{"id":"https://openalex.org/W4415310464","doi":"https://doi.org/10.5802/jcm.142","title":"On the efficiency of a posteriori error estimators for parabolic partial differential equations in the energy norm","display_name":"On the efficiency of a posteriori error estimators for parabolic partial differential equations in the energy norm","publication_year":2026,"publication_date":"2026-03-02","ids":{"openalex":"https://openalex.org/W4415310464","doi":"https://doi.org/10.5802/jcm.142"},"language":"en","primary_location":{"id":"doi:10.5802/jcm.142","doi":"https://doi.org/10.5802/jcm.142","is_oa":true,"landing_page_url":"https://doi.org/10.5802/jcm.142","pdf_url":"https://smai-jcm.centre-mersenne.org/item/10.5802/jcm.142.pdf","source":{"id":"https://openalex.org/S4210224905","display_name":"SMAI Journal of Computational Mathematics","issn_l":"2426-8399","issn":["2426-8399"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","jufo-1","norway-2","poland-20","russia-white-list-2"],"host_organization":"https://openalex.org/P4361727469","host_organization_name":"Cellule MathDoc/CEDRAM","host_organization_lineage":["https://openalex.org/P4361727469","https://openalex.org/P4310311411"],"host_organization_lineage_names":["Cellule MathDoc/CEDRAM","National Center for Scientific Research"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The SMAI Journal of computational mathematics","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["arxiv","crossref","datacite","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://smai-jcm.centre-mersenne.org/item/10.5802/jcm.142.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5003137738","display_name":"Iain Smears","orcid":"https://orcid.org/0000-0003-0005-4327","observed_orcids":["https://orcid.org/0000-0003-0005-4327"]},"institutions":[{"id":"https://openalex.org/I45129253","display_name":"University College London","ror":"https://ror.org/02jx3x895","country_code":"GB","type":"education","lineage":["https://openalex.org/I124357947","https://openalex.org/I45129253"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Iain Smears","raw_affiliation_strings":["Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom","institution_ids":["https://openalex.org/I45129253"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5003137738"],"corresponding_institution_ids":["https://openalex.org/I45129253"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.01545644,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"12","issue":null,"first_page":"27","last_page":"42"},"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":0.9984999895095825,"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":0.9984999895095825,"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.00019999999494757503,"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"}},{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":9.999999747378752e-05,"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/a-posteriori-error-estimators","display_name":"a posteriori error estimators","score":0.9100000262260437},{"id":"https://openalex.org/keywords/efficiency","display_name":"efficiency","score":0.6959999799728394},{"id":"https://openalex.org/keywords/parabolic-partial-differential-equations","display_name":"parabolic equations","score":0.9829999804496765},{"id":"https://openalex.org/keywords/heat-equation","display_name":"heat equation","score":0.9860000014305115},{"id":"https://openalex.org/keywords/backward-euler-method","display_name":"backward Euler method","score":0.8849999904632568},{"id":"https://openalex.org/keywords/conforming-finite-elements","display_name":"conforming finite elements","score":0.9100000262260437},{"id":"https://openalex.org/keywords/energy-norm","display_name":"energy norm","score":0.996999979019165},{"id":"https://openalex.org/keywords/numerical-solution","display_name":"numerical solution","score":0.546999990940094}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8197000026702881},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.7821999788284302},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.7095000147819519},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.618399977684021},{"id":"https://openalex.org/C768646","wikidata":"https://www.wikidata.org/wiki/Q2736820","display_name":"Backward Euler method","level":3,"score":0.6176000237464905},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.6157000064849854},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.5928999781608582},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.5702999830245972},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.5669000148773193},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.5598999857902527},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.41260001063346863},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.3926999866962433},{"id":"https://openalex.org/C202787564","wikidata":"https://www.wikidata.org/wiki/Q6510488","display_name":"Heat equation","level":2,"score":0.39070001244544983},{"id":"https://openalex.org/C75380026","wikidata":"https://www.wikidata.org/wiki/Q868454","display_name":"Euler method","level":3,"score":0.37630000710487366},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.34439998865127563},{"id":"https://openalex.org/C186867907","wikidata":"https://www.wikidata.org/wiki/Q1575416","display_name":"Parabolic partial differential equation","level":3,"score":0.3208000063896179},{"id":"https://openalex.org/C38409319","wikidata":"https://www.wikidata.org/wiki/Q375175","display_name":"Euler equations","level":2,"score":0.31709998846054077},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.31470000743865967},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3100999891757965},{"id":"https://openalex.org/C3018824978","wikidata":"https://www.wikidata.org/wiki/Q2894891","display_name":"Error analysis","level":2,"score":0.29829999804496765},{"id":"https://openalex.org/C2986878899","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical approximation","level":3,"score":0.2912999987602234},{"id":"https://openalex.org/C17095337","wikidata":"https://www.wikidata.org/wiki/Q2375229","display_name":"Piecewise linear function","level":2,"score":0.26089999079704285},{"id":"https://openalex.org/C53846429","wikidata":"https://www.wikidata.org/wiki/Q186475","display_name":"Partial derivative","level":2,"score":0.2549999952316284}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.5802/jcm.142","doi":"https://doi.org/10.5802/jcm.142","is_oa":true,"landing_page_url":"https://doi.org/10.5802/jcm.142","pdf_url":"https://smai-jcm.centre-mersenne.org/item/10.5802/jcm.142.pdf","source":{"id":"https://openalex.org/S4210224905","display_name":"SMAI Journal of Computational Mathematics","issn_l":"2426-8399","issn":["2426-8399"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","jufo-1","norway-2","poland-20","russia-white-list-2"],"host_organization":"https://openalex.org/P4361727469","host_organization_name":"Cellule MathDoc/CEDRAM","host_organization_lineage":["https://openalex.org/P4361727469","https://openalex.org/P4310311411"],"host_organization_lineage_names":["Cellule MathDoc/CEDRAM","National Center for Scientific Research"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The SMAI Journal of computational mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2507.13188","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2507.13188","pdf_url":"https://arxiv.org/pdf/2507.13188","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:eprints.ucl.ac.uk.OAI2:10219545","doi":null,"is_oa":true,"landing_page_url":"https://discovery.ucl.ac.uk/id/eprint/10219545/","pdf_url":"https://discovery.ucl.ac.uk/10219545/1/energy_norm_estimator_rps.pdf","source":{"id":"https://openalex.org/S4306400024","display_name":"UCL Discovery (University College London)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I45129253","host_organization_name":"University College London","host_organization_lineage":["https://openalex.org/I45129253"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SMAI Journal of Computational Mathematics (2026) (In press).","raw_type":"Article"},{"id":"doi:10.48550/arxiv.2507.13188","doi":"https://doi.org/10.48550/arxiv.2507.13188","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2507.13188","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5802/jcm.142","doi":"https://doi.org/10.5802/jcm.142","is_oa":true,"landing_page_url":"https://doi.org/10.5802/jcm.142","pdf_url":"https://smai-jcm.centre-mersenne.org/item/10.5802/jcm.142.pdf","source":{"id":"https://openalex.org/S4210224905","display_name":"SMAI Journal of Computational Mathematics","issn_l":"2426-8399","issn":["2426-8399"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","jufo-1","norway-2","poland-20","russia-white-list-2"],"host_organization":"https://openalex.org/P4361727469","host_organization_name":"Cellule MathDoc/CEDRAM","host_organization_lineage":["https://openalex.org/P4361727469","https://openalex.org/P4310311411"],"host_organization_lineage_names":["Cellule MathDoc/CEDRAM","National Center for Scientific Research"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The SMAI Journal of computational mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4415310464.pdf","grobid_xml":"https://content.openalex.org/works/W4415310464.grobid-xml"},"referenced_works_count":26,"referenced_works":["https://openalex.org/W100357283","https://openalex.org/W1534061075","https://openalex.org/W1855169650","https://openalex.org/W1973664641","https://openalex.org/W1978399516","https://openalex.org/W1985988195","https://openalex.org/W2006443697","https://openalex.org/W2015788535","https://openalex.org/W2016012726","https://openalex.org/W2025163266","https://openalex.org/W2037701755","https://openalex.org/W2038626747","https://openalex.org/W2064416388","https://openalex.org/W2067951086","https://openalex.org/W2079138830","https://openalex.org/W2093049475","https://openalex.org/W2157441081","https://openalex.org/W2300734653","https://openalex.org/W2331190391","https://openalex.org/W2429258439","https://openalex.org/W2463821656","https://openalex.org/W2528152073","https://openalex.org/W2539693652","https://openalex.org/W2562668947","https://openalex.org/W2599521302","https://openalex.org/W2963044644"],"related_works":[],"abstract_inverted_index":{"For":[0],"the":[1,5,26,36,40,44,48,51,63,66,74,81],"model":[2],"problem":[3],"of":[4,28,39,65,76,83,85],"heat":[6],"equation":[7],"discretized":[8],"by":[9],"an":[10],"implicit":[11],"Euler":[12],"method":[13,21],"in":[14,22],"time":[15],"and":[16,56],"a":[17,29],"conforming":[18],"finite":[19],"element":[20],"space,":[23],"we":[24],"prove":[25],"efficiency":[27,64],"posteriori":[30],"error":[31],"estimators":[32,67],"with":[33],"respect":[34],"to":[35],"energy":[37],"norm":[38],"error,":[41],"when":[42],"considering":[43],"numerical":[45,86],"solution":[46],"as":[47],"average":[49],"between":[50],"usual":[52],"continuous":[53],"piecewise":[54,57],"affine-in-time":[55],"constant-in-time":[58],"reconstructions.":[59],"This":[60],"illustrates":[61],"how":[62],"is":[68],"not":[69],"only":[70],"possibly":[71],"dependent":[72],"on":[73,80],"choice":[75,82],"norm,":[77],"but":[78],"also":[79],"notion":[84],"solution.":[87]},"counts_by_year":[],"updated_date":"2026-10-14T16:39:01.820549","created_date":"2025-10-18T00:00:00"}
