{"id":"https://openalex.org/W2192231559","doi":"https://doi.org/10.1137/15m1006489","title":"An $h$-$p$ Version of the Continuous Petrov--Galerkin Finite Element Method for Volterra Integro-Differential Equations with Smooth and Nonsmooth Kernels","display_name":"An $h$-$p$ Version of the Continuous Petrov--Galerkin Finite Element Method for Volterra Integro-Differential Equations with Smooth and Nonsmooth Kernels","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W2192231559","doi":"https://doi.org/10.1137/15m1006489","mag":"2192231559"},"language":"en","primary_location":{"id":"doi:10.1137/15m1006489","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1006489","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/A5084338992","display_name":"Lijun Yi","orcid":"https://orcid.org/0000-0002-2922-5508"},"institutions":[{"id":"https://openalex.org/I21945476","display_name":"Shanghai Normal University","ror":"https://ror.org/01cxqmw89","country_code":"CN","type":"education","lineage":["https://openalex.org/I21945476"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Lijun Yi","raw_affiliation_strings":["Shanghai Normal University"],"affiliations":[{"raw_affiliation_string":"Shanghai Normal University","institution_ids":["https://openalex.org/I21945476"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100885624","display_name":"Benqi Guo","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Benqi Guo","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5084338992"],"corresponding_institution_ids":["https://openalex.org/I21945476"],"apc_list":null,"apc_paid":null,"fwci":2.0119,"has_fulltext":false,"cited_by_count":72,"citation_normalized_percentile":{"value":0.85491902,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"53","issue":"6","first_page":"2677","last_page":"2704"},"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9997000098228455,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9995999932289124,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8929213285446167},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5629647970199585},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.5517246723175049},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.5483671426773071},{"id":"https://openalex.org/keywords/gravitational-singularity","display_name":"Gravitational singularity","score":0.5288689732551575},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.4780298173427582},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.45420074462890625},{"id":"https://openalex.org/keywords/degree-of-a-polynomial","display_name":"Degree of a polynomial","score":0.45134586095809937},{"id":"https://openalex.org/keywords/petrov\u2013galerkin-method","display_name":"Petrov\u2013Galerkin method","score":0.4403395354747772},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.42158740758895874},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.40086114406585693},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3600633144378662}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8929213285446167},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5629647970199585},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.5517246723175049},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.5483671426773071},{"id":"https://openalex.org/C12843","wikidata":"https://www.wikidata.org/wiki/Q201721","display_name":"Gravitational singularity","level":2,"score":0.5288689732551575},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.4780298173427582},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.45420074462890625},{"id":"https://openalex.org/C25878781","wikidata":"https://www.wikidata.org/wiki/Q1473607","display_name":"Degree of a polynomial","level":3,"score":0.45134586095809937},{"id":"https://openalex.org/C2778540877","wikidata":"https://www.wikidata.org/wiki/Q7179155","display_name":"Petrov\u2013Galerkin method","level":3,"score":0.4403395354747772},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.42158740758895874},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.40086114406585693},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3600633144378662},{"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/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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/15m1006489","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1006489","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":32,"referenced_works":["https://openalex.org/W191236717","https://openalex.org/W205381016","https://openalex.org/W1966192083","https://openalex.org/W1978612347","https://openalex.org/W1978958071","https://openalex.org/W1986006981","https://openalex.org/W1988932741","https://openalex.org/W1990416353","https://openalex.org/W1994678449","https://openalex.org/W1996037589","https://openalex.org/W2012062138","https://openalex.org/W2013693242","https://openalex.org/W2014617375","https://openalex.org/W2017571276","https://openalex.org/W2020988386","https://openalex.org/W2022694169","https://openalex.org/W2028224531","https://openalex.org/W2031567600","https://openalex.org/W2041588624","https://openalex.org/W2044385163","https://openalex.org/W2050380784","https://openalex.org/W2063164826","https://openalex.org/W2071166325","https://openalex.org/W2084872583","https://openalex.org/W2086119896","https://openalex.org/W2089266075","https://openalex.org/W2094840682","https://openalex.org/W2145736329","https://openalex.org/W2151276461","https://openalex.org/W2556715742","https://openalex.org/W3120038842","https://openalex.org/W4250243937"],"related_works":["https://openalex.org/W2055696851","https://openalex.org/W4288262055","https://openalex.org/W2967076701","https://openalex.org/W2990320605","https://openalex.org/W2883560644","https://openalex.org/W4321472236","https://openalex.org/W4389325935","https://openalex.org/W2167486107","https://openalex.org/W2132433239","https://openalex.org/W2086298877"],"abstract_inverted_index":{"We":[0,23],"present":[1],"an":[2],"$h$-$p$":[3,68],"version":[4,69],"of":[5,70,83],"the":[6,30,42,53,67,71,109],"continuous":[7],"Petrov--Galerkin":[8],"(CPG)":[9],"finite":[10],"element":[11],"method":[12,73],"for":[13,66,88],"linear":[14],"Volterra":[15],"integro-differential":[16],"equations":[17],"with":[18,39,74,90],"smooth":[19],"and":[20,33,45,98],"nonsmooth":[21],"kernels.":[22],"establish":[24],"a":[25],"priori":[26],"error":[27],"estimates":[28],"in":[29],"$L^2$-,":[31],"$H^1$-,":[32],"$L^\\infty$-norms":[34],"that":[35,59,80],"are":[36,105],"completely":[37],"explicit":[38],"respect":[40],"to":[41,107],"local":[43],"discretization":[44],"regularity":[46],"parameters.":[47],"For":[48],"singular":[49,55],"solutions":[50,89],"caused":[51],"by":[52,93],"weakly":[54],"kernels,":[56],"we":[57,78],"prove":[58],"algebraic":[60],"convergence":[61,84],"rates":[62,82],"can":[63,85],"be":[64,86],"achieved":[65,87],"CPG":[72],"quasi-uniform":[75],"meshes.":[76],"Moreover,":[77],"show":[79],"exponential":[81],"start-up":[91],"singularities":[92],"using":[94],"geometric":[95],"time":[96],"partitions":[97],"linearly":[99],"increasing":[100],"polynomial":[101],"degrees.":[102],"Numerical":[103],"experiments":[104],"provided":[106],"illustrate":[108],"theoretical":[110],"results.":[111]},"counts_by_year":[{"year":2025,"cited_by_count":12},{"year":2024,"cited_by_count":16},{"year":2023,"cited_by_count":11},{"year":2022,"cited_by_count":7},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":10},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":4}],"updated_date":"2026-03-30T08:08:38.191290","created_date":"2025-10-10T00:00:00"}