{"id":"https://openalex.org/W2107692173","doi":"https://doi.org/10.1137/120901301","title":"Is the Helmholtz Equation Really Sign-Indefinite?","display_name":"Is the Helmholtz Equation Really Sign-Indefinite?","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W2107692173","doi":"https://doi.org/10.1137/120901301","mag":"2107692173"},"language":"en","primary_location":{"id":"doi:10.1137/120901301","is_oa":false,"landing_page_url":"https://doi.org/10.1137/120901301","pdf_url":null,"source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Review","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://researchportal.bath.ac.uk/en/publications/ade9e3bf-abbc-42b7-9989-dc8e4983e662","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5045318480","display_name":"Andrea Moiola","orcid":"https://orcid.org/0000-0002-6251-4440"},"institutions":[{"id":"https://openalex.org/I71052956","display_name":"University of Reading","ror":"https://ror.org/05v62cm79","country_code":"GB","type":"education","lineage":["https://openalex.org/I71052956"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Andrea Moiola","raw_affiliation_strings":["University of Reading,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Reading,","institution_ids":["https://openalex.org/I71052956"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5083248653","display_name":"Euan A. Spence","orcid":"https://orcid.org/0000-0003-1236-4592"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Euan A. Spence","raw_affiliation_strings":["Department of Mathematical Sciences"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":8.1003,"has_fulltext":true,"cited_by_count":75,"citation_normalized_percentile":{"value":0.97890562,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":99},"biblio":{"volume":"56","issue":"2","first_page":"274","last_page":"312"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10514","display_name":"Numerical methods in engineering","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T10514","display_name":"Numerical methods in engineering","score":1.0,"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"}},{"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/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"}}],"keywords":[{"id":"https://openalex.org/keywords/helmholtz-equation","display_name":"Helmholtz equation","score":0.8519400358200073},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7285830974578857},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.6693809032440186},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5346751809120178},{"id":"https://openalex.org/keywords/sign","display_name":"Sign (mathematics)","score":0.4806350767612457},{"id":"https://openalex.org/keywords/bilinear-form","display_name":"Bilinear form","score":0.4792247414588928},{"id":"https://openalex.org/keywords/dirichlet-problem","display_name":"Dirichlet problem","score":0.46824920177459717},{"id":"https://openalex.org/keywords/dirichlet-boundary-condition","display_name":"Dirichlet boundary condition","score":0.45839154720306396},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.45643484592437744},{"id":"https://openalex.org/keywords/helmholtz-free-energy","display_name":"Helmholtz free energy","score":0.4329547584056854},{"id":"https://openalex.org/keywords/wavenumber","display_name":"Wavenumber","score":0.4308188557624817},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.41133785247802734},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.19106602668762207}],"concepts":[{"id":"https://openalex.org/C18591234","wikidata":"https://www.wikidata.org/wiki/Q860615","display_name":"Helmholtz equation","level":3,"score":0.8519400358200073},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7285830974578857},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6693809032440186},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5346751809120178},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.4806350767612457},{"id":"https://openalex.org/C8828549","wikidata":"https://www.wikidata.org/wiki/Q837924","display_name":"Bilinear form","level":2,"score":0.4792247414588928},{"id":"https://openalex.org/C2779560616","wikidata":"https://www.wikidata.org/wiki/Q1192869","display_name":"Dirichlet problem","level":3,"score":0.46824920177459717},{"id":"https://openalex.org/C110167270","wikidata":"https://www.wikidata.org/wiki/Q1193699","display_name":"Dirichlet boundary condition","level":3,"score":0.45839154720306396},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.45643484592437744},{"id":"https://openalex.org/C27592594","wikidata":"https://www.wikidata.org/wiki/Q865821","display_name":"Helmholtz free energy","level":2,"score":0.4329547584056854},{"id":"https://openalex.org/C121130766","wikidata":"https://www.wikidata.org/wiki/Q192510","display_name":"Wavenumber","level":2,"score":0.4308188557624817},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.41133785247802734},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.19106602668762207},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","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}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1137/120901301","is_oa":false,"landing_page_url":"https://doi.org/10.1137/120901301","pdf_url":null,"source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Review","raw_type":"journal-article"},{"id":"pmh:oai:centaur.reading.ac.uk:34175","is_oa":false,"landing_page_url":"https://centaur.reading.ac.uk/view/creators/90005242.html>","pdf_url":null,"source":{"id":"https://openalex.org/S4306402273","display_name":"CentAUR (University of Reading)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I71052956","host_organization_name":"University of Reading","host_organization_lineage":["https://openalex.org/I71052956"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":null,"raw_type":"PeerReviewed"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.298.7097","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.298.7097","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.640.2269","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.640.2269","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.personal.reading.ac.uk/~st904897/Moiola-Spence--SiamReview2014-IsTheHelmholtzEquationReallySign-indefinite.pdf","raw_type":"text"},{"id":"pmh:oai:opus.bath.ac.uk:40162","is_oa":false,"landing_page_url":"http://opus.bath.ac.uk/40162/1/sirev_final.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306401631","display_name":"The University of Bath Online Publications Store (The 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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Articles"},{"id":"pmh:oai:purehost.bath.ac.uk:openaire/ade9e3bf-abbc-42b7-9989-dc8e4983e662","is_oa":true,"landing_page_url":"https://researchportal.bath.ac.uk/en/publications/ade9e3bf-abbc-42b7-9989-dc8e4983e662","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":"Moiola, A & Spence, E A 2014, 'Is the helmholtz equation really sign-indefinite?', Siam Review, vol. 56, no. 2, pp. 274-312. https://doi.org/10.1137/120901301","raw_type":"article"},{"id":"pmh:oai:purehost.bath.ac.uk:publications/ade9e3bf-abbc-42b7-9989-dc8e4983e662","is_oa":false,"landing_page_url":"http://www.scopus.com/inward/record.url?scp=84901342997&partnerID=8YFLogxK","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":""}],"best_oa_location":{"id":"pmh:oai:purehost.bath.ac.uk:openaire/ade9e3bf-abbc-42b7-9989-dc8e4983e662","is_oa":true,"landing_page_url":"https://researchportal.bath.ac.uk/en/publications/ade9e3bf-abbc-42b7-9989-dc8e4983e662","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":"Moiola, A & Spence, E A 2014, 'Is the helmholtz equation really sign-indefinite?', Siam Review, vol. 56, no. 2, pp. 274-312. https://doi.org/10.1137/120901301","raw_type":"article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4579076251","display_name":null,"funder_award_id":"EP/1025995/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G7599046667","display_name":null,"funder_award_id":"EP/I025995/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320310862","display_name":"Colorado School of Mines","ror":"https://ror.org/04raf6v53"},{"id":"https://openalex.org/F4320310925","display_name":"Universit\u00e9 de Lorraine","ror":"https://ror.org/04vfs2w97"},{"id":"https://openalex.org/F4320320286","display_name":"University College London","ror":"https://ror.org/02jx3x895"},{"id":"https://openalex.org/F4320320322","display_name":"University of Bath","ror":"https://ror.org/002h8g185"},{"id":"https://openalex.org/F4320320326","display_name":"University of Reading","ror":"https://ror.org/05v62cm79"},{"id":"https://openalex.org/F4320321652","display_name":"Eidgen\u00f6ssische Technische Hochschule Z\u00fcrich","ror":"https://ror.org/05a28rw58"},{"id":"https://openalex.org/F4320322839","display_name":"Technische Universit\u00e4t Wien","ror":"https://ror.org/04d836q62"},{"id":"https://openalex.org/F4320322874","display_name":"Universit\u00e0 degli Studi di Pavia","ror":"https://ror.org/00s6t1f81"},{"id":"https://openalex.org/F4320323830","display_name":"Universit\u00e9 de Gen\u00e8ve","ror":"https://ror.org/01swzsf04"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":75,"referenced_works":["https://openalex.org/W4509327","https://openalex.org/W8052360","https://openalex.org/W147220820","https://openalex.org/W592368443","https://openalex.org/W1492326914","https://openalex.org/W1504642953","https://openalex.org/W1520256736","https://openalex.org/W1550543785","https://openalex.org/W1555557982","https://openalex.org/W1579884124","https://openalex.org/W1582734821","https://openalex.org/W1586623601","https://openalex.org/W1593038947","https://openalex.org/W1602383637","https://openalex.org/W1604809219","https://openalex.org/W1678850202","https://openalex.org/W1725492752","https://openalex.org/W1965980050","https://openalex.org/W1967387621","https://openalex.org/W1967469744","https://openalex.org/W1969600280","https://openalex.org/W1969745760","https://openalex.org/W1970622200","https://openalex.org/W1975183790","https://openalex.org/W1979692628","https://openalex.org/W1980928855","https://openalex.org/W1987862116","https://openalex.org/W1996774337","https://openalex.org/W2003803951","https://openalex.org/W2004993994","https://openalex.org/W2005436751","https://openalex.org/W2005672272","https://openalex.org/W2009961626","https://openalex.org/W2017071204","https://openalex.org/W2020818415","https://openalex.org/W2025124289","https://openalex.org/W2029342920","https://openalex.org/W2029966922","https://openalex.org/W2030596198","https://openalex.org/W2031621501","https://openalex.org/W2032941427","https://openalex.org/W2037475392","https://openalex.org/W2038870672","https://openalex.org/W2050690821","https://openalex.org/W2056171461","https://openalex.org/W2063679404","https://openalex.org/W2064666322","https://openalex.org/W2068395392","https://openalex.org/W2077458918","https://openalex.org/W2080212747","https://openalex.org/W2080631773","https://openalex.org/W2080668001","https://openalex.org/W2084991940","https://openalex.org/W2087828859","https://openalex.org/W2091222631","https://openalex.org/W2092011174","https://openalex.org/W2095572167","https://openalex.org/W2103142061","https://openalex.org/W2113949199","https://openalex.org/W2115850109","https://openalex.org/W2133248904","https://openalex.org/W2151832515","https://openalex.org/W2152583013","https://openalex.org/W2152619933","https://openalex.org/W2156154693","https://openalex.org/W2168644075","https://openalex.org/W2168914934","https://openalex.org/W2289301134","https://openalex.org/W2326586272","https://openalex.org/W2470546557","https://openalex.org/W2494040823","https://openalex.org/W2963251793","https://openalex.org/W3021722416","https://openalex.org/W3106381385","https://openalex.org/W3207230352"],"related_works":["https://openalex.org/W4365420316","https://openalex.org/W3193896743","https://openalex.org/W3176862315","https://openalex.org/W3120022531","https://openalex.org/W1988524666","https://openalex.org/W4245646221","https://openalex.org/W2171059325","https://openalex.org/W2032673468","https://openalex.org/W2306590305","https://openalex.org/W4378378759"],"abstract_inverted_index":{"The":[0],"usual":[1],"variational":[2,68,87,161],"(or":[3],"weak)":[4],"formulations":[5,88,129,165],"of":[6,27,49,62,89,113,130,139,150],"the":[7,13,16,28,47,50,63,67,78,85,90,97,107,110,131,137,148,159,170],"Helmholtz":[8,51,91,132,171],"equation":[9,52,92,133,172],"are":[10,93],"sign-indefinite":[11],"in":[12,40,75,135],"sense":[14],"that":[15,58],"bilinear":[17],"forms":[18],"cannot":[19,70],"be":[20,71],"bounded":[21,41],"below":[22],"by":[23,173,179],"a":[24,36,140,151],"positive":[25],"multiple":[26],"appropriate":[29],"norm":[30],"squared.":[31],"This":[32,101],"is":[33,53,80,99],"often":[34],"for":[35,82,105],"good":[37],"reason,":[38],"since":[39],"domains":[42],"under":[43],"certain":[44],"boundary":[45,145,156],"conditions":[46,146],"solution":[48,79],"not":[54],"unique":[55,81],"at":[56],"wavenumbers":[57],"correspond":[59],"to":[60],"eigenvalues":[61],"Laplacian,":[64],"and":[65,109,177],"thus":[66],"problem":[69],"sign-definite.":[72],"However,":[73],"even":[74],"cases":[76],"where":[77],"all":[83],"wavenumbers,":[84],"standard":[86,160],"still":[94],"indefinite":[95],"when":[96],"wavenumber":[98],"large.":[100],"indefiniteness":[102],"has":[103],"implications":[104],"both":[106],"analysis":[108],"practical":[111],"implementation":[112],"finite":[114],"element":[115],"methods.":[116],"In":[117],"this":[118],"paper":[119],"we":[120],"introduce":[121],"new":[122,164],"sign-definite":[123],"(also":[124],"called":[125],"coercive":[126],"or":[127,147],"elliptic)":[128],"posed":[134],"either":[136],"interior":[138],"star-shaped":[141,152],"domain":[142,153],"with":[143,154],"impedance":[144],"exterior":[149],"Dirichlet":[155],"conditions.":[157],"Like":[158],"formulations,":[162],"these":[163],"arise":[166],"from":[167],"simply":[168],"multiplying":[169],"particular":[174],"test":[175],"functions":[176],"integrating":[178],"parts.":[180]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":7},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":10},{"year":2019,"cited_by_count":7},{"year":2018,"cited_by_count":9},{"year":2017,"cited_by_count":5},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":7}],"updated_date":"2026-06-19T17:40:00.097472","created_date":"2025-10-10T00:00:00"}