{"id":"https://openalex.org/W2026688642","doi":"https://doi.org/10.1137/0917011","title":"Multigrid Waveform Relaxation on Spatial Finite Element Meshes: The Discrete-Time Case","display_name":"Multigrid Waveform Relaxation on Spatial Finite Element Meshes: The Discrete-Time Case","publication_year":1996,"publication_date":"1996-01-01","ids":{"openalex":"https://openalex.org/W2026688642","doi":"https://doi.org/10.1137/0917011","mag":"2026688642"},"language":"en","primary_location":{"id":"doi:10.1137/0917011","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0917011","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"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 Scientific Computing","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/A5113784264","display_name":"Jan Jan\u00dfen","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Jan Janssen","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5062993526","display_name":"Stefan Vandewalle","orcid":"https://orcid.org/0000-0002-8988-2374"},"institutions":[{"id":"https://openalex.org/I99464096","display_name":"KU Leuven","ror":"https://ror.org/05f950310","country_code":"BE","type":"education","lineage":["https://openalex.org/I99464096"]},{"id":"https://openalex.org/I122411786","display_name":"California Institute of Technology","ror":"https://ror.org/05dxps055","country_code":"US","type":"education","lineage":["https://openalex.org/I122411786"]}],"countries":["BE","US"],"is_corresponding":false,"raw_author_name":"Stefan Vandewalle","raw_affiliation_strings":["California Institute of Technology, Applied Mathematics 217-50, Pasadena, CA 91125","Katholieke Universiteit Leuven, Department of Computer Science, Celestijnenlaan 200A, B-3001 Heverlee, Belgium"],"affiliations":[{"raw_affiliation_string":"California Institute of Technology, Applied Mathematics 217-50, Pasadena, CA 91125","institution_ids":["https://openalex.org/I122411786"]},{"raw_affiliation_string":"Katholieke Universiteit Leuven, Department of Computer Science, Celestijnenlaan 200A, B-3001 Heverlee, Belgium","institution_ids":["https://openalex.org/I99464096"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5113784264"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.1632,"has_fulltext":false,"cited_by_count":36,"citation_normalized_percentile":{"value":0.76745996,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"17","issue":"1","first_page":"133","last_page":"155"},"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":0.9998000264167786,"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":0.9998000264167786,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9986000061035156,"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/T11263","display_name":"Electromagnetic Simulation and Numerical Methods","score":0.9970999956130981,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/multigrid-method","display_name":"Multigrid method","score":0.8555747866630554},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.7940280437469482},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7230650782585144},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.6699380278587341},{"id":"https://openalex.org/keywords/waveform","display_name":"Waveform","score":0.6214675307273865},{"id":"https://openalex.org/keywords/polygon-mesh","display_name":"Polygon mesh","score":0.620654821395874},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.6083845496177673},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.6063910126686096},{"id":"https://openalex.org/keywords/relaxation","display_name":"Relaxation (psychology)","score":0.5876110196113586},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.5508201718330383},{"id":"https://openalex.org/keywords/numerical-partial-differential-equations","display_name":"Numerical partial differential equations","score":0.5139920115470886},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.477855384349823},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4501979351043701},{"id":"https://openalex.org/keywords/discrete-time-and-continuous-time","display_name":"Discrete time and continuous time","score":0.43059998750686646},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.42570245265960693},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.3998104929924011},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1807161569595337},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.16796809434890747}],"concepts":[{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.8555747866630554},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.7940280437469482},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7230650782585144},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.6699380278587341},{"id":"https://openalex.org/C197424946","wikidata":"https://www.wikidata.org/wiki/Q1165717","display_name":"Waveform","level":3,"score":0.6214675307273865},{"id":"https://openalex.org/C31487907","wikidata":"https://www.wikidata.org/wiki/Q1154597","display_name":"Polygon mesh","level":2,"score":0.620654821395874},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.6083845496177673},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.6063910126686096},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.5876110196113586},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.5508201718330383},{"id":"https://openalex.org/C97826883","wikidata":"https://www.wikidata.org/wiki/Q7069659","display_name":"Numerical partial differential equations","level":3,"score":0.5139920115470886},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.477855384349823},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4501979351043701},{"id":"https://openalex.org/C55689738","wikidata":"https://www.wikidata.org/wiki/Q15963867","display_name":"Discrete time and continuous time","level":2,"score":0.43059998750686646},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.42570245265960693},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.3998104929924011},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1807161569595337},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.16796809434890747},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","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/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0},{"id":"https://openalex.org/C77805123","wikidata":"https://www.wikidata.org/wiki/Q161272","display_name":"Social psychology","level":1,"score":0.0},{"id":"https://openalex.org/C554190296","wikidata":"https://www.wikidata.org/wiki/Q47528","display_name":"Radar","level":2,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1137/0917011","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0917011","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"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 Scientific Computing","raw_type":"journal-article"},{"id":"pmh:oai:authors.library.caltech.edu:9406","is_oa":false,"landing_page_url":"https://authors.library.caltech.edu/9406/","pdf_url":null,"source":{"id":"https://openalex.org/S4306402161","display_name":"CaltechAUTHORS (California Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I122411786","host_organization_name":"California Institute of Technology","host_organization_lineage":["https://openalex.org/I122411786"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.515.3364","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.515.3364","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://authors.library.caltech.edu/9405/1/JANsiamjna96.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.54.6539","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.6539","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ftp://softlib.rice.edu/pub/CRPC-TRs/reports/CRPC-TR94450.ps.gz","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.56.9899","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.9899","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.kuleuven.ac.be/~stefan/Papers/mwrdisc.ps.gz","raw_type":"text"},{"id":"pmh:oai:lirias2repo.kuleuven.be:123456789/99833","is_oa":false,"landing_page_url":"http://gateway.newisiknowledge.com/gateway/Gateway.cgi?&GWVersion=2&SrcAuth=INSPEC&SrcApp=PRODUCT_NAME&CKEY=JANS0133960017SJ&SrcURL=WOS_RETURN_URL&DestLinkType=FullRecord&DestApp=INSPEC&SrcDesc=RETURN_ALT_TEXT&SrcAppSID=APP_SID","pdf_url":null,"source":{"id":"https://openalex.org/S4306401954","display_name":"Lirias (KU Leuven)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I99464096","host_organization_name":"KU Leuven","host_organization_lineage":["https://openalex.org/I99464096"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Scientific Computing, vol. 17 (1), Art.No. 1, (133-155)","raw_type":"info:eu-repo/semantics/publishedVersion"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1523875969","display_name":null,"funder_award_id":"9120008","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7973944762","display_name":null,"funder_award_id":"CCR-9120008","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":22,"referenced_works":["https://openalex.org/W212594717","https://openalex.org/W625917541","https://openalex.org/W1501940787","https://openalex.org/W1572639095","https://openalex.org/W1964039844","https://openalex.org/W1967595729","https://openalex.org/W1987418883","https://openalex.org/W1987455271","https://openalex.org/W1995664184","https://openalex.org/W2022396161","https://openalex.org/W2036597275","https://openalex.org/W2062543345","https://openalex.org/W2065884217","https://openalex.org/W2082621526","https://openalex.org/W2117919754","https://openalex.org/W2140779530","https://openalex.org/W2151317657","https://openalex.org/W2241115326","https://openalex.org/W2414547273","https://openalex.org/W4230055019","https://openalex.org/W4232499108","https://openalex.org/W4293515545"],"related_works":["https://openalex.org/W2077078424","https://openalex.org/W2182328658","https://openalex.org/W1519938374","https://openalex.org/W1537807642","https://openalex.org/W3022010538","https://openalex.org/W2525918155","https://openalex.org/W2167386686","https://openalex.org/W2785989279","https://openalex.org/W2400926413","https://openalex.org/W4232510292"],"abstract_inverted_index":{"The":[0,25,58,67],"efficiency":[1],"of":[2,28,43,52],"numerically":[3],"solving":[4],"time-dependent":[5],"partial":[6],"differential":[7,45],"equations":[8,46],"on":[9,20],"parallel":[10],"computers":[11],"can":[12],"be":[13],"greatly":[14],"improved":[15],"by":[16,48,84],"computing":[17],"the":[18,33,63],"solution":[19],"many":[21],"time":[22],"levels":[23],"simultaneously.":[24],"theoretical":[26],"properties":[27],"one":[29],"such":[30],"method,":[31,38],"namely":[32],"discrete-time":[34],"multigrid":[35],"waveform":[36],"relaxation":[37],"are":[39,60],"investigated":[40],"for":[41,71],"systems":[42],"ordinary":[44],"obtained":[47,83],"spatial":[49],"finite-element":[50],"discretisation":[51],"linear":[53],"parabolic":[54],"initial-boundary":[55],"value":[56],"problems.":[57],"results":[59,82],"compared":[61],"to":[62],"corresponding":[64],"continuous-time":[65],"results.":[66],"theory":[68],"is":[69],"illustrated":[70],"a":[72,75],"one-dimensional":[73],"and":[74,79],"two-dimensional":[76],"model":[77],"problem":[78],"checked":[80],"against":[81],"numerical":[85],"experiments.":[86]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":2},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}