{"id":"https://openalex.org/W1970912568","doi":"https://doi.org/10.1137/070710858","title":"Galerkin Methods for Parabolic and Schr\u00f6dinger Equations with Dynamical Boundary Conditions and Applications to Underwater Acoustics","display_name":"Galerkin Methods for Parabolic and Schr\u00f6dinger Equations with Dynamical Boundary Conditions and Applications to Underwater Acoustics","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W1970912568","doi":"https://doi.org/10.1137/070710858","mag":"1970912568"},"language":"en","primary_location":{"id":"doi:10.1137/070710858","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070710858","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Journal on Numerical Analysis","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/A5015893485","display_name":"Dimitra C. Antonopoulou","orcid":"https://orcid.org/0000-0003-1511-6623"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"D. C. Antonopoulou","raw_affiliation_strings":["danton@tem.uoc.gr#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"danton@tem.uoc.gr#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5113573316","display_name":"Vassilios A. Dougalis","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"V. A. Dougalis","raw_affiliation_strings":["doug@math.uoa.gr#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"doug@math.uoa.gr#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5075381508","display_name":"Georgios E. Zouraris","orcid":"https://orcid.org/0000-0003-0234-0175"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"G. E. Zouraris","raw_affiliation_strings":["zouraris@math.uoc.gr#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"zouraris@math.uoc.gr#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.2821,"has_fulltext":false,"cited_by_count":12,"citation_normalized_percentile":{"value":0.78811548,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"47","issue":"4","first_page":"2752","last_page":"2781"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9994999766349792,"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9994999766349792,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9980000257492065,"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/T11205","display_name":"Numerical methods in inverse problems","score":0.9970999956130981,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7816088199615479},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.69767165184021},{"id":"https://openalex.org/keywords/helmholtz-equation","display_name":"Helmholtz equation","score":0.6382960081100464},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.631546139717102},{"id":"https://openalex.org/keywords/dissipative-system","display_name":"Dissipative system","score":0.6121328473091125},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.5504848957061768},{"id":"https://openalex.org/keywords/parabolic-partial-differential-equation","display_name":"Parabolic partial differential equation","score":0.48689761757850647},{"id":"https://openalex.org/keywords/underwater-acoustics","display_name":"Underwater acoustics","score":0.43890780210494995},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.38942816853523254},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.36707353591918945},{"id":"https://openalex.org/keywords/underwater","display_name":"Underwater","score":0.20578891038894653},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.13645151257514954}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7816088199615479},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.69767165184021},{"id":"https://openalex.org/C18591234","wikidata":"https://www.wikidata.org/wiki/Q860615","display_name":"Helmholtz equation","level":3,"score":0.6382960081100464},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.631546139717102},{"id":"https://openalex.org/C99692599","wikidata":"https://www.wikidata.org/wiki/Q596577","display_name":"Dissipative system","level":2,"score":0.6121328473091125},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.5504848957061768},{"id":"https://openalex.org/C186867907","wikidata":"https://www.wikidata.org/wiki/Q1575416","display_name":"Parabolic partial differential equation","level":3,"score":0.48689761757850647},{"id":"https://openalex.org/C67467970","wikidata":"https://www.wikidata.org/wiki/Q1292425","display_name":"Underwater acoustics","level":3,"score":0.43890780210494995},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.38942816853523254},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.36707353591918945},{"id":"https://openalex.org/C98083399","wikidata":"https://www.wikidata.org/wiki/Q3246517","display_name":"Underwater","level":2,"score":0.20578891038894653},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.13645151257514954},{"id":"https://openalex.org/C111368507","wikidata":"https://www.wikidata.org/wiki/Q43518","display_name":"Oceanography","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/C127313418","wikidata":"https://www.wikidata.org/wiki/Q1069","display_name":"Geology","level":0,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/070710858","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070710858","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Journal on Numerical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.315.8174","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.315.8174","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/0904.3900v1.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.8600000143051147,"id":"https://metadata.un.org/sdg/14","display_name":"Life below water"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W1492326914","https://openalex.org/W1589236075","https://openalex.org/W1591593323","https://openalex.org/W1633205748","https://openalex.org/W1963947443","https://openalex.org/W1978066710","https://openalex.org/W1979770726","https://openalex.org/W1989395541","https://openalex.org/W1996959056","https://openalex.org/W1997031923","https://openalex.org/W2015961792","https://openalex.org/W2028186988","https://openalex.org/W2039374955","https://openalex.org/W2049167397","https://openalex.org/W2056282805","https://openalex.org/W2076986342","https://openalex.org/W2077070913","https://openalex.org/W2081344277","https://openalex.org/W2082621526","https://openalex.org/W2325041346","https://openalex.org/W2464114944","https://openalex.org/W2506667719","https://openalex.org/W2568772621","https://openalex.org/W3204708751","https://openalex.org/W4210970566","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2371200319","https://openalex.org/W2367609594","https://openalex.org/W2350320737","https://openalex.org/W2391455523","https://openalex.org/W2739518152","https://openalex.org/W2091416718","https://openalex.org/W2735258363","https://openalex.org/W2369806658","https://openalex.org/W2386706262","https://openalex.org/W1968480952"],"abstract_inverted_index":{"In":[0,118],"this":[1],"paper":[2],"we":[3],"consider":[4],"Galerkin-finite":[5],"element":[6],"methods":[7],"that":[8,103],"approximate":[9],"the":[10,43,54,61,67,80,83,105,119],"solutions":[11],"of":[12,31,34,86,93,131],"initial-boundary-value":[13],"problems":[14],"in":[15,36,71,79,82],"one":[16],"space":[17],"dimension":[18],"for":[19,42,127],"parabolic":[20,121],"and":[21,38,46,110,133],"Schr\u00f6dinger":[22,55],"evolution":[23],"equations":[24],"with":[25,89,100],"dynamical":[26,106,128],"boundary":[27,108,129],"conditions.":[28],"Error":[29],"estimates":[30],"optimal":[32],"rates":[33],"convergence":[35],"$L^2$":[37],"$H^1$":[39],"are":[40,111,125],"proved":[41],"associated":[44],"semidiscrete":[45],"fully":[47],"discrete":[48],"Crank\u2013Nicolson\u2013Galerkin":[49],"approximations.":[50,117],"The":[51],"problem":[52],"involving":[53],"equation":[56],"is":[57,98],"motivated":[58],"by":[59],"considering":[60],"standard":[62],"\u201cparabolic\u201d":[63],"(paraxial)":[64],"approximation":[65],"to":[66,74,113],"Helmholtz":[68],"equation,":[69],"used":[70],"underwater":[72],"acoustics":[73],"model":[75,97],"long-range":[76],"sound":[77],"propagation":[78],"sea,":[81],"specific":[84],"case":[85],"a":[87,90],"domain":[88],"rigid":[91],"bottom":[92,107],"variable":[94],"topography.":[95],"This":[96],"contrasted":[99],"alternative":[101],"ones":[102],"avoid":[104],"condition":[109],"shown":[112],"yield":[114],"qualitatively":[115],"better":[116],"(real)":[120],"case,":[122],"numerical":[123],"approximations":[124],"considered":[126],"conditions":[130],"reactive":[132],"dissipative":[134],"type.":[135]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2026-07-15T18:14:33.161393","created_date":"2025-10-10T00:00:00"}
