{"id":"https://openalex.org/W1982519991","doi":"https://doi.org/10.1137/040621247","title":"Anisotropic Mesh Refinement: The Conditioning of Galerkin Boundary Element Matrices and Simple Preconditioners","display_name":"Anisotropic Mesh Refinement: The Conditioning of Galerkin Boundary Element Matrices and Simple Preconditioners","publication_year":2006,"publication_date":"2006-01-01","ids":{"openalex":"https://openalex.org/W1982519991","doi":"https://doi.org/10.1137/040621247","mag":"1982519991"},"language":"en","primary_location":{"id":"doi:10.1137/040621247","is_oa":false,"landing_page_url":"https://doi.org/10.1137/040621247","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/A5064637859","display_name":"Ivan G. Graham","orcid":"https://orcid.org/0000-0002-5730-676X"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Ivan G. Graham","raw_affiliation_strings":["Department of Mathematical Sciences"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5072328086","display_name":"William McLean","orcid":"https://orcid.org/0000-0002-7133-2884"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"William McLean","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5064637859"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.0358,"has_fulltext":false,"cited_by_count":36,"citation_normalized_percentile":{"value":0.75003739,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":100},"biblio":{"volume":"44","issue":"4","first_page":"1487","last_page":"1513"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9998000264167786,"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"}},"topics":[{"id":"https://openalex.org/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9998000264167786,"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"}},{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9995999932289124,"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/T11263","display_name":"Electromagnetic Simulation and Numerical Methods","score":0.9990000128746033,"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":0.8206379413604736},{"id":"https://openalex.org/keywords/degree-of-a-polynomial","display_name":"Degree of a polynomial","score":0.5785386562347412},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.5700141787528992},{"id":"https://openalex.org/keywords/condition-number","display_name":"Condition number","score":0.5077637434005737},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5046678781509399},{"id":"https://openalex.org/keywords/stiffness-matrix","display_name":"Stiffness matrix","score":0.49791431427001953},{"id":"https://openalex.org/keywords/bilinear-form","display_name":"Bilinear form","score":0.46614566445350647},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.43650826811790466},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.43348443508148193},{"id":"https://openalex.org/keywords/diagonal","display_name":"Diagonal","score":0.42840877175331116},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.428339421749115},{"id":"https://openalex.org/keywords/sobolev-space","display_name":"Sobolev space","score":0.41143161058425903},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.40740731358528137},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.3448581099510193},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.16543740034103394}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8206379413604736},{"id":"https://openalex.org/C25878781","wikidata":"https://www.wikidata.org/wiki/Q1473607","display_name":"Degree of a polynomial","level":3,"score":0.5785386562347412},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.5700141787528992},{"id":"https://openalex.org/C84545080","wikidata":"https://www.wikidata.org/wiki/Q1147936","display_name":"Condition number","level":3,"score":0.5077637434005737},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5046678781509399},{"id":"https://openalex.org/C14198674","wikidata":"https://www.wikidata.org/wiki/Q3851930","display_name":"Stiffness matrix","level":3,"score":0.49791431427001953},{"id":"https://openalex.org/C8828549","wikidata":"https://www.wikidata.org/wiki/Q837924","display_name":"Bilinear form","level":2,"score":0.46614566445350647},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.43650826811790466},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.43348443508148193},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.42840877175331116},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.428339421749115},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.41143161058425903},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.40740731358528137},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.3448581099510193},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.16543740034103394},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/040621247","is_oa":false,"landing_page_url":"https://doi.org/10.1137/040621247","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W152165499","https://openalex.org/W591292934","https://openalex.org/W1602383637","https://openalex.org/W1978321922","https://openalex.org/W1982115588","https://openalex.org/W1986120564","https://openalex.org/W1989056711","https://openalex.org/W2032916960","https://openalex.org/W2047292749","https://openalex.org/W2080086345","https://openalex.org/W2159595645"],"related_works":["https://openalex.org/W1847062822","https://openalex.org/W2950578642","https://openalex.org/W2038509953","https://openalex.org/W2065603880","https://openalex.org/W2339294360","https://openalex.org/W2118042923","https://openalex.org/W2069302992","https://openalex.org/W2902703932","https://openalex.org/W3043679535","https://openalex.org/W1506381275"],"abstract_inverted_index":{"In":[0,96],"this":[1,114],"paper":[2],"we":[3,73,98],"obtain":[4,99],"upper":[5],"and":[6,177,187,190],"lower":[7],"bounds":[8,85,101,136],"on":[9,36,125],"the":[10,13,49,76,80,87,103,110,118,146,154,159,201],"spectrum":[11],"of":[12,27,40,79,121],"stiffness":[14,106],"matrix":[15],"arising":[16],"from":[17,92],"a":[18,28,37,126,165],"finite":[19,50],"element":[20,51],"Galerkin":[21],"approximation":[22],"(using":[23],"nodal":[24],"basis":[25],"functions)":[26],"bounded,":[29],"symmetric":[30],"bilinear":[31],"form":[32],"which":[33,89,108],"is":[34,47,53,82],"elliptic":[35],"Sobolev":[38],"space":[39],"real":[41],"index":[42],"$m\\in[-1,1]$.":[43],"The":[44],"key":[45],"point":[46],"that":[48,62,75,134,141],"mesh":[52,94,150,167],"required":[54],"to":[55,149,153],"be":[56],"neither":[57],"quasi\u2010uniform":[58,166],"nor":[59],"shape\u2010regular,":[60],"so":[61],"our":[63,135],"theory":[64],"allows":[65],"anisotropic":[66,93],"meshes":[67],"often":[68],"used":[69],"in":[70,129,158],"practice.":[71],"(However,":[72],"assume":[74],"polynomial":[77],"degree":[78],"elements":[81],"fixed.)":[83],"Our":[84,169],"indicate":[86,109],"ill\u2010conditioning":[88,147],"can":[90],"arise":[91],"refinement.":[95,168],"addition":[97],"spectral":[100],"for":[102,164,200],"diagonally":[104],"scaled":[105],"matrix,":[107],"improvement":[111],"provided":[112],"by":[113,175],"simple":[115],"preconditioning.":[116],"For":[117],"special":[119],"case":[120],"boundary":[122],"integral":[123],"operators":[124],"two\u2010dimensional":[127],"screen":[128],"$\\R^3$,":[130],"numerical":[131],"experiments":[132],"show":[133],"are":[137],"sharp.":[138],"We":[139],"find":[140],"diagonal":[142],"scaling":[143],"essentially":[144],"removes":[145],"due":[148],"degeneracy,":[151],"leading":[152],"same":[155],"asymptotic":[156],"growth":[157],"condition":[160],"number":[161],"as":[162],"arises":[163],"results":[170],"thus":[171],"generalize":[172],"earlier":[173],"work":[174],"Bank":[176],"Scott":[178],"[SIAM":[179,192],"J.":[180,193],"Numer.":[181,194],"Anal.,":[182,195],"26":[183],"(1989),":[184],"pp.":[185,198],"1383\u20131394]":[186],"Ainsworth,":[188],"McLean,":[189],"Tran":[191],"36":[196],"(1999),":[197],"1901\u20131932]":[199],"shape\u2010regular":[202],"case.":[203]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":4},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2026-03-14T08:43:22.919905","created_date":"2025-10-10T00:00:00"}