{"id":"https://openalex.org/W2045354979","doi":"https://doi.org/10.1137/s0036142902397090","title":"A Posteriori Error Estimates for Discontinuous Galerkin Time-Stepping Method for Optimal Control Problems Governed by Parabolic Equations","display_name":"A Posteriori Error Estimates for Discontinuous Galerkin Time-Stepping Method for Optimal Control Problems Governed by Parabolic Equations","publication_year":2004,"publication_date":"2004-01-01","ids":{"openalex":"https://openalex.org/W2045354979","doi":"https://doi.org/10.1137/s0036142902397090","mag":"2045354979"},"language":"en","primary_location":{"id":"doi:10.1137/s0036142902397090","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036142902397090","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/A5100373261","display_name":"Wenbin Liu","orcid":"https://orcid.org/0000-0002-9469-7499"},"institutions":[{"id":"https://openalex.org/I20581793","display_name":"University of Kent","ror":"https://ror.org/00xkeyj56","country_code":"GB","type":"education","lineage":["https://openalex.org/I20581793"]},{"id":"https://openalex.org/I4610292","display_name":"Xiangtan University","ror":"https://ror.org/00xsfaz62","country_code":"CN","type":"education","lineage":["https://openalex.org/I4610292"]}],"countries":["CN","GB"],"is_corresponding":false,"raw_author_name":"Wenbin Liu","raw_affiliation_strings":["University of Kent","XiangTan University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Kent","institution_ids":["https://openalex.org/I20581793"]},{"raw_affiliation_string":"XiangTan University","institution_ids":["https://openalex.org/I4610292"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5087806562","display_name":"Heping Ma","orcid":"https://orcid.org/0000-0002-8891-7635"},"institutions":[{"id":"https://openalex.org/I113940042","display_name":"Shanghai University","ror":"https://ror.org/006teas31","country_code":"CN","type":"education","lineage":["https://openalex.org/I113940042"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Heping Ma","raw_affiliation_strings":["Shanghai University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shanghai University","institution_ids":["https://openalex.org/I113940042"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5058967726","display_name":"Tao Tang","orcid":"https://orcid.org/0000-0003-2919-2774"},"institutions":[{"id":"https://openalex.org/I530967","display_name":"Toronto Metropolitan University","ror":"https://ror.org/05g13zd79","country_code":"CA","type":"education","lineage":["https://openalex.org/I530967"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Tao Tang","raw_affiliation_strings":["Department of Mathematics"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics","institution_ids":["https://openalex.org/I530967"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5053053206","display_name":"Ningning Yan","orcid":"https://orcid.org/0000-0003-3072-7262"},"institutions":[{"id":"https://openalex.org/I4210120485","display_name":"Academy of Mathematics and Systems Science","ror":"https://ror.org/02jkmyk67","country_code":"CN","type":"facility","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210120485"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ningning Yan","raw_affiliation_strings":["CAS - Institute of Systems Science Beijing"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"CAS - Institute of Systems Science Beijing","institution_ids":["https://openalex.org/I4210120485"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.4029,"has_fulltext":false,"cited_by_count":73,"citation_normalized_percentile":{"value":0.63685571,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"42","issue":"3","first_page":"1032","last_page":"1061"},"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9993000030517578,"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"}},{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9789999723434448,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/discontinuous-galerkin-method","display_name":"Discontinuous Galerkin method","score":0.836676836013794},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.7960555553436279},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7909103631973267},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.7537781000137329},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.6333892345428467},{"id":"https://openalex.org/keywords/a-priori-and-a-posteriori","display_name":"A priori and a posteriori","score":0.6093137264251709},{"id":"https://openalex.org/keywords/optimal-control","display_name":"Optimal control","score":0.5798419117927551},{"id":"https://openalex.org/keywords/parabolic-partial-differential-equation","display_name":"Parabolic partial differential equation","score":0.5609804391860962},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.48870256543159485},{"id":"https://openalex.org/keywords/obstacle-problem","display_name":"Obstacle problem","score":0.4686027765274048},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.45882725715637207},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.3338358998298645},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3284974694252014},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.16881772875785828}],"concepts":[{"id":"https://openalex.org/C92244383","wikidata":"https://www.wikidata.org/wiki/Q428273","display_name":"Discontinuous Galerkin method","level":3,"score":0.836676836013794},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.7960555553436279},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7909103631973267},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.7537781000137329},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.6333892345428467},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.6093137264251709},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.5798419117927551},{"id":"https://openalex.org/C186867907","wikidata":"https://www.wikidata.org/wiki/Q1575416","display_name":"Parabolic partial differential equation","level":3,"score":0.5609804391860962},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.48870256543159485},{"id":"https://openalex.org/C132369183","wikidata":"https://www.wikidata.org/wiki/Q3406250","display_name":"Obstacle problem","level":3,"score":0.4686027765274048},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.45882725715637207},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.3338358998298645},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3284974694252014},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.16881772875785828},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"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/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/s0036142902397090","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036142902397090","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:8501","is_oa":false,"landing_page_url":"https://kar.kent.ac.uk/8501/","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":45,"referenced_works":["https://openalex.org/W89839489","https://openalex.org/W126240089","https://openalex.org/W132080054","https://openalex.org/W144579591","https://openalex.org/W594510519","https://openalex.org/W1188154435","https://openalex.org/W1534828741","https://openalex.org/W1543766063","https://openalex.org/W1588974174","https://openalex.org/W1597209545","https://openalex.org/W1968702788","https://openalex.org/W1974032925","https://openalex.org/W1976468712","https://openalex.org/W1976565091","https://openalex.org/W1984479582","https://openalex.org/W1984777675","https://openalex.org/W1986879407","https://openalex.org/W1994336408","https://openalex.org/W1995404822","https://openalex.org/W1996778903","https://openalex.org/W2000334666","https://openalex.org/W2014303698","https://openalex.org/W2017955504","https://openalex.org/W2022977680","https://openalex.org/W2030580548","https://openalex.org/W2038626747","https://openalex.org/W2041523977","https://openalex.org/W2048341997","https://openalex.org/W2053632598","https://openalex.org/W2059709516","https://openalex.org/W2069454819","https://openalex.org/W2071036364","https://openalex.org/W2092796751","https://openalex.org/W2093581986","https://openalex.org/W2099279998","https://openalex.org/W2121623735","https://openalex.org/W2153328986","https://openalex.org/W2160520936","https://openalex.org/W2429258439","https://openalex.org/W2508983562","https://openalex.org/W4205634890","https://openalex.org/W4230954636","https://openalex.org/W4233755870","https://openalex.org/W4252060206","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W623557396","https://openalex.org/W2120874885","https://openalex.org/W2162026849","https://openalex.org/W2394687667","https://openalex.org/W2040975642","https://openalex.org/W3083171925","https://openalex.org/W2147612446","https://openalex.org/W3039794337","https://openalex.org/W2386706262","https://openalex.org/W2096445841"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3],"examine":[4],"the":[5,24,32,36,44,55,58,64,81],"discontinuous":[6,25],"Galerkin":[7],"(DG)":[8],"finite":[9,26,38],"element":[10,27,39],"approximation":[11],"to":[12],"convex":[13],"distributed":[14],"optimal":[15],"control":[16,59,74],"problems":[17,72],"governed":[18],"by":[19],"linear":[20],"parabolic":[21],"equations,":[22],"where":[23],"method":[28,40],"is":[29,41,69],"used":[30,42],"for":[31,43,53],"time":[33],"discretization":[34],"and":[35,57],"conforming":[37],"space":[45,68],"discretization.":[46],"We":[47],"derive":[48],"a":[49],"posteriori":[50],"error":[51,90],"estimates":[52,91],"both":[54],"state":[56],"approximation,":[60],"assuming":[61],"only":[62],"that":[63],"underlying":[65],"mesh":[66],"in":[67,86],"nondegenerate.":[70],"For":[71],"with":[73],"constraints":[75],"of":[76],"obstacle":[77],"type,":[78],"which":[79],"are":[80,92],"kind":[82],"most":[83],"frequently":[84],"met":[85],"applications,":[87],"further":[88],"improved":[89],"obtained.":[93]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":5},{"year":2015,"cited_by_count":9},{"year":2014,"cited_by_count":6},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":3}],"updated_date":"2026-07-27T08:26:11.824852","created_date":"2025-10-10T00:00:00"}