{"id":"https://openalex.org/W4242649875","doi":"https://doi.org/10.1137/s0036142999365072","title":"Preconditioning Legendre Spectral Collocation Methods for Elliptic Problems II: Finite Element Operators","display_name":"Preconditioning Legendre Spectral Collocation Methods for Elliptic Problems II: Finite Element Operators","publication_year":2001,"publication_date":"2001-01-01","ids":{"openalex":"https://openalex.org/W4242649875","doi":"https://doi.org/10.1137/s0036142999365072"},"language":"en","primary_location":{"id":"doi:10.1137/s0036142999365072","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036142999365072","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/A5091007664","display_name":"Seymour V. Parter","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Seymour V. Parter","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5091007664"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.6794,"has_fulltext":false,"cited_by_count":12,"citation_normalized_percentile":{"value":0.77270307,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"39","issue":"1","first_page":"348","last_page":"362"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10514","display_name":"Numerical methods in engineering","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"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/T10514","display_name":"Numerical methods in engineering","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"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/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/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9994999766349792,"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/mathematics","display_name":"Mathematics","score":0.7598891854286194},{"id":"https://openalex.org/keywords/legendre-polynomials","display_name":"Legendre polynomials","score":0.6243467330932617},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5065401792526245},{"id":"https://openalex.org/keywords/lambda","display_name":"Lambda","score":0.4896090626716614},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.48300015926361084},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4477165937423706},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.435308575630188},{"id":"https://openalex.org/keywords/dirichlet-boundary-condition","display_name":"Dirichlet boundary condition","score":0.41101783514022827},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.23688635230064392},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.21497905254364014},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.09023886919021606}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7598891854286194},{"id":"https://openalex.org/C111458787","wikidata":"https://www.wikidata.org/wiki/Q215405","display_name":"Legendre polynomials","level":2,"score":0.6243467330932617},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5065401792526245},{"id":"https://openalex.org/C2778113609","wikidata":"https://www.wikidata.org/wiki/Q10897","display_name":"Lambda","level":2,"score":0.4896090626716614},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.48300015926361084},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4477165937423706},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.435308575630188},{"id":"https://openalex.org/C110167270","wikidata":"https://www.wikidata.org/wiki/Q1193699","display_name":"Dirichlet boundary condition","level":3,"score":0.41101783514022827},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.23688635230064392},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.21497905254364014},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.09023886919021606},{"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/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"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":1,"locations":[{"id":"doi:10.1137/s0036142999365072","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036142999365072","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":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":15,"referenced_works":["https://openalex.org/W77471398","https://openalex.org/W1982005385","https://openalex.org/W1985144286","https://openalex.org/W1990038465","https://openalex.org/W2002952441","https://openalex.org/W2013242019","https://openalex.org/W2033702923","https://openalex.org/W2036614641","https://openalex.org/W2039461168","https://openalex.org/W2041231713","https://openalex.org/W2065006139","https://openalex.org/W2069736604","https://openalex.org/W2073761720","https://openalex.org/W3021821037","https://openalex.org/W4298060601"],"related_works":["https://openalex.org/W2736737644","https://openalex.org/W3106446297","https://openalex.org/W2385225694","https://openalex.org/W2966956638","https://openalex.org/W2374325362","https://openalex.org/W2074688623","https://openalex.org/W1913294887","https://openalex.org/W2214763494","https://openalex.org/W2951695430","https://openalex.org/W2079298033"],"abstract_inverted_index":{"In":[0,49],"1979":[1],"Orszag":[2],"and":[3,29,34,41,110,139,150],"Morchoisne":[4],"independently":[5],"proposed":[6],"a":[7,56,195],"finite-difference":[8,33],"preconditioning":[9,37,63],"of":[10,15,38,60,64,135,156,168,198],"the":[11,16,20,32,70,132,142,147,153,157,165,199],"Chebyshev":[12,40],"collocation":[13,44,67,143,212],"discretization":[14],"Poisson":[17],"equation.":[18],"Over":[19],"years":[21],"there":[22,80],"has":[23],"been":[24],"intensive":[25],"research,":[26],"both":[27,39],"experimental":[28],"theoretical,":[30],"on":[31,55,194],"finite":[35,61,158],"element":[36,62,159],"Legendre":[42,65],"spectral":[43,66],"methods":[45,68],"for":[46,69],"this":[47,50],"problem.":[48],"work":[51],"we":[52],"present":[53],"results":[54,192],"frequently":[57],"employed":[58],"variant":[59],"Helmholtz":[71],"equation":[72],"with":[73],"Dirichlet":[74],"boundary":[75],"conditions.":[76],"We":[77],"show":[78],"that":[79,91,164],"exist":[81],"two":[82],"positive":[83],"constants":[84],"$0":[85],"<":[86,88],"\\Lambda_0":[87,106,179],"\\Lambda_1$":[89],"such":[90],"$$":[92,109,111,123,173,182,183,190],"{\\rm":[93,127,174],"Re}":[94],"\\{":[95],"(W_{n,m}":[96,100,116],"A_{n,m}":[97,113],"U,U)_{\\ell_2}":[98,103,114,119],"/":[99,115,202],"M^{-1}_{n,m}":[101,117],"S_{n,m}":[102,118],"\\}":[104],"\\geq":[105,178],">":[107,180],"0":[108],"|(W_{n,m}":[112],"|":[120,184,186],"\\leq":[121,187],"\\Lambda_1,":[122],"where":[124,207],"$W_{n,m}":[125],"=":[126],"diagonal}":[128],"(w_k":[129],"\\tilde":[130],"w_j)$,":[131],"diagonal":[133],"matrix":[134,155],"Legendre--Gauss--Lobatto":[136],"(LGL)":[137],"weights,":[138],"$A_{n,m}$":[140],"is":[141,146,152],"matrix.":[144],"$M_{n,m}$":[145],"mass":[148],"matrix,":[149],"$S_{n,m}$":[151],"stiffness":[154],"method.":[160],"These":[161,191],"estimates":[162],"imply":[163],"eigenvalues":[166],"$\\gamma_k$":[167],"$S^{-1}_{n,m}":[169],"M_{n,m}":[170],"A_{n,m}$":[171],"satisfy":[172],"Re":[175],"}":[176],"\\gamma_k":[177,185],"0,":[181],"\\Lambda_1":[188],".":[189],"depend":[193],"thorough":[196],"study":[197],"ratio":[200],"$2w_k":[201],"[":[203],"x_{k+1}":[204],"-":[205],"x_{k-1}]$,":[206],"$\\{":[208],"x_k\\}$":[209],"are":[210],"LGL":[211],"points.":[213]},"counts_by_year":[{"year":2016,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}