{"id":"https://openalex.org/W7154599315","doi":"https://doi.org/10.48550/arxiv.2604.13889","title":"A Two-Level Additive Schwarz Method for Computing Interior Multiple and Clustered Eigenvalues of Symmetric Elliptic Operators","display_name":"A Two-Level Additive Schwarz Method for Computing Interior Multiple and Clustered Eigenvalues of Symmetric Elliptic Operators","publication_year":2026,"publication_date":"2026-04-15","ids":{"openalex":"https://openalex.org/W7154599315","doi":"https://doi.org/10.48550/arxiv.2604.13889"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2604.13889","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.13889","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2604.13889","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5133753345","display_name":"Qigang Liang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Liang, Qigang","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5133816902","display_name":"Xuejun Xu","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xu, Xuejun","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"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":0.8044999837875366,"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":0.8044999837875366,"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":0.12280000001192093,"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.025699999183416367,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.7825000286102295},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.6869999766349792},{"id":"https://openalex.org/keywords/additive-schwarz-method","display_name":"Additive Schwarz method","score":0.6819999814033508},{"id":"https://openalex.org/keywords/schwarz-alternating-method","display_name":"Schwarz alternating method","score":0.6445000171661377},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5435000061988831},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5217999815940857},{"id":"https://openalex.org/keywords/monotonic-function","display_name":"Monotonic function","score":0.4887000024318695},{"id":"https://openalex.org/keywords/elliptic-operator","display_name":"Elliptic operator","score":0.4684999883174896},{"id":"https://openalex.org/keywords/uniform-boundedness","display_name":"Uniform boundedness","score":0.4587000012397766}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8238000273704529},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.7825000286102295},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.6869999766349792},{"id":"https://openalex.org/C174819683","wikidata":"https://www.wikidata.org/wiki/Q4681340","display_name":"Additive Schwarz method","level":4,"score":0.6819999814033508},{"id":"https://openalex.org/C31278502","wikidata":"https://www.wikidata.org/wiki/Q7433186","display_name":"Schwarz alternating method","level":4,"score":0.6445000171661377},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5435000061988831},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5217999815940857},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.4887000024318695},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.47920000553131104},{"id":"https://openalex.org/C70610323","wikidata":"https://www.wikidata.org/wiki/Q427625","display_name":"Elliptic operator","level":2,"score":0.4684999883174896},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.4587000012397766},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.40880000591278076},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.40139999985694885},{"id":"https://openalex.org/C192702615","wikidata":"https://www.wikidata.org/wiki/Q5283998","display_name":"Divide-and-conquer eigenvalue algorithm","level":3,"score":0.39089998602867126},{"id":"https://openalex.org/C2505209","wikidata":"https://www.wikidata.org/wiki/Q1671724","display_name":"Inverse iteration","level":3,"score":0.3806999921798706},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.36899998784065247},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.3418000042438507},{"id":"https://openalex.org/C198880260","wikidata":"https://www.wikidata.org/wiki/Q5289813","display_name":"Domain decomposition methods","level":3,"score":0.3327000141143799},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.32190001010894775},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.2953000068664551},{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.27959999442100525},{"id":"https://openalex.org/C200288055","wikidata":"https://www.wikidata.org/wiki/Q2621792","display_name":"Element (criminal law)","level":2,"score":0.27250000834465027},{"id":"https://openalex.org/C164866538","wikidata":"https://www.wikidata.org/wiki/Q367351","display_name":"Cluster (spacecraft)","level":2,"score":0.2711000144481659},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.265500009059906},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.25859999656677246},{"id":"https://openalex.org/C84545080","wikidata":"https://www.wikidata.org/wiki/Q1147936","display_name":"Condition number","level":3,"score":0.2578999996185303}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2604.13889","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.13889","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2604.13889","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.13889","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,111],"propose":[4],"an":[5],"efficient":[6],"two-level":[7,63],"additive":[8,64],"Schwarz":[9,65],"method":[10,46,125],"for":[11,99],"solving":[12,92],"large-scale":[13],"eigenvalue":[14,95],"problems":[15],"arising":[16],"from":[17,147],"the":[18,39,56,62,67,123,133,139,145,151,168,173,177,193,198],"finite":[19],"element":[20],"discretization":[21],"of":[22,74,88,102,122,135,150,167],"symmetric":[23],"elliptic":[24],"operators,":[25],"which":[26,191],"may":[27],"compute":[28],"efficiently":[29],"more":[30,194],"interior":[31],"multiple":[32],"and":[33,91,109,142,172],"clustered":[34],"eigenvalues":[35,75,153],"other":[36,68],"than":[37],"only":[38,83],"first":[40],"several":[41],"smallest":[42],"eigenvalues.":[43,103,179],"The":[44,161,180],"proposed":[45,124],"is":[47,53,69,126,132,138,165],"parallel":[48,89],"in":[49,79,159],"two":[50],"ways:":[51],"one":[52],"to":[54,70,117,154,186,197],"solve":[55,71],"preconditioned":[57],"Jacobi-Davidson":[58],"correction":[59],"equations":[60],"by":[61,128],"preconditioner,":[66],"different":[72],"clusters":[73],"(see":[76,156],"Figure":[77,157],"1":[78],"Introduction)":[80],"simultaneously.":[81],"It":[82],"requires":[84],"computing":[85],"a":[86,93,100,113],"series":[87],"subproblems":[90],"small-dimensional":[94],"problem":[96],"per":[97],"iteration":[98],"cluster":[101],"Based":[104],"on":[105],"some":[106],"new":[107],"estimates":[108],"tools,":[110],"provide":[112],"rigorous":[114],"theoretical":[115],"analysis":[116],"prove":[118],"that":[119],"convergence":[120],"factor":[121],"bounded":[127],"$\u03b3=c(H)\u03c1(\\frac\u03b4{H},d_{m}^{-},d_{M}^{+})$,":[129],"where":[130],"$H$":[131],"diameter":[134],"subdomains,":[136],"$\u03b4$":[137],"overlapping":[140],"size":[141,171],"$d_{m}^{-},d_{M}^{+}$":[143],"are":[144,206],"distances":[146],"both":[148],"ends":[149],"targeted":[152,178],"others":[155],"2":[158],"Introduction).":[160],"positive":[162],"number":[163],"$\u03c1(\\frac\u03b4{H},d_{m}^{-},d_{M}^{+})<1$":[164],"independent":[166],"fine":[169],"mesh":[170],"internal":[174],"gaps":[175],"among":[176],"$H$-dependent":[181],"constant":[182],"$c(H)$":[183],"decreases":[184],"monotonically":[185],"1,":[187],"as":[188],"$H\\to":[189],"0$,":[190],"means":[192],"subdomains":[195],"lead":[196],"better":[199],"convergence.":[200],"Numerical":[201],"results":[202],"supporting":[203],"our":[204],"theory":[205],"given.":[207]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2026-04-17T00:00:00"}