{"id":"https://openalex.org/W3209791911","doi":"https://doi.org/10.1090/mcom/3772","title":"Convergence of restricted additive Schwarz with impedance transmission conditions for discretised Helmholtz problems","display_name":"Convergence of restricted additive Schwarz with impedance transmission conditions for discretised Helmholtz problems","publication_year":2022,"publication_date":"2022-10-06","ids":{"openalex":"https://openalex.org/W3209791911","doi":"https://doi.org/10.1090/mcom/3772","mag":"3209791911"},"language":"en","primary_location":{"id":"doi:10.1090/mcom/3772","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/3772","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2110.14495","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5076270422","display_name":"Shihua Gong","orcid":"https://orcid.org/0000-0003-3650-2283"},"institutions":[{"id":"https://openalex.org/I4210116924","display_name":"Chinese University of Hong Kong, Shenzhen","ror":"https://ror.org/02d5ks197","country_code":"CN","type":"education","lineage":["https://openalex.org/I177725633","https://openalex.org/I180726961","https://openalex.org/I4210116924"]},{"id":"https://openalex.org/I51601045","display_name":"University of Bath","ror":"https://ror.org/002h8g185","country_code":"GB","type":"education","lineage":["https://openalex.org/I51601045"]}],"countries":["CN","GB"],"is_corresponding":true,"raw_author_name":"Shihua Gong","raw_affiliation_strings":["Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom","School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom","institution_ids":["https://openalex.org/I51601045"]},{"raw_affiliation_string":"School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China","institution_ids":["https://openalex.org/I4210116924"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5064637859","display_name":"Ivan G. Graham","orcid":"https://orcid.org/0000-0002-5730-676X"},"institutions":[{"id":"https://openalex.org/I51601045","display_name":"University of Bath","ror":"https://ror.org/002h8g185","country_code":"GB","type":"education","lineage":["https://openalex.org/I51601045"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Ivan Graham","raw_affiliation_strings":["Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom","institution_ids":["https://openalex.org/I51601045"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5083248653","display_name":"Euan A. Spence","orcid":"https://orcid.org/0000-0003-1236-4592"},"institutions":[{"id":"https://openalex.org/I51601045","display_name":"University of Bath","ror":"https://ror.org/002h8g185","country_code":"GB","type":"education","lineage":["https://openalex.org/I51601045"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Euan Spence","raw_affiliation_strings":["Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom","institution_ids":["https://openalex.org/I51601045"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5076270422"],"corresponding_institution_ids":["https://openalex.org/I4210116924","https://openalex.org/I51601045"],"apc_list":null,"apc_paid":null,"fwci":1.6614,"has_fulltext":true,"cited_by_count":13,"citation_normalized_percentile":{"value":0.79505536,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"92","issue":"339","first_page":"175","last_page":"215"},"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":0.9998999834060669,"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":0.9998999834060669,"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":0.9998000264167786,"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"}},{"id":"https://openalex.org/T10514","display_name":"Numerical methods in engineering","score":0.9997000098228455,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/schwarz-alternating-method","display_name":"Schwarz alternating method","score":0.9354409575462341},{"id":"https://openalex.org/keywords/additive-schwarz-method","display_name":"Additive Schwarz method","score":0.8677111864089966},{"id":"https://openalex.org/keywords/preconditioner","display_name":"Preconditioner","score":0.8395861983299255},{"id":"https://openalex.org/keywords/domain-decomposition-methods","display_name":"Domain decomposition methods","score":0.7815762758255005},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7333685159683228},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.5938770174980164},{"id":"https://openalex.org/keywords/helmholtz-equation","display_name":"Helmholtz equation","score":0.5385677814483643},{"id":"https://openalex.org/keywords/helmholtz-free-energy","display_name":"Helmholtz free energy","score":0.5262136459350586},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5145353674888611},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.44199568033218384},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.43565016984939575},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.4202958941459656},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.401278555393219},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.35575830936431885},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.34098586440086365},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.19152948260307312}],"concepts":[{"id":"https://openalex.org/C31278502","wikidata":"https://www.wikidata.org/wiki/Q7433186","display_name":"Schwarz alternating method","level":4,"score":0.9354409575462341},{"id":"https://openalex.org/C174819683","wikidata":"https://www.wikidata.org/wiki/Q4681340","display_name":"Additive Schwarz method","level":4,"score":0.8677111864089966},{"id":"https://openalex.org/C167431342","wikidata":"https://www.wikidata.org/wiki/Q1754327","display_name":"Preconditioner","level":3,"score":0.8395861983299255},{"id":"https://openalex.org/C198880260","wikidata":"https://www.wikidata.org/wiki/Q5289813","display_name":"Domain decomposition methods","level":3,"score":0.7815762758255005},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7333685159683228},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.5938770174980164},{"id":"https://openalex.org/C18591234","wikidata":"https://www.wikidata.org/wiki/Q860615","display_name":"Helmholtz equation","level":3,"score":0.5385677814483643},{"id":"https://openalex.org/C27592594","wikidata":"https://www.wikidata.org/wiki/Q865821","display_name":"Helmholtz free energy","level":2,"score":0.5262136459350586},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5145353674888611},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.44199568033218384},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.43565016984939575},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4202958941459656},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.401278555393219},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.35575830936431885},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.34098586440086365},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.19152948260307312},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","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},{"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1090/mcom/3772","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/3772","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2110.14495","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2110.14495","pdf_url":"https://arxiv.org/pdf/2110.14495","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":null},{"id":"pmh:oai:purehost.bath.ac.uk:openaire_cris_publications/caa5d8ac-1d35-477c-83ae-d1d880fd6b83","is_oa":true,"landing_page_url":"https://researchportal.bath.ac.uk/en/publications/caa5d8ac-1d35-477c-83ae-d1d880fd6b83","pdf_url":"https://purehost.bath.ac.uk/ws/files/239061853/2110.14495.pdf","source":{"id":"https://openalex.org/S4377196294","display_name":"Pure (University of Bath)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I51601045","host_organization_name":"University of Bath","host_organization_lineage":["https://openalex.org/I51601045"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Gong, S, Graham, I G & Spence, E A 2023, 'Convergence of Restricted Additive Schwarz with impedance transmission conditions for discretised Helmholtz problems', Mathematics of Computation, vol. 92, no. 339, pp. 175-215. https://doi.org/10.1090/mcom/3772","raw_type":"article"},{"id":"pmh:oai:purehost.bath.ac.uk:openaire/caa5d8ac-1d35-477c-83ae-d1d880fd6b83","is_oa":true,"landing_page_url":"https://researchportal.bath.ac.uk/files/239061853/2110.14495.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4377196294","display_name":"Pure (University of Bath)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I51601045","host_organization_name":"University of Bath","host_organization_lineage":["https://openalex.org/I51601045"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Gong, S, Graham, I G & Spence, E A 2023, 'Convergence of Restricted Additive Schwarz with impedance transmission conditions for discretised Helmholtz problems', Mathematics of Computation, vol. 92, no. 339, pp. 175-215. https://doi.org/10.1090/mcom/3772","raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2110.14495","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2110.14495","pdf_url":"https://arxiv.org/pdf/2110.14495","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":null},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1847181937","display_name":"At the interface between semiclassical analysis and numerical analysis of wave propagation problems","funder_award_id":"EP/R005591/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G2547005790","display_name":null,"funder_award_id":"EP/S003975/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G8333796729","display_name":"Fast solvers for frequency-domain wave-scattering problems and applications","funder_award_id":"EP/S003975/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G8386841362","display_name":null,"funder_award_id":"EP/R005591/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":45,"referenced_works":["https://openalex.org/W1602383637","https://openalex.org/W1780245888","https://openalex.org/W1970622200","https://openalex.org/W1986812736","https://openalex.org/W1990258816","https://openalex.org/W2003803951","https://openalex.org/W2004475615","https://openalex.org/W2008215882","https://openalex.org/W2008747183","https://openalex.org/W2009985306","https://openalex.org/W2024381649","https://openalex.org/W2029342920","https://openalex.org/W2035238652","https://openalex.org/W2041226551","https://openalex.org/W2042199958","https://openalex.org/W2071928229","https://openalex.org/W2084463157","https://openalex.org/W2161243591","https://openalex.org/W2488658169","https://openalex.org/W2531211378","https://openalex.org/W2592456637","https://openalex.org/W2767278683","https://openalex.org/W2806526478","https://openalex.org/W2902526927","https://openalex.org/W2949839419","https://openalex.org/W2963292939","https://openalex.org/W3009935049","https://openalex.org/W3010737952","https://openalex.org/W3026323655","https://openalex.org/W3029481163","https://openalex.org/W3041771677","https://openalex.org/W3087680391","https://openalex.org/W3091209979","https://openalex.org/W3104324886","https://openalex.org/W3111536438","https://openalex.org/W3112570415","https://openalex.org/W3137034750","https://openalex.org/W3167951494","https://openalex.org/W3208590247","https://openalex.org/W3209791911","https://openalex.org/W4232597038","https://openalex.org/W4242794827","https://openalex.org/W4245007249","https://openalex.org/W4250985830","https://openalex.org/W6824878655"],"related_works":["https://openalex.org/W1986812736","https://openalex.org/W2902154408","https://openalex.org/W2963172443","https://openalex.org/W1623009832","https://openalex.org/W2471168230","https://openalex.org/W2773710110","https://openalex.org/W4229577910","https://openalex.org/W1554397852","https://openalex.org/W1877902382","https://openalex.org/W2940255121"],"abstract_inverted_index":{"The":[0,77,177],"Restricted":[1,14],"Additive":[2,15],"Schwarz":[3,16,99,109,171],"method":[4,110],"with":[5],"impedance":[6],"transmission":[7],"conditions,":[8],"also":[9,64],"known":[10],"as":[11,33,38,60,65,202],"the":[12,52,103,129,134,158,166,170,182,226],"Optimised":[13],"(ORAS)":[17],"method,":[18,27,100,172],"is":[19,88,161],"a":[20,39,55,66,95,138,195],"simple":[21],"overlapping":[22,124],"one-level":[23],"parallel":[24,97,123],"domain":[25,84,125,148],"decomposition":[26,126],"which":[28,191],"has":[29],"been":[30],"successfully":[31],"used":[32],"an":[34,61,89],"iterative":[35,62,98],"solver":[36],"and":[37,119,133],"preconditioner":[40],"for":[41,51,58,128,147],"discretised":[42],"Helmholtz":[43,71,130],"boundary-value":[44],"problems.":[45],"In":[46,145],"this":[47,143],"paper,":[48],"we":[49,154,192],"give,":[50],"first":[53],"time,":[54],"convergence":[56,167],"analysis":[57,78],"ORAS":[59,87,164,183],"solver\u2014and":[63],"preconditioner\u2014for":[67],"nodal":[68],"finite":[69,91],"element":[70,92],"systems":[72],"of":[73,94,122,142,169,174,187],"any":[74],"polynomial":[75,175],"order.":[76,176],"starts":[79],"by":[80],"showing":[81],"(for":[82],"general":[83],"decompositions)":[85],"that":[86],"unconventional":[90],"approximation":[93],"classical":[96],"formulated":[101],"at":[102],"PDE":[104],"(non-discrete)":[105],"level.":[106],"This":[107],"non-discrete":[108,231],"was":[111],"recently":[112],"analysed":[113],"in":[114,150,152,185,225],"[Gong,":[115],"Gander,":[116],"Graham,":[117],"Lafontaine,":[118],"Spence,":[120],"<italic>Convergence":[121],"methods":[127],"equation</italic>":[131],"],":[132],"present":[135],"paper":[136],"gives":[137],"corresponding":[139],"discrete":[140,188],"version":[141],"analysis.":[144],"particular,":[146],"decompositions":[149],"strips":[151],"2-d,":[153],"show":[155],"that,":[156],"when":[157],"mesh":[159],"size":[160],"small":[162],"enough,":[163],"inherits":[165],"properties":[168],"independent":[173],"proof":[178],"relies":[179],"on":[180],"characterising":[181],"iteration":[184],"terms":[186],"\u2018impedance-to-impedance":[189],"maps\u2019,":[190],"prove":[193],"(via":[194],"novel":[196],"weighted":[197],"finite-element":[198],"error":[199],"analysis)":[200],"converge":[201],"<inline-formula":[203],"content-type=\"math/mathml\">":[204],"<mml:math":[205],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[206],"alttext=\"h":[207],"right-arrow":[208],"0\">":[209],"<mml:semantics>":[210],"<mml:mrow>":[211],"<mml:mi>h</mml:mi>":[212],"<mml:mo":[213],"stretchy=\"false\">":[214],"\u2192":[215],"</mml:mo>":[216],"<mml:mn>0</mml:mn>":[217],"</mml:mrow>":[218],"<mml:annotation":[219],"encoding=\"application/x-tex\">h\\rightarrow":[220],"0</mml:annotation>":[221],"</mml:semantics>":[222],"</mml:math>":[223],"</inline-formula>":[224],"operator":[227],"norm":[228],"to":[229],"their":[230],"counterparts.":[232]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":4},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":4}],"updated_date":"2026-04-23T06:14:38.165362","created_date":"2022-11-30T00:00:00"}