{"id":"https://openalex.org/W2008808718","doi":"https://doi.org/10.1007/s002110100275","title":"Algebraic theory of multiplicative Schwarz methods","display_name":"Algebraic theory of multiplicative Schwarz methods","publication_year":2001,"publication_date":"2001-10-01","ids":{"openalex":"https://openalex.org/W2008808718","doi":"https://doi.org/10.1007/s002110100275","mag":"2008808718"},"language":"en","primary_location":{"id":"doi:10.1007/s002110100275","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s002110100275","pdf_url":null,"source":{"id":"https://openalex.org/S4210205683","display_name":"Numerische Mathematik","issn_l":"0029-599X","issn":["0029-599X","0945-3245"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerische Mathematik","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/A5053109099","display_name":"Michele Benzi","orcid":"https://orcid.org/0000-0001-6099-0294"},"institutions":[{"id":"https://openalex.org/I150468666","display_name":"Emory University","ror":"https://ror.org/03czfpz43","country_code":"US","type":"education","lineage":["https://openalex.org/I150468666"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Michele Benzi","raw_affiliation_strings":["Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA; e-mail: benzi@mathcs.emory.edu\n, , , , , , US","Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA; e-mail: benzi@mathcs.emory.edu , US"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA; e-mail: benzi@mathcs.emory.edu\n, , , , , , US","institution_ids":["https://openalex.org/I150468666"]},{"raw_affiliation_string":"Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA; e-mail: benzi@mathcs.emory.edu , US","institution_ids":["https://openalex.org/I150468666"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5072635213","display_name":"Andreas Frommer","orcid":"https://orcid.org/0000-0002-6901-0114"},"institutions":[{"id":"https://openalex.org/I167360494","display_name":"University of Wuppertal","ror":"https://ror.org/00613ak93","country_code":"DE","type":"education","lineage":["https://openalex.org/I167360494"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Andreas Frommer","raw_affiliation_strings":["Fachbereich Mathematik, Universit\u00e4t Wuppertal, 42097 Wuppertal, Germany; e-mail: frommer@math.uni-wuppertal.de\n, , , , , , DE","Fachbereich Mathematik, Universit\u00e4t Wuppertal, 42097 Wuppertal, Germany; e-mail: frommer@math.uni-wuppertal.de#N#, , DE"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Fachbereich Mathematik, Universit\u00e4t Wuppertal, 42097 Wuppertal, Germany; e-mail: frommer@math.uni-wuppertal.de\n, , , , , , DE","institution_ids":["https://openalex.org/I167360494"]},{"raw_affiliation_string":"Fachbereich Mathematik, Universit\u00e4t Wuppertal, 42097 Wuppertal, Germany; e-mail: frommer@math.uni-wuppertal.de#N#, , DE","institution_ids":["https://openalex.org/I167360494"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5082240312","display_name":"Reinhard Nabben","orcid":"https://orcid.org/0000-0001-5076-9895"},"institutions":[{"id":"https://openalex.org/I20121455","display_name":"Bielefeld University","ror":"https://ror.org/02hpadn98","country_code":"DE","type":"education","lineage":["https://openalex.org/I20121455"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Reinhard Nabben","raw_affiliation_strings":["Fakult\u00e4t f\u00fcr Mathematik, Universit\u00e4t Bielefeld, Postfach 10 01 31, 33501 Bielefeld,Germany; e-mail: nabben@mathematik.uni-bielefeld.de\n, , , , , , DE","Fakult\u00e4t f\u00fcr Mathematik, Universit\u00e4t Bielefeld, Postfach 10 01 31, 33501 Bielefeld,Germany; e-mail: nabben@mathematik.uni-bielefeld.de#N#, , DE"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Fakult\u00e4t f\u00fcr Mathematik, Universit\u00e4t Bielefeld, Postfach 10 01 31, 33501 Bielefeld,Germany; e-mail: nabben@mathematik.uni-bielefeld.de\n, , , , , , DE","institution_ids":["https://openalex.org/I20121455"]},{"raw_affiliation_string":"Fakult\u00e4t f\u00fcr Mathematik, Universit\u00e4t Bielefeld, Postfach 10 01 31, 33501 Bielefeld,Germany; e-mail: nabben@mathematik.uni-bielefeld.de#N#, , DE","institution_ids":["https://openalex.org/I20121455"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5047875137","display_name":"Daniel B. Szyld","orcid":"https://orcid.org/0000-0001-8010-0391"},"institutions":[{"id":"https://openalex.org/I84392919","display_name":"Temple University","ror":"https://ror.org/00kx1jb78","country_code":"US","type":"education","lineage":["https://openalex.org/I84392919"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Daniel B. Szyld","raw_affiliation_strings":["Department of Mathematics, Temple University, Philadelphia, PA 19122-6094, USA; e-mail: szyld@math.temple.edu\n, , , , , , US","Department of Mathematics, Temple University, Philadelphia, PA 19122-6094, USA; e-mail: szyld@math.temple.edu , US"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Temple University, Philadelphia, PA 19122-6094, USA; e-mail: szyld@math.temple.edu\n, , , , , , US","institution_ids":["https://openalex.org/I84392919"]},{"raw_affiliation_string":"Department of Mathematics, Temple University, Philadelphia, PA 19122-6094, USA; e-mail: szyld@math.temple.edu , US","institution_ids":["https://openalex.org/I84392919"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":null,"fwci":5.9527,"has_fulltext":false,"cited_by_count":79,"citation_normalized_percentile":{"value":0.96262025,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"89","issue":"4","first_page":"605","last_page":"639"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":1.0,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"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":0.9980999827384949,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8607715964317322},{"id":"https://openalex.org/keywords/schwarz-alternating-method","display_name":"Schwarz alternating method","score":0.8033546805381775},{"id":"https://openalex.org/keywords/invertible-matrix","display_name":"Invertible matrix","score":0.7478089928627014},{"id":"https://openalex.org/keywords/additive-schwarz-method","display_name":"Additive Schwarz method","score":0.7109374403953552},{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.6708288192749023},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.6160810589790344},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5856874585151672},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5385915040969849},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5256598591804504},{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.49456140398979187},{"id":"https://openalex.org/keywords/coefficient-matrix","display_name":"Coefficient matrix","score":0.47828349471092224},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.32097429037094116},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.26907044649124146},{"id":"https://openalex.org/keywords/domain-decomposition-methods","display_name":"Domain decomposition methods","score":0.16093596816062927},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.11119288206100464}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8607715964317322},{"id":"https://openalex.org/C31278502","wikidata":"https://www.wikidata.org/wiki/Q7433186","display_name":"Schwarz alternating method","level":4,"score":0.8033546805381775},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.7478089928627014},{"id":"https://openalex.org/C174819683","wikidata":"https://www.wikidata.org/wiki/Q4681340","display_name":"Additive Schwarz method","level":4,"score":0.7109374403953552},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.6708288192749023},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.6160810589790344},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5856874585151672},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5385915040969849},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5256598591804504},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.49456140398979187},{"id":"https://openalex.org/C60866291","wikidata":"https://www.wikidata.org/wiki/Q5140577","display_name":"Coefficient matrix","level":3,"score":0.47828349471092224},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.32097429037094116},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.26907044649124146},{"id":"https://openalex.org/C198880260","wikidata":"https://www.wikidata.org/wiki/Q5289813","display_name":"Domain decomposition methods","level":3,"score":0.16093596816062927},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.11119288206100464},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","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}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1007/s002110100275","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s002110100275","pdf_url":null,"source":{"id":"https://openalex.org/S4210205683","display_name":"Numerische Mathematik","issn_l":"0029-599X","issn":["0029-599X","0945-3245"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerische Mathematik","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.130.3364","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.130.3364","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.temple.edu/~szyld/papers/multShw_NM.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.134.2873","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.134.2873","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.mathcs.emory.edu/~benzi/Web_papers/msm.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.33.4674","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.4674","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.uni-wuppertal.de/org/SciComp/Preprints/SC0007.ps.gz","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.35.910","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.35.910","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.c3.lanl.gov/~benzi/Web_papers/schwarz.ps.gz","raw_type":"text"},{"id":"pmh:oai:ricerca.sns.it:11384/75229","is_oa":false,"landing_page_url":"http://hdl.handle.net/11384/75229","pdf_url":null,"source":{"id":"https://openalex.org/S7407050981","display_name":"Scuola Normale Superiore di Pisa","issn_l":null,"issn":[],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":49,"referenced_works":["https://openalex.org/W105972517","https://openalex.org/W207685730","https://openalex.org/W1500921805","https://openalex.org/W1540854488","https://openalex.org/W1542243431","https://openalex.org/W1608601220","https://openalex.org/W1965647439","https://openalex.org/W1970261865","https://openalex.org/W1976758549","https://openalex.org/W1988048764","https://openalex.org/W1988478906","https://openalex.org/W1992208469","https://openalex.org/W1992381145","https://openalex.org/W1994696552","https://openalex.org/W1995124160","https://openalex.org/W2000582360","https://openalex.org/W2005124301","https://openalex.org/W2005470604","https://openalex.org/W2010067982","https://openalex.org/W2016499627","https://openalex.org/W2017085690","https://openalex.org/W2018834578","https://openalex.org/W2021257096","https://openalex.org/W2043369156","https://openalex.org/W2049664825","https://openalex.org/W2051234298","https://openalex.org/W2052114329","https://openalex.org/W2053157162","https://openalex.org/W2059356446","https://openalex.org/W2062246266","https://openalex.org/W2071688940","https://openalex.org/W2083261109","https://openalex.org/W2083883802","https://openalex.org/W2093184664","https://openalex.org/W2101196233","https://openalex.org/W2106898404","https://openalex.org/W2106990738","https://openalex.org/W2110275812","https://openalex.org/W2114226919","https://openalex.org/W2119862562","https://openalex.org/W2126036157","https://openalex.org/W2155444267","https://openalex.org/W2162241008","https://openalex.org/W2314343349","https://openalex.org/W2315566124","https://openalex.org/W2330756801","https://openalex.org/W2588081979","https://openalex.org/W3035235874","https://openalex.org/W3185776723"],"related_works":["https://openalex.org/W2902154408","https://openalex.org/W2159100661","https://openalex.org/W2000290814","https://openalex.org/W2028890215","https://openalex.org/W3839471","https://openalex.org/W2339825459","https://openalex.org/W2138840323","https://openalex.org/W1994696552","https://openalex.org/W1647103664","https://openalex.org/W2042271947"],"abstract_inverted_index":null,"counts_by_year":[{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
