":[104],"control.":[108],"Finally,":[109],"some":[110],"illustrative":[111],"numerical":[112],"examples":[113],"presented":[115],"demonstrate":[117],"estimates.":[120]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":8},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":5},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":9},{"year":2015,"cited_by_count":11},{"year":2014,"cited_by_count":8},{"year":2013,"cited_by_count":10},{"year":2012,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}