{"id":"https://openalex.org/W2084718774","doi":"https://doi.org/10.1137/080724381","title":"A Symmetric Nodal Conservative Finite Element Method for the Darcy Equation","display_name":"A Symmetric Nodal Conservative Finite Element Method for the Darcy Equation","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W2084718774","doi":"https://doi.org/10.1137/080724381","mag":"2084718774"},"language":"en","primary_location":{"id":"doi:10.1137/080724381","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080724381","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/A5025324947","display_name":"Gabriel R. Barrenechea","orcid":"https://orcid.org/0000-0003-4490-678X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Gabriel R. Barrenechea","raw_affiliation_strings":["gabriel.barrenechea@strath.ac.uk#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"gabriel.barrenechea@strath.ac.uk#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5108440518","display_name":"Leopoldo P. Franca","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Leopoldo P. Franca","raw_affiliation_strings":["leo.franca@cudenver.edu#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"leo.franca@cudenver.edu#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5059362886","display_name":"Fr\u00e9d\u00e9ric Valentin","orcid":"https://orcid.org/0000-0002-5056-7529"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Fr\u00e9d\u00e9ric Valentin","raw_affiliation_strings":["valentin@lncc.br#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"valentin@lncc.br#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.0412,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.78402215,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"47","issue":"5","first_page":"3652","last_page":"3677"},"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":1.0,"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":1.0,"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":0.9997000098228455,"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/T11263","display_name":"Electromagnetic Simulation and Numerical Methods","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic 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":0.8256782293319702},{"id":"https://openalex.org/keywords/petrov\u2013galerkin-method","display_name":"Petrov\u2013Galerkin method","score":0.6081179976463318},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.5724700093269348},{"id":"https://openalex.org/keywords/basis-function","display_name":"Basis function","score":0.5675225853919983},{"id":"https://openalex.org/keywords/interpolation","display_name":"Interpolation (computer graphics)","score":0.5453507304191589},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5247009992599487},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.4884251654148102},{"id":"https://openalex.org/keywords/method-of-mean-weighted-residuals","display_name":"Method of mean weighted residuals","score":0.45644575357437134},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4539903402328491},{"id":"https://openalex.org/keywords/weight-function","display_name":"Weight function","score":0.4335537552833557},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4301181435585022},{"id":"https://openalex.org/keywords/darcy\u2013weisbach-equation","display_name":"Darcy\u2013Weisbach equation","score":0.42508310079574585},{"id":"https://openalex.org/keywords/residual","display_name":"Residual","score":0.4205928444862366},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4180786907672882},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.41154199838638306},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.10381117463111877},{"id":"https://openalex.org/keywords/image","display_name":"Image (mathematics)","score":0.09733256697654724},{"id":"https://openalex.org/keywords/porous-medium","display_name":"Porous medium","score":0.07704281806945801}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8256782293319702},{"id":"https://openalex.org/C2778540877","wikidata":"https://www.wikidata.org/wiki/Q7179155","display_name":"Petrov\u2013Galerkin method","level":3,"score":0.6081179976463318},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.5724700093269348},{"id":"https://openalex.org/C5917680","wikidata":"https://www.wikidata.org/wiki/Q2621825","display_name":"Basis function","level":2,"score":0.5675225853919983},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.5453507304191589},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5247009992599487},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.4884251654148102},{"id":"https://openalex.org/C159674985","wikidata":"https://www.wikidata.org/wiki/Q6823718","display_name":"Method of mean weighted residuals","level":4,"score":0.45644575357437134},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4539903402328491},{"id":"https://openalex.org/C134466208","wikidata":"https://www.wikidata.org/wiki/Q1520657","display_name":"Weight function","level":2,"score":0.4335537552833557},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4301181435585022},{"id":"https://openalex.org/C181757486","wikidata":"https://www.wikidata.org/wiki/Q1142285","display_name":"Darcy\u2013Weisbach equation","level":4,"score":0.42508310079574585},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.4205928444862366},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4180786907672882},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.41154199838638306},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.10381117463111877},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.09733256697654724},{"id":"https://openalex.org/C105569014","wikidata":"https://www.wikidata.org/wiki/Q3271208","display_name":"Porous medium","level":3,"score":0.07704281806945801},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C6648577","wikidata":"https://www.wikidata.org/wiki/Q622669","display_name":"Porosity","level":2,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","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/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","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/C187320778","wikidata":"https://www.wikidata.org/wiki/Q1349130","display_name":"Geotechnical engineering","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/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/080724381","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080724381","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:strathprints.strath.ac.uk:15098","is_oa":false,"landing_page_url":"https://strathprints.strath.ac.uk/view/author/566211.html>","pdf_url":null,"source":{"id":"https://openalex.org/S4306402226","display_name":"Strathprints: The University of Strathclyde institutional repository (University of Strathclyde)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I181647926","host_organization_name":"University of Strathclyde","host_organization_lineage":["https://openalex.org/I181647926"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.552.7406","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.552.7406","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.lncc.br/~valentin/docs/BarFraVal.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.554.3779","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.554.3779","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.maths.strath.ac.uk/downloads/publications/paper7.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Life in Land","score":0.6899999976158142,"id":"https://metadata.un.org/sdg/15"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W136185420","https://openalex.org/W1601728065","https://openalex.org/W1674659126","https://openalex.org/W1852082788","https://openalex.org/W1973590664","https://openalex.org/W1977399978","https://openalex.org/W1978573864","https://openalex.org/W1985247669","https://openalex.org/W1997640799","https://openalex.org/W2005788407","https://openalex.org/W2016534590","https://openalex.org/W2030826026","https://openalex.org/W2041589227","https://openalex.org/W2043176577","https://openalex.org/W2051682834","https://openalex.org/W2053342859","https://openalex.org/W2061148553","https://openalex.org/W2074481576","https://openalex.org/W2078892983","https://openalex.org/W2094442825","https://openalex.org/W2113267855","https://openalex.org/W2123597308","https://openalex.org/W2132648167","https://openalex.org/W2149400269","https://openalex.org/W2155879256","https://openalex.org/W2156609537","https://openalex.org/W2199169705","https://openalex.org/W3149667697"],"related_works":["https://openalex.org/W2388596822","https://openalex.org/W2257371197","https://openalex.org/W2372688601","https://openalex.org/W2439864307","https://openalex.org/W2349100274","https://openalex.org/W2349777921","https://openalex.org/W2390957183","https://openalex.org/W1969632866","https://openalex.org/W831202728","https://openalex.org/W2361179119"],"abstract_inverted_index":{"This":[0],"work":[1],"introduces":[2],"and":[3,42,81,111,117],"analyzes":[4],"novel":[5],"stable":[6],"Petrov\u2013Galerkin":[7,29],"enriched":[8],"methods":[9],"(PGEM)":[10],"for":[11,78,86,102],"the":[12,17,47,71,79,87,93,97,109],"Darcy":[13],"problem,":[14],"based":[15],"on":[16],"simplest":[18],"but":[19],"unstable":[20],"continuous":[21],"$\\mathbb{P}_1/\\mathbb{P}_0$":[22],"pair.":[23],"Stability":[24],"is":[25,52],"recovered":[26],"inside":[27],"a":[28],"framework":[30],"where":[31],"elementwise":[32],"dependent":[33],"residual":[34],"functions,":[35,38,56],"named":[36],"multiscale":[37,57,65],"enrich":[39],"both":[40],"velocity":[41,48,80],"pressure":[43],"trial":[44],"spaces.":[45,105],"Unlike":[46],"test":[49],"space":[50],"that":[51,108],"augmented":[53],"with":[54],"bubble-like":[55],"functions":[58,66,77,85],"correct":[59],"edge":[60],"residuals":[61],"as":[62],"well.":[63],"The":[64,89],"turn":[67],"out":[68],"to":[69,95],"be":[70],"well-known":[72],"lowest":[73],"order":[74],"Raviart\u2013Thomas":[75],"basis":[76],"discontinuous":[82],"quadratics":[83],"polynomial":[84],"pressure.":[88],"enrichment":[90],"strategy":[91],"suggests":[92],"way":[94],"recover":[96],"local":[98],"mass":[99],"conservation":[100],"property":[101],"nodal-based":[103],"interpolation":[104],"We":[106],"prove":[107],"method":[110],"its":[112],"symmetric":[113],"version":[114],"are":[115],"well-posed":[116],"achieve":[118],"optimal":[119],"error":[120],"estimates":[121],"in":[122],"natural":[123],"norms.":[124],"Numerical":[125],"validations":[126],"confirm":[127],"claimed":[128],"theoretical":[129],"results.":[130]},"counts_by_year":[{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2013,"cited_by_count":2}],"updated_date":"2026-07-02T09:51:11.867554","created_date":"2025-10-10T00:00:00"}
