{"id":"https://openalex.org/W2196996317","doi":"https://doi.org/10.1090/mcom/3190","title":"Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption","display_name":"Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption","publication_year":2016,"publication_date":"2016-06-28","ids":{"openalex":"https://openalex.org/W2196996317","doi":"https://doi.org/10.1090/mcom/3190","mag":"2196996317"},"language":"en","primary_location":{"id":"doi:10.1090/mcom/3190","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/3190","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":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1507.02097","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","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":"I. Graham","raw_affiliation_strings":["Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom","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"]},{"raw_affiliation_string":"Department of Mathematical Sciences University of Bath  Bath BA2 7AY United Kingdom","institution_ids":["https://openalex.org/I51601045"]}]},{"author_position":"middle","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":"E. Spence","raw_affiliation_strings":["Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom","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"]},{"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/A5034786931","display_name":"Eero Vainikko","orcid":"https://orcid.org/0000-0003-1902-9383"},"institutions":[{"id":"https://openalex.org/I56085075","display_name":"University of Tartu","ror":"https://ror.org/03z77qz90","country_code":"EE","type":"education","lineage":["https://openalex.org/I56085075"]}],"countries":["EE"],"is_corresponding":false,"raw_author_name":"E. Vainikko","raw_affiliation_strings":["Institute of Computer Science, University of Tartu, 50409, Estonia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Computer Science, University of Tartu, 50409, Estonia","institution_ids":["https://openalex.org/I56085075"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.3088,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.58340981,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"86","issue":"307","first_page":"2089","last_page":"2127"},"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":0.9997000098228455,"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":0.9997000098228455,"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":0.9997000098228455,"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"}},{"id":"https://openalex.org/T10514","display_name":"Numerical methods in engineering","score":0.9993000030517578,"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/helmholtz-equation","display_name":"Helmholtz equation","score":0.7633122205734253},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6641262769699097},{"id":"https://openalex.org/keywords/preconditioner","display_name":"Preconditioner","score":0.6517229080200195},{"id":"https://openalex.org/keywords/domain-decomposition-methods","display_name":"Domain decomposition methods","score":0.634850025177002},{"id":"https://openalex.org/keywords/generalized-minimal-residual-method","display_name":"Generalized minimal residual method","score":0.6252437233924866},{"id":"https://openalex.org/keywords/helmholtz-free-energy","display_name":"Helmholtz free energy","score":0.6088238954544067},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.48596203327178955},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.45314109325408936},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4274476170539856},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4272116422653198},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.38843631744384766},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.19670352339744568},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.1419420838356018},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.10127094388008118}],"concepts":[{"id":"https://openalex.org/C18591234","wikidata":"https://www.wikidata.org/wiki/Q860615","display_name":"Helmholtz equation","level":3,"score":0.7633122205734253},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6641262769699097},{"id":"https://openalex.org/C167431342","wikidata":"https://www.wikidata.org/wiki/Q1754327","display_name":"Preconditioner","level":3,"score":0.6517229080200195},{"id":"https://openalex.org/C198880260","wikidata":"https://www.wikidata.org/wiki/Q5289813","display_name":"Domain decomposition methods","level":3,"score":0.634850025177002},{"id":"https://openalex.org/C155332342","wikidata":"https://www.wikidata.org/wiki/Q1432976","display_name":"Generalized minimal residual method","level":3,"score":0.6252437233924866},{"id":"https://openalex.org/C27592594","wikidata":"https://www.wikidata.org/wiki/Q865821","display_name":"Helmholtz free energy","level":2,"score":0.6088238954544067},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.48596203327178955},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.45314109325408936},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4274476170539856},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4272116422653198},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.38843631744384766},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.19670352339744568},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.1419420838356018},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.10127094388008118},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1090/mcom/3190","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/3190","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:1507.02097","is_oa":true,"landing_page_url":"https://arxiv.org/abs/1507.02097","pdf_url":"https://arxiv.org/pdf/1507.02097","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":"text"},{"id":"mag:2196996317","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1507.02097.pdf","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:purehost.bath.ac.uk:openaire/4b265f8a-a0c8-4a61-aa46-a47937e7031d","is_oa":true,"landing_page_url":"https://researchportal.bath.ac.uk/en/publications/4b265f8a-a0c8-4a61-aa46-a47937e7031d","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":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Graham, I G, Spence, E A & Vainikko, E 2017, 'Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption', Mathematics of Computation, vol. 86, no. 307, pp. 2089-2127. https://doi.org/10.1090/mcom/3190","raw_type":"article"},{"id":"doi:10.48550/arxiv.1507.02097","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1507.02097","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":"pmh:oai:arXiv.org:1507.02097","is_oa":true,"landing_page_url":"https://arxiv.org/abs/1507.02097","pdf_url":"https://arxiv.org/pdf/1507.02097","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":33,"referenced_works":["https://openalex.org/W582364689","https://openalex.org/W591292934","https://openalex.org/W1210265535","https://openalex.org/W1548589512","https://openalex.org/W1586623601","https://openalex.org/W1604809219","https://openalex.org/W1673309917","https://openalex.org/W1969745760","https://openalex.org/W1976057809","https://openalex.org/W1986629428","https://openalex.org/W1987152170","https://openalex.org/W2009985306","https://openalex.org/W2030947573","https://openalex.org/W2035238652","https://openalex.org/W2041226551","https://openalex.org/W2057709785","https://openalex.org/W2068395392","https://openalex.org/W2084073638","https://openalex.org/W2084463157","https://openalex.org/W2095572167","https://openalex.org/W2095956181","https://openalex.org/W2098102734","https://openalex.org/W2108572708","https://openalex.org/W2115272362","https://openalex.org/W2117293438","https://openalex.org/W2142094322","https://openalex.org/W2158511063","https://openalex.org/W2161243591","https://openalex.org/W2165762596","https://openalex.org/W2169549054","https://openalex.org/W2469720646","https://openalex.org/W2952652025","https://openalex.org/W3123875103"],"related_works":["https://openalex.org/W2963726548","https://openalex.org/W2469720646","https://openalex.org/W1979817432","https://openalex.org/W2767278683","https://openalex.org/W2184482746","https://openalex.org/W2964987866","https://openalex.org/W2147235190","https://openalex.org/W2949776645","https://openalex.org/W2949331335","https://openalex.org/W2982389629","https://openalex.org/W2951143782","https://openalex.org/W2337689163","https://openalex.org/W2033258523","https://openalex.org/W2166217501","https://openalex.org/W2982609610","https://openalex.org/W2048559463","https://openalex.org/W640280325","https://openalex.org/W2891319366","https://openalex.org/W2331290794","https://openalex.org/W1685985699"],"abstract_inverted_index":{"In":[0],"this":[1,131],"paper":[2,463],"we":[3,464],"give":[4,410],"new":[5,242],"results":[6],"on":[7],"domain":[8,424],"decomposition":[9,425],"preconditioners":[10,161,429],"for":[11,162,195,202,244,253,307,314,367,404,415,430,470],"GMRES":[12,409],"when":[13],"computing":[14],"piecewise":[15],"linear":[16],"finite-element":[17],"approximations":[18,129,426],"of":[19,130,251,261,266,268,400,461,482],"the":[20,163,203,209,245,262,269,308,368,372,388,392,402,416,431,459,462,471],"Helmholtz":[21,165,205,433,473],"equation":[22,132,211,247],"<inline-formula":[23,96,134,168,213,276,290,318,435,484],"content-type=\"math/mathml\">":[24,97,135,169,214,277,291,319,436,485],"<mml:math":[25,98,136,170,215,278,292,320,437],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[26,99,137,171,216,279,293,321,438],"alttext=\"minus":[27],"normal":[28,37],"upper":[29,103],"Delta":[30],"u":[31,41,80],"minus":[32],"left-parenthesis":[33],"k":[34,326],"squared":[35],"plus":[36],"i":[38],"epsilon":[39,323],"right-parenthesis":[40],"equals":[42,173,440],"f\">":[43],"<mml:semantics>":[44,105,141,175,220,281,295,328,442],"<mml:mrow>":[45,106,142,176,221,329,443],"<mml:mo>":[46,54,110,343],"\u2212":[47,55],"</mml:mo>":[48,56,112,345],"<mml:mi":[49,66,115],"mathvariant=\"normal\">":[50],"\u0394":[51],"</mml:mi>":[52,71,109,145,179,224,298,337,446],"<mml:mi>u</mml:mi>":[53,74],"<mml:mo":[57,72,332,340],"stretchy=\"false\">(</mml:mo>":[58],"<mml:msup>":[59,346],"<mml:mi>k</mml:mi>":[60,282,347],"<mml:mn>2</mml:mn>":[61,348],"</mml:msup>":[62,349],"<mml:mo>+</mml:mo>":[63],"<mml:mrow":[64,113,330,338],"class=\"MJX-TeXAtom-ORD\">":[65,114,331,339],"mathvariant=\"normal\">i</mml:mi>":[67],"</mml:mrow>":[68,77,117,118,148,182,227,334,342,350,449],"<mml:mi>":[69,107,143,177,222,296,335,444],"\u03b5":[70,108,144,178,223,297,336,445],"stretchy=\"false\">)</mml:mo>":[73],"<mml:mo>=</mml:mo>":[75,180,447],"<mml:mi>f</mml:mi>":[76],"<mml:annotation":[78,119,149,183,228,283,299,351,450],"encoding=\"application/x-tex\">-\\Delta":[79],"-":[81,288],"(k^2+":[82],"\\mathrm":[83],"{i}":[84],"\\varepsilon":[85],")u":[86],"=":[87,152,185,452],"f</mml:annotation>":[88],"</mml:semantics>":[89,124,154,187,232,285,301,355,454],"</mml:math>":[90,125,155,188,233,286,302,356,455],"</inline-formula>":[91,126,156,189,234,287,303,357,456],",":[92,358],"with":[93,133],"absorption":[94],"parameter":[95],"alttext=\"epsilon":[100,138,172,217,439],"element-of":[101],"double-struck":[102],"R\">":[104],"\u2208":[111],"mathvariant=\"double-struck\">R</mml:mi>":[116],"encoding=\"application/x-tex\">\\varepsilon":[120,150,184,229,451],"\\in":[121],"\\mathbb":[122],"{R}</mml:annotation>":[123],".":[127,457],"Multigrid":[128],"not-equals":[139,218],"0\">":[140,174,219,441],"<mml:mo>\u2260</mml:mo>":[146,225],"<mml:mn>0</mml:mn>":[147,181,226,448],"\\not":[151,230],"0</mml:annotation>":[153,186,453],"are":[157,378,384],"commonly":[158],"used":[159],"as":[160,428],"pure":[164,204,432,472],"case":[166,418,434],"(":[167,212],").":[190],"However":[191],"a":[192,241,275,396,466],"rigorous":[193],"theory":[194,243,273,393,414],"such":[196],"(so-called":[197],"\u201cshifted":[198],"Laplace\u201d)":[199],"preconditioners,":[200],"either":[201],"equation,":[206],"or":[207,255],"even":[208],"absorptive":[210,246,417],"=0</mml:annotation>":[231],"),":[235],"is":[236],"still":[237],"missing.":[238],"We":[239],"present":[240],"that":[248,316,386,475],"provides":[249],"rates":[250],"convergence":[252],"(left-":[254],"right-)":[256],"preconditioned":[257,270],"GMRES,":[258,401],"via":[259],"estimates":[260],"norm":[263],"and":[264,289,312,374,407],"field":[265],"values":[267],"matrix.":[271],"This":[272],"uses":[274],"alttext=\"k\">":[280],"encoding=\"application/x-tex\">k</mml:annotation>":[284],"alttext=\"epsilon\">":[294],"encoding=\"application/x-tex\">\\varepsilon</mml:annotation>":[300],"-explicit":[304],"coercivity":[305],"result":[306],"underlying":[309],"sesquilinear":[310],"form":[311],"shows,":[313],"example,":[315],"if":[317],"alttext=\"StartAbsoluteValue":[322],"EndAbsoluteValue":[324],"tilde":[325],"squared\">":[327],"stretchy=\"false\">|</mml:mo>":[333,341],"\u223c":[344],"encoding=\"application/x-tex\">|\\varepsilon":[352],"|\\sim":[353],"k^2</mml:annotation>":[354],"then":[359],"classical":[360,408],"overlapping":[361],"Additive":[362],"Schwarz":[363],"will":[364],"perform":[365,427],"optimally":[366],"damped":[369],"problem,":[370],"provided":[371],"subdomain":[373],"coarse":[375],"mesh":[376],"diameters":[377],"carefully":[379],"chosen.":[380],"Extensive":[381],"numerical":[382],"experiments":[383,403],"given":[385],"support":[387],"theoretical":[389],"results.":[390,412],"While":[391],"applies":[394],"to":[395],"certain":[397],"weighted":[398,406],"variant":[399],"both":[405],"comparable":[411],"The":[413],"gives":[419],"insight":[420],"into":[421],"how":[422],"its":[423],"At":[458],"end":[460],"propose":[465],"(scalable)":[467],"multilevel":[468],"preconditioner":[469],"problem":[474],"has":[476],"an":[477],"empirical":[478],"computation":[479],"time":[480],"complexity":[481],"about":[483]},"counts_by_year":[{"year":2019,"cited_by_count":1}],"updated_date":"2026-06-21T07:57:09.225873","created_date":"2025-10-10T00:00:00"}