{"id":"https://openalex.org/W1998453302","doi":"https://doi.org/10.1002/nla.709","title":"Finite element methods for quantum electrodynamics using a Helmholtz decomposition of the gauge field","display_name":"Finite element methods for quantum electrodynamics using a Helmholtz decomposition of the gauge field","publication_year":2010,"publication_date":"2010-03-03","ids":{"openalex":"https://openalex.org/W1998453302","doi":"https://doi.org/10.1002/nla.709","mag":"1998453302"},"language":"en","primary_location":{"id":"doi:10.1002/nla.709","doi":"https://doi.org/10.1002/nla.709","is_oa":true,"landing_page_url":"https://doi.org/10.1002/nla.709","pdf_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/nla.709","source":{"id":"https://openalex.org/S60324941","display_name":"Numerical Linear Algebra with Applications","issn_l":"1070-5325","issn":["1070-5325","1099-1506"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","jufo-1","ki-jl-1","norway-1"],"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Linear Algebra with Applications","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/nla.709","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5081032069","display_name":"C. Ketelsen","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I188538660","display_name":"University of Colorado Boulder","ror":"https://ror.org/02ttsq026","country_code":"US","type":"education","lineage":["https://openalex.org/I188538660"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"C. Ketelsen","raw_affiliation_strings":["Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309-0526, U.S.A","Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309\u20100526, U.S.A"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309-0526, U.S.A","institution_ids":["https://openalex.org/I188538660"]},{"raw_affiliation_string":"Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309\u20100526, U.S.A","institution_ids":["https://openalex.org/I188538660"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5008694561","display_name":"Thomas A. Manteuffel","orcid":"https://orcid.org/0000-0002-8897-4146","observed_orcids":["https://orcid.org/0000-0002-8897-4146"]},"institutions":[{"id":"https://openalex.org/I188538660","display_name":"University of Colorado Boulder","ror":"https://ror.org/02ttsq026","country_code":"US","type":"education","lineage":["https://openalex.org/I188538660"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"T. Manteuffel","raw_affiliation_strings":["Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309-0526, U.S.A","Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309\u20100526, U.S.A"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309-0526, U.S.A","institution_ids":["https://openalex.org/I188538660"]},{"raw_affiliation_string":"Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309\u20100526, U.S.A","institution_ids":["https://openalex.org/I188538660"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5046048300","display_name":"Stephen F. McCormick","orcid":"https://orcid.org/0000-0002-6327-790X","observed_orcids":["https://orcid.org/0000-0002-6327-790X"]},"institutions":[{"id":"https://openalex.org/I188538660","display_name":"University of Colorado Boulder","ror":"https://ror.org/02ttsq026","country_code":"US","type":"education","lineage":["https://openalex.org/I188538660"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"S. McCormick","raw_affiliation_strings":["Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309-0526, U.S.A","Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309\u20100526, U.S.A"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309-0526, U.S.A","institution_ids":["https://openalex.org/I188538660"]},{"raw_affiliation_string":"Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309\u20100526, U.S.A","institution_ids":["https://openalex.org/I188538660"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5047719926","display_name":"John W. Ruge","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I188538660","display_name":"University of Colorado Boulder","ror":"https://ror.org/02ttsq026","country_code":"US","type":"education","lineage":["https://openalex.org/I188538660"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"J. Ruge","raw_affiliation_strings":["Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309-0526, U.S.A","Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309\u20100526, U.S.A"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309-0526, U.S.A","institution_ids":["https://openalex.org/I188538660"]},{"raw_affiliation_string":"Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309\u20100526, U.S.A","institution_ids":["https://openalex.org/I188538660"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5081032069"],"corresponding_institution_ids":["https://openalex.org/I188538660"],"apc_list":{"value":4880,"currency":"USD","value_usd":4880},"apc_paid":null,"fwci":0.3139,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.60014304,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"17","issue":"2-3","first_page":"539","last_page":"556"},"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":0.9014000296592712,"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":0.9014000296592712,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.0689999982714653,"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/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.007000000216066837,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/quantum-electrodynamics","display_name":"quantum electrodynamics","score":0.9409999847412109},{"id":"https://openalex.org/keywords/dirac-equation","display_name":"Dirac equation","score":0.9810000061988831},{"id":"https://openalex.org/keywords/helmholtz-decomposition","display_name":"Helmholtz decomposition","score":0.8769999742507935},{"id":"https://openalex.org/keywords/gauge-fields","display_name":"gauge fields","score":0.6949999928474426},{"id":"https://openalex.org/keywords/u-1-gauge-theory","display_name":"U(1) gauge theory","score":0.7789999842643738},{"id":"https://openalex.org/keywords/adaptive-multigrid","display_name":"adaptive multigrid","score":0.7860000133514404},{"id":"https://openalex.org/keywords/two-dimensional-dirac-equation","display_name":"two-dimensional Dirac equation","score":0.6330000162124634},{"id":"https://openalex.org/keywords/operator-transformation","display_name":"operator transformation","score":0.3149999976158142}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5510266423225403},{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.5225610136985779},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.46682208776474},{"id":"https://openalex.org/C181830111","wikidata":"https://www.wikidata.org/wiki/Q214850","display_name":"Gauge theory","level":2,"score":0.4513964056968689},{"id":"https://openalex.org/C81174988","wikidata":"https://www.wikidata.org/wiki/Q6786849","display_name":"Mathematical descriptions of the electromagnetic field","level":5,"score":0.4462170898914337},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.43739235401153564},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.4343727231025696},{"id":"https://openalex.org/C18591234","wikidata":"https://www.wikidata.org/wiki/Q860615","display_name":"Helmholtz equation","level":3,"score":0.4174308180809021},{"id":"https://openalex.org/C31593994","wikidata":"https://www.wikidata.org/wiki/Q832289","display_name":"Gauge fixing","level":4,"score":0.41566187143325806},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.40697431564331055},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.3767250180244446},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3665642738342285},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.34576791524887085},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.29627275466918945},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.14363238215446472},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.13243386149406433},{"id":"https://openalex.org/C40263606","wikidata":"https://www.wikidata.org/wiki/Q105580","display_name":"Gauge boson","level":3,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1002/nla.709","doi":"https://doi.org/10.1002/nla.709","is_oa":true,"landing_page_url":"https://doi.org/10.1002/nla.709","pdf_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/nla.709","source":{"id":"https://openalex.org/S60324941","display_name":"Numerical Linear Algebra with Applications","issn_l":"1070-5325","issn":["1070-5325","1099-1506"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","jufo-1","ki-jl-1","norway-1"],"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Linear Algebra with Applications","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.485.2718","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.485.2718","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://amath.colorado.edu/pub/multigrid/qed2.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1002/nla.709","doi":"https://doi.org/10.1002/nla.709","is_oa":true,"landing_page_url":"https://doi.org/10.1002/nla.709","pdf_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/nla.709","source":{"id":"https://openalex.org/S60324941","display_name":"Numerical Linear Algebra with Applications","issn_l":"1070-5325","issn":["1070-5325","1099-1506"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","jufo-1","ki-jl-1","norway-1"],"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Linear Algebra with Applications","raw_type":"journal-article"},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G1702950523","display_name":null,"funder_award_id":"W-7405","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G1911031706","display_name":null,"funder_award_id":"B568677","funder_id":"https://openalex.org/F4320338286","funder_display_name":"Lawrence Livermore National Laboratory"},{"id":"https://openalex.org/G222768804","display_name":null,"funder_award_id":"DE\u2010FG02\u2010","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G2713861829","display_name":null,"funder_award_id":"W\u20107405\u2010ENG\u201036","funder_id":"https://openalex.org/F4320338304","funder_display_name":"Los Alamos National Laboratory"},{"id":"https://openalex.org/G4489581436","display_name":"FIRST ORDER SYSTEM LEAST SQUARES (FOSLS):  FUNDAMENTALS AND APPLICATIONS (FAST ALGORITHMS FOR TRANSPORT MODELS)","funder_award_id":"FG0203ER25574","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G4784282273","display_name":"Collaborative Research:   Multigrid QCD at the Petascale","funder_award_id":"0749317","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5323443009","display_name":"TOWARDS OPTIMAL PETASCALE SIMULATIONS (TOPS)","funder_award_id":"FC02-06ER25784","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G6115782557","display_name":null,"funder_award_id":"W-7405-ENG-36","funder_id":"https://openalex.org/F4320338286","funder_display_name":"Lawrence Livermore National Laboratory"},{"id":"https://openalex.org/G628297446","display_name":"TOWARDS OPTIMAL PETASCALE SIMULATIONS (TOPS)","funder_award_id":"DEFC0206ER25784","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G697799863","display_name":"Collaborative Research:  Enhanced Least-Squares Methods for PIV Analysis","funder_award_id":"0811275","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G797745481","display_name":null,"funder_award_id":"W-7405-ENG-36","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G842327514","display_name":null,"funder_award_id":"LA-UR-08-04368","funder_id":"https://openalex.org/F4320338304","funder_display_name":"Los Alamos National Laboratory"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306084","display_name":"U.S. Department of Energy","ror":"https://ror.org/01bj3aw27"},{"id":"https://openalex.org/F4320309370","display_name":"University of Pennsylvania","ror":"https://ror.org/00b30xv10"},{"id":"https://openalex.org/F4320310419","display_name":"Pennsylvania State University","ror":"https://ror.org/04p491231"},{"id":"https://openalex.org/F4320338286","display_name":"Lawrence Livermore National Laboratory","ror":"https://ror.org/041nk4h53"},{"id":"https://openalex.org/F4320338304","display_name":"Los Alamos National Laboratory","ror":"https://ror.org/01e41cf67"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1998453302.pdf","grobid_xml":"https://content.openalex.org/works/W1998453302.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W3250287","https://openalex.org/W89660469","https://openalex.org/W170849929","https://openalex.org/W575488288","https://openalex.org/W1667125974","https://openalex.org/W1967983557","https://openalex.org/W1972266198","https://openalex.org/W1984450486","https://openalex.org/W1988199152","https://openalex.org/W2041827155","https://openalex.org/W2051172645","https://openalex.org/W2052959638","https://openalex.org/W2132292023","https://openalex.org/W2320705814","https://openalex.org/W4230108662"],"related_works":["https://openalex.org/W2963616678","https://openalex.org/W4226523763","https://openalex.org/W3102372055","https://openalex.org/W1553422289","https://openalex.org/W2796948780","https://openalex.org/W4252777094","https://openalex.org/W2016246005","https://openalex.org/W1939056692","https://openalex.org/W1989168897","https://openalex.org/W2065512531"],"abstract_inverted_index":{"Abstract":[0],"The":[1],"Dirac":[2,26],"equation":[3],"of":[4,17,23,30,45,73,89,97,138,144],"quantum":[5],"electrodynamics":[6],"describes":[7],"the":[8,18,24,60,69,74,90,98,103,112,127,139,145],"interaction":[9],"between":[10],"electrons":[11],"and":[12,50,119,135],"photons.":[13],"Large\u2010scale":[14],"numerical":[15,133],"simulations":[16],"theory":[19],"require":[20],"repeated":[21],"solution":[22],"two\u2010dimensional":[25],"equation,":[27],"a":[28,38,95],"system":[29,47,75,104,114],"two":[31],"first\u2010order":[32],"partial":[33],"differential":[34],"equations":[35,92],"coupled":[36],"to":[37,125],"background":[39,61],"U(1)":[40],"gauge":[41,67],"field.":[42,62],"Traditional":[43],"discretizations":[44],"this":[46],"are":[48,76],"sparse":[49],"highly":[51,77],"structured,":[52],"but":[53],"contain":[54],"random":[55],"complex":[56],"entries":[57],"introduced":[58],"by":[59,94],"For":[63],"even":[64],"mildly":[65],"disordered":[66],"fields,":[68],"near":[70],"kernel":[71],"components":[72],"oscillatory,":[78],"rendering":[79],"standard":[80],"multilevel":[81],"methods":[82],"ineffective.":[83],"We":[84,110,131],"consider":[85],"an":[86],"alternate":[87],"formulation":[88,141],"governing":[91],"obtained":[93],"transformation":[96],"continuum":[99],"operator":[100],"that":[101],"decouples":[102],"into":[105],"separate":[106],"scalar":[107],"diffusion\u2010like":[108],"equations.":[109],"discretize":[111],"transformed":[113,140],"using":[115],"least\u2010squares":[116],"finite":[117],"elements":[118],"use":[120],"adaptive":[121],"smoothed":[122],"aggregation":[123],"multigrid":[124],"solve":[126],"resulting":[128],"linear":[129],"system.":[130],"present":[132],"results":[134],"discuss":[136],"implications":[137],"in":[142],"terms":[143],"physical":[146],"theory.":[147],"Copyright":[148],"\u00a9":[149],"2010":[150],"John":[151],"Wiley":[152],"&":[153],"Sons,":[154],"Ltd.":[155]},"counts_by_year":[],"updated_date":"2026-09-22T06:50:58.402377","created_date":"2025-10-10T00:00:00"}