{"id":"https://openalex.org/W2102136117","doi":"https://doi.org/10.1002/(sici)1099-1506(199605/06)3:3<205::aid-nla79>3.0.co;2-k","title":"Convergence Estimates of Multilevel Additive and Multiplicative Algorithms for Non-symmetric and Indefinite Problems","display_name":"Convergence Estimates of Multilevel Additive and Multiplicative Algorithms for Non-symmetric and Indefinite Problems","publication_year":1996,"publication_date":"1996-05-01","ids":{"openalex":"https://openalex.org/W2102136117","doi":"https://doi.org/10.1002/(sici)1099-1506(199605/06)3:3<205::aid-nla79>3.0.co;2-k","mag":"2102136117"},"language":"en","primary_location":{"id":"doi:10.1002/(sici)1099-1506(199605/06)3:3<205::aid-nla79>3.0.co;2-k","is_oa":false,"landing_page_url":"https://doi.org/10.1002/(sici)1099-1506(199605/06)3:3<205::aid-nla79>3.0.co;2-k","pdf_url":null,"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,"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"},"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/A5023481731","display_name":"Zhiqiang Cai","orcid":"https://orcid.org/0000-0002-3775-7248"},"institutions":[{"id":"https://openalex.org/I219193219","display_name":"Purdue University West Lafayette","ror":"https://ror.org/02dqehb95","country_code":"US","type":"education","lineage":["https://openalex.org/I219193219"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Zhiqiang Cai","raw_affiliation_strings":["Department of Mathematics, Purdue University, 1395 Mathematical Science Building, West Lafayette, IN 47907-1395, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Purdue University, 1395 Mathematical Science Building, West Lafayette, IN 47907-1395, USA","institution_ids":["https://openalex.org/I219193219"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5018604216","display_name":"Chen-Yao G. Lai","orcid":null},"institutions":[{"id":"https://openalex.org/I148099254","display_name":"National Chung Cheng University","ror":"https://ror.org/0028v3876","country_code":"TW","type":"education","lineage":["https://openalex.org/I148099254"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Chen-Yao G. Lai","raw_affiliation_strings":["Department of Mathematics, National Chung Cheng University, No. 160, San-Hsing Village, Ming-Hsiung, Chia-Yi 621, Taiwan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, National Chung Cheng University, No. 160, San-Hsing Village, Ming-Hsiung, Chia-Yi 621, Taiwan","institution_ids":["https://openalex.org/I148099254"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5023481731"],"corresponding_institution_ids":["https://openalex.org/I219193219"],"apc_list":{"value":4430,"currency":"USD","value_usd":4430},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.21001671,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"3","issue":"3","first_page":"205","last_page":"220"},"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":0.9998000264167786,"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":0.9998000264167786,"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":0.9988999962806702,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.996999979019165,"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/multigrid-method","display_name":"Multigrid method","score":0.8693307042121887},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.801331639289856},{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.7163344621658325},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6690555214881897},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5891892313957214},{"id":"https://openalex.org/keywords/reduction","display_name":"Reduction (mathematics)","score":0.519991934299469},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5032758116722107},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.45834845304489136},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.13742539286613464},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.13470721244812012},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11255282163619995}],"concepts":[{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.8693307042121887},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.801331639289856},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.7163344621658325},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6690555214881897},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5891892313957214},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.519991934299469},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5032758116722107},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.45834845304489136},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.13742539286613464},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.13470721244812012},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11255282163619995},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1002/(sici)1099-1506(199605/06)3:3<205::aid-nla79>3.0.co;2-k","is_oa":false,"landing_page_url":"https://doi.org/10.1002/(sici)1099-1506(199605/06)3:3<205::aid-nla79>3.0.co;2-k","pdf_url":null,"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,"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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":21,"referenced_works":["https://openalex.org/W672314457","https://openalex.org/W785641167","https://openalex.org/W1586623601","https://openalex.org/W1982701660","https://openalex.org/W1990671420","https://openalex.org/W1997558267","https://openalex.org/W1997831216","https://openalex.org/W2017085690","https://openalex.org/W2017934981","https://openalex.org/W2041523977","https://openalex.org/W2060615213","https://openalex.org/W2068395392","https://openalex.org/W2071928229","https://openalex.org/W2072463419","https://openalex.org/W2072598943","https://openalex.org/W2093538658","https://openalex.org/W2094813161","https://openalex.org/W2140153041","https://openalex.org/W2153127767","https://openalex.org/W6635154111","https://openalex.org/W6660938892"],"related_works":["https://openalex.org/W2077078424","https://openalex.org/W2182328658","https://openalex.org/W1519938374","https://openalex.org/W84272617","https://openalex.org/W613282331","https://openalex.org/W2092341060","https://openalex.org/W2625595466","https://openalex.org/W2349420447","https://openalex.org/W2952295721","https://openalex.org/W3009176749"],"abstract_inverted_index":{"New":[0],"uniform":[1,41],"estimates":[2],"for":[3,8],"multigrid":[4,24],"algorithms":[5,26,39],"are":[6,16],"established":[7],"certain":[9],"non-symmetric":[10],"indefinite":[11],"problems.":[12],"In":[13],"particular,":[14],"we":[15],"concerned":[17],"with":[18],"the":[19,47,58],"simple":[20],"additive":[21],"algorithm":[22],"and":[23,51],"(V(1,0)-cycle)":[25],"given":[27],"in":[28],"[5].":[29],"We":[30],"prove,":[31],"without":[32],"full":[33],"elliptic":[34],"regularity":[35],"assumption,":[36],"that":[37,57],"these":[38],"have":[40],"reduction":[42],"per":[43],"iteration,":[44],"independent":[45],"of":[46,53],"finest":[48],"mesh":[49,60],"size":[50,61],"number":[52],"refinement":[54],"levels,":[55],"provided":[56],"coarsest":[59],"is":[62],"sufficiently":[63],"small.":[64]},"counts_by_year":[{"year":2020,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}