{"id":"https://openalex.org/W4386492126","doi":"https://doi.org/10.1515/cmam-2024-0021","title":"Analysis of a Combined Spherical Harmonics and Discontinuous Galerkin Discretization for the Boltzmann Transport Equation","display_name":"Analysis of a Combined Spherical Harmonics and Discontinuous Galerkin Discretization for the Boltzmann Transport Equation","publication_year":2024,"publication_date":"2024-10-01","ids":{"openalex":"https://openalex.org/W4386492126","doi":"https://doi.org/10.1515/cmam-2024-0021"},"language":"en","primary_location":{"id":"doi:10.1515/cmam-2024-0021","is_oa":false,"landing_page_url":"https://doi.org/10.1515/cmam-2024-0021","pdf_url":null,"source":{"id":"https://openalex.org/S2765013998","display_name":"Computational Methods in Applied Mathematics","issn_l":"1609-4840","issn":["1609-4840","1609-9389"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310313990","host_organization_name":"De Gruyter","host_organization_lineage":["https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Methods in Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://hal.science/hal-04196435/document","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5025580221","display_name":"Kenneth Assogba","orcid":"https://orcid.org/0000-0002-0635-7508"},"institutions":[{"id":"https://openalex.org/I142476485","display_name":"\u00c9cole Polytechnique","ror":"https://ror.org/05hy3tk52","country_code":"FR","type":"education","lineage":["https://openalex.org/I142476485","https://openalex.org/I4210145102"]},{"id":"https://openalex.org/I2738703131","display_name":"Commissariat \u00e0 l'\u00c9nergie Atomique et aux \u00c9nergies Alternatives","ror":"https://ror.org/00jjx8s55","country_code":"FR","type":"government","lineage":["https://openalex.org/I2738703131"]},{"id":"https://openalex.org/I277688954","display_name":"Universit\u00e9 Paris-Saclay","ror":"https://ror.org/03xjwb503","country_code":"FR","type":"education","lineage":["https://openalex.org/I277688954"]},{"id":"https://openalex.org/I4210107641","display_name":"Centre de Math\u00e9matiques Appliqu\u00e9es de l'\u00c9cole polytechnique","ror":"https://ror.org/012e1xn46","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I142476485","https://openalex.org/I4210107641","https://openalex.org/I4210141950","https://openalex.org/I4210145102"]},{"id":"https://openalex.org/I4210128565","display_name":"CEA Paris-Saclay","ror":"https://ror.org/03n15ch10","country_code":"FR","type":"government","lineage":["https://openalex.org/I2738703131","https://openalex.org/I277688954","https://openalex.org/I4210128565"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Kenneth Assogba","raw_affiliation_strings":["Universit\u00e9 Paris-Saclay , CEA, Service d\u2019\u00c9tudes des R\u00e9acteurs et de Math\u00e9matiques Appliqu\u00e9es 91191 Gif-sur-Yvette , France","LLPR - Laboratoire de Logiciels pour la Physique des R\u00e9acteurs (France)","CMAP - Centre de Math\u00e9matiques Appliqu\u00e9es - Ecole Polytechnique (\u00c9cole Polytechnique Route de Saclay 91128 Palaiseau Cedex - France)"],"raw_orcid":"https://orcid.org/0000-0002-0635-7508","affiliations":[{"raw_affiliation_string":"Universit\u00e9 Paris-Saclay , CEA, Service d\u2019\u00c9tudes des R\u00e9acteurs et de Math\u00e9matiques Appliqu\u00e9es 91191 Gif-sur-Yvette , France","institution_ids":["https://openalex.org/I277688954","https://openalex.org/I2738703131","https://openalex.org/I4210128565"]},{"raw_affiliation_string":"LLPR - Laboratoire de Logiciels pour la Physique des R\u00e9acteurs (France)","institution_ids":[]},{"raw_affiliation_string":"CMAP - Centre de Math\u00e9matiques Appliqu\u00e9es - Ecole Polytechnique (\u00c9cole Polytechnique Route de Saclay 91128 Palaiseau Cedex - France)","institution_ids":["https://openalex.org/I4210107641","https://openalex.org/I142476485"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5042296753","display_name":"Gr\u00e9goire Allaire","orcid":"https://orcid.org/0000-0002-8213-1092"},"institutions":[{"id":"https://openalex.org/I142476485","display_name":"\u00c9cole Polytechnique","ror":"https://ror.org/05hy3tk52","country_code":"FR","type":"education","lineage":["https://openalex.org/I142476485","https://openalex.org/I4210145102"]},{"id":"https://openalex.org/I4210107641","display_name":"Centre de Math\u00e9matiques Appliqu\u00e9es de l'\u00c9cole polytechnique","ror":"https://ror.org/012e1xn46","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I142476485","https://openalex.org/I4210107641","https://openalex.org/I4210141950","https://openalex.org/I4210145102"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Gr\u00e9goire Allaire","raw_affiliation_strings":["CMAP , \u00c9cole polytechnique , 52830 Institut Polytechnique de Paris , 91120 Palaiseau , France","CMAP - Centre de Math\u00e9matiques Appliqu\u00e9es - Ecole Polytechnique (\u00c9cole Polytechnique Route de Saclay 91128 Palaiseau Cedex - France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"CMAP , \u00c9cole polytechnique , 52830 Institut Polytechnique de Paris , 91120 Palaiseau , France","institution_ids":["https://openalex.org/I142476485"]},{"raw_affiliation_string":"CMAP - Centre de Math\u00e9matiques Appliqu\u00e9es - Ecole Polytechnique (\u00c9cole Polytechnique Route de Saclay 91128 Palaiseau Cedex - France)","institution_ids":["https://openalex.org/I4210107641","https://openalex.org/I142476485"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5041886851","display_name":"Lahbib Bourhrara","orcid":"https://orcid.org/0009-0007-8607-2708"},"institutions":[{"id":"https://openalex.org/I2738703131","display_name":"Commissariat \u00e0 l'\u00c9nergie Atomique et aux \u00c9nergies Alternatives","ror":"https://ror.org/00jjx8s55","country_code":"FR","type":"government","lineage":["https://openalex.org/I2738703131"]},{"id":"https://openalex.org/I277688954","display_name":"Universit\u00e9 Paris-Saclay","ror":"https://ror.org/03xjwb503","country_code":"FR","type":"education","lineage":["https://openalex.org/I277688954"]},{"id":"https://openalex.org/I4210128565","display_name":"CEA Paris-Saclay","ror":"https://ror.org/03n15ch10","country_code":"FR","type":"government","lineage":["https://openalex.org/I2738703131","https://openalex.org/I277688954","https://openalex.org/I4210128565"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Lahbib Bourhrara","raw_affiliation_strings":["Universit\u00e9 Paris-Saclay , CEA, Service d\u2019\u00c9tudes des R\u00e9acteurs et de Math\u00e9matiques Appliqu\u00e9es 91191 Gif-sur-Yvette , France","LLPR - Laboratoire de Logiciels pour la Physique des R\u00e9acteurs (France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Universit\u00e9 Paris-Saclay , CEA, Service d\u2019\u00c9tudes des R\u00e9acteurs et de Math\u00e9matiques Appliqu\u00e9es 91191 Gif-sur-Yvette , France","institution_ids":["https://openalex.org/I277688954","https://openalex.org/I2738703131","https://openalex.org/I4210128565"]},{"raw_affiliation_string":"LLPR - Laboratoire de Logiciels pour la Physique des R\u00e9acteurs (France)","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5025580221"],"corresponding_institution_ids":["https://openalex.org/I142476485","https://openalex.org/I2738703131","https://openalex.org/I277688954","https://openalex.org/I4210107641","https://openalex.org/I4210128565"],"apc_list":null,"apc_paid":null,"fwci":0.7137,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.60542487,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":95,"max":96},"biblio":{"volume":"25","issue":"2","first_page":"287","last_page":"311"},"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":0.9998000264167786,"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":0.9998000264167786,"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":0.9990000128746033,"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/T11751","display_name":"Lattice Boltzmann Simulation Studies","score":0.9986000061035156,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/spherical-harmonics","display_name":"Spherical harmonics","score":0.8084816336631775},{"id":"https://openalex.org/keywords/boltzmann-equation","display_name":"Boltzmann equation","score":0.7665350437164307},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.705803632736206},{"id":"https://openalex.org/keywords/spin-weighted-spherical-harmonics","display_name":"Spin-weighted spherical harmonics","score":0.6412882804870605},{"id":"https://openalex.org/keywords/discontinuous-galerkin-method","display_name":"Discontinuous Galerkin method","score":0.6181176900863647},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.5682305097579956},{"id":"https://openalex.org/keywords/zonal-spherical-harmonics","display_name":"Zonal spherical harmonics","score":0.49079450964927673},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.45947614312171936},{"id":"https://openalex.org/keywords/boltzmann-constant","display_name":"Boltzmann constant","score":0.4423520565032959},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.42830508947372437},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.42227524518966675},{"id":"https://openalex.org/keywords/solid-harmonics","display_name":"Solid harmonics","score":0.41155725717544556},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.38693898916244507},{"id":"https://openalex.org/keywords/harmonics","display_name":"Harmonics","score":0.3849031329154968},{"id":"https://openalex.org/keywords/vector-spherical-harmonics","display_name":"Vector spherical harmonics","score":0.36721330881118774},{"id":"https://openalex.org/keywords/classical-mechanics","display_name":"Classical mechanics","score":0.346557080745697},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.14075806736946106},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.06596970558166504},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.06302720308303833}],"concepts":[{"id":"https://openalex.org/C3768446","wikidata":"https://www.wikidata.org/wiki/Q877100","display_name":"Spherical harmonics","level":2,"score":0.8084816336631775},{"id":"https://openalex.org/C165995430","wikidata":"https://www.wikidata.org/wiki/Q891653","display_name":"Boltzmann equation","level":2,"score":0.7665350437164307},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.705803632736206},{"id":"https://openalex.org/C20569695","wikidata":"https://www.wikidata.org/wiki/Q7577353","display_name":"Spin-weighted spherical harmonics","level":4,"score":0.6412882804870605},{"id":"https://openalex.org/C92244383","wikidata":"https://www.wikidata.org/wiki/Q428273","display_name":"Discontinuous Galerkin method","level":3,"score":0.6181176900863647},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.5682305097579956},{"id":"https://openalex.org/C39176672","wikidata":"https://www.wikidata.org/wiki/Q8073864","display_name":"Zonal spherical harmonics","level":4,"score":0.49079450964927673},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.45947614312171936},{"id":"https://openalex.org/C35304006","wikidata":"https://www.wikidata.org/wiki/Q5962","display_name":"Boltzmann constant","level":2,"score":0.4423520565032959},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.42830508947372437},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.42227524518966675},{"id":"https://openalex.org/C199056591","wikidata":"https://www.wikidata.org/wiki/Q7557904","display_name":"Solid harmonics","level":4,"score":0.41155725717544556},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38693898916244507},{"id":"https://openalex.org/C188414643","wikidata":"https://www.wikidata.org/wiki/Q3001183","display_name":"Harmonics","level":3,"score":0.3849031329154968},{"id":"https://openalex.org/C77237485","wikidata":"https://www.wikidata.org/wiki/Q7917842","display_name":"Vector spherical harmonics","level":3,"score":0.36721330881118774},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.346557080745697},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.14075806736946106},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.06596970558166504},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.06302720308303833},{"id":"https://openalex.org/C165801399","wikidata":"https://www.wikidata.org/wiki/Q25428","display_name":"Voltage","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1515/cmam-2024-0021","is_oa":false,"landing_page_url":"https://doi.org/10.1515/cmam-2024-0021","pdf_url":null,"source":{"id":"https://openalex.org/S2765013998","display_name":"Computational Methods in Applied Mathematics","issn_l":"1609-4840","issn":["1609-4840","1609-9389"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310313990","host_organization_name":"De Gruyter","host_organization_lineage":["https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Methods in Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:HAL:hal-04196435v1","is_oa":true,"landing_page_url":"https://hal.science/hal-04196435","pdf_url":"https://hal.science/hal-04196435/document","source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"2023","raw_type":"Preprints, Working Papers, ..."},{"id":"pmh:oai:HAL:hal-05004796v1","is_oa":false,"landing_page_url":"https://hal.science/hal-05004796","pdf_url":null,"source":{"id":"https://openalex.org/S4406922466","display_name":"SPIRE - Sciences Po Institutional REpository","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Computational Methods in Applied Mathematics, 2024, &#x27E8;10.1515/cmam-2024-0021&#x27E9;","raw_type":"Journal articles"}],"best_oa_location":{"id":"pmh:oai:HAL:hal-04196435v1","is_oa":true,"landing_page_url":"https://hal.science/hal-04196435","pdf_url":"https://hal.science/hal-04196435/document","source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"2023","raw_type":"Preprints, Working Papers, ..."},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2444547639","display_name":"International PhD programme in NUMERICal Simulation","funder_award_id":"800945","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G7842005466","display_name":null,"funder_award_id":"Horizon 2020","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"}],"funders":[{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4386492126.pdf","grobid_xml":"https://content.openalex.org/works/W4386492126.grobid-xml"},"referenced_works_count":38,"referenced_works":["https://openalex.org/W69485787","https://openalex.org/W202835528","https://openalex.org/W628726389","https://openalex.org/W1971281740","https://openalex.org/W1979692628","https://openalex.org/W1980466118","https://openalex.org/W1985660996","https://openalex.org/W1997308972","https://openalex.org/W1998004012","https://openalex.org/W2003576589","https://openalex.org/W2005867846","https://openalex.org/W2009609449","https://openalex.org/W2012071009","https://openalex.org/W2014660679","https://openalex.org/W2055247282","https://openalex.org/W2069084681","https://openalex.org/W2077809296","https://openalex.org/W2084944524","https://openalex.org/W2093224916","https://openalex.org/W2125822172","https://openalex.org/W2130350269","https://openalex.org/W2138233193","https://openalex.org/W2143638843","https://openalex.org/W2150642785","https://openalex.org/W2275400975","https://openalex.org/W2961754903","https://openalex.org/W2990455384","https://openalex.org/W3039179570","https://openalex.org/W3199713112","https://openalex.org/W4245820193","https://openalex.org/W4249949325","https://openalex.org/W4294836170","https://openalex.org/W4296457427","https://openalex.org/W4310650116","https://openalex.org/W4318769421","https://openalex.org/W4388311457","https://openalex.org/W4398143480","https://openalex.org/W4400225973"],"related_works":["https://openalex.org/W2083807145","https://openalex.org/W4239999982","https://openalex.org/W2122307093","https://openalex.org/W4313065247","https://openalex.org/W2611466209","https://openalex.org/W4312647289","https://openalex.org/W4243714467","https://openalex.org/W2103011624","https://openalex.org/W2171909906","https://openalex.org/W4235358950"],"abstract_inverted_index":{"Abstract":[0],"In":[1,81],"[L.":[2],"Bourhrara,":[3],"A":[4],"new":[5],"numerical":[6,37,106,156,180],"method":[7,17,50],"for":[8,51,163,175,251,351],"solving":[9],"the":[10,15,19,52,55,73,78,82,89,93,98,111,136,144,155,160,213,216,275,352],"Boltzmann":[11,56,94],"transport":[12,57],"equation":[13,58],"using":[14],"PN":[16],"and":[18,25,45,100,169,204,270,277,341],"discontinuous":[20,46,192],"finite":[21,48,137,194,293],"elements":[22,195],"on":[23,40,183,198],"unstructured":[24],"curved":[26],"meshes,":[27],"J.":[28],"Comput.":[29],"Phys.":[30],"397":[31],"2019,":[32],"Article":[33],"ID":[34],"108801],":[35],"a":[36,41,67,118,131,141,151,165,199,205,252,298,347],"scheme":[38],"based":[39,182],"combined":[42],"spherical":[43,206],"harmonics":[44,207],"Galerkin":[47,193],"element":[49,138],"resolution":[53],"of":[54,62,92,104,124,143,150,201,209,220,259,301,363],"is":[59,65,70,115,135,140,219],"proposed.":[60],"One":[61],"its":[63],"features":[64],"that":[66,134],"streamline":[68],"weight":[69],"added":[71],"to":[72,76,88,129,274],"test":[74],"function":[75],"obtain":[77,297],"variational":[79,113],"formulation.":[80],"present":[83],"paper,":[84],"restricting":[85],"our":[86,176,179],"attention":[87],"advective":[90],"part":[91],"equation,":[95],"we":[96,295],"prove":[97],"convergence":[99,217,299,349],"provide":[101],"error":[102,161],"estimates":[103,172],"this":[105,109],"scheme.":[107],"To":[108],"end,":[110],"original":[112],"formulation":[114],"restated":[116],"in":[117,158,338],"broken":[119,125,145],"functional":[120,126,146],"space.":[121,147],"The":[122,148],"use":[123],"spaces":[127],"enables":[128],"build":[130],"conforming":[132,152],"approximation,":[133],"space":[139],"subspace":[142],"setting":[149],"approximation":[153,208],"simplifies":[154],"analysis,":[157],"particular":[159],"estimates,":[162],"which":[164,255],"C\u00e9a\u2019s":[166],"type":[167],"lemma":[168],"standard":[170],"interpolation":[171],"are":[173,344],"sufficient":[174],"analysis.":[177],"For":[178,281],"scheme,":[181],"<m:math":[184,222,261,282,303,353,367],"xmlns:m=\"http://www.w3.org/1998/Math/MathML\">":[185,223,262,283,304,354,368],"<m:msup>":[186,232,240,313,321,355,377,385],"<m:mi>\u2119</m:mi>":[187],"<m:mi>k</m:mi>":[188,242,264,285,388],"</m:msup>":[189,238,243,319,327,358,383,392],"</m:math>":[190,249,268,289,333,359,398],"{\\mathbb{P}^{k}}":[191],"(in":[196,212],"space)":[197],"mesh":[200],"size":[202],"h":[203],"order":[210,221,260,302],"N":[211],"angular":[214,278],"variable),":[215],"rate":[218],"<m:mrow>":[224,228,231,234,263,284,305,309,312,315,369,373,376,379,387],"<m:mi":[225,306,370],"mathvariant=\"script\">\ud835\udcaa</m:mi>":[226,307,371],"<m:mo>\u2062</m:mo>":[227,308,372],"<m:mo":[229,245,310,329,374,394],"stretchy=\"false\">(</m:mo>":[230,311,375],"<m:mi>N</m:mi>":[233,314,378],"<m:mo>-</m:mo>":[235,316,380],"<m:mi>t</m:mi>":[236,317,381],"</m:mrow>":[237,244,247,248,267,288,318,328,331,332,382,391,393,396,397],"<m:mo>+</m:mo>":[239,265,320,384,389],"<m:mi>h</m:mi>":[241,322,386],"stretchy=\"false\">)</m:mo>":[246,330,395],"{\\mathcal{O}(N^{-t}+h^{k})}":[250],"smooth":[253],"solution":[254],"admits":[256],"partial":[257],"derivatives":[258],"<m:mn>1</m:mn>":[266,324,390],"{k+1}":[269],"t":[271],"with":[272],"respect":[273],"spatial":[276],"variables,":[279],"respectively.":[280],"<m:mo>=</m:mo>":[286],"<m:mn>0</m:mn>":[287],"{k=0}":[290],"(piecewise":[291],"constant":[292],"elements)":[294],"also":[296],"result":[300],"<m:mfrac>":[323],"<m:mn>2</m:mn>":[325,357],"</m:mfrac>":[326],"{\\mathcal{O}(N^{-t}+h^{\\frac{1}{2}})}":[334],".":[335,400],"Numerical":[336],"experiments":[337],"one,":[339],"two":[340],"three":[342],"dimensions":[343],"provided,":[345],"showing":[346],"better":[348],"behavior":[350],"<m:mi>L</m:mi>":[356],"{L^{2}}":[360],"-norm,":[361],"typically":[362],"one":[364],"more":[365],"order,":[366],"{\\mathcal{O}(N^{-t}+h^{k+1})}":[399]},"counts_by_year":[{"year":2025,"cited_by_count":2}],"updated_date":"2026-04-27T08:22:11.395708","created_date":"2025-10-10T00:00:00"}
