{"id":"https://openalex.org/W1585842086","doi":"https://doi.org/10.1137/15m1008117","title":"A $C^1$ Virtual Element Method for the Cahn--Hilliard Equation with Polygonal Meshes","display_name":"A $C^1$ Virtual Element Method for the Cahn--Hilliard Equation with Polygonal Meshes","publication_year":2016,"publication_date":"2016-01-01","ids":{"openalex":"https://openalex.org/W1585842086","doi":"https://doi.org/10.1137/15m1008117","mag":"1585842086"},"language":"en","primary_location":{"id":"doi:10.1137/15m1008117","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1008117","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":true,"oa_status":"green","oa_url":"http://hdl.handle.net/2434/391198","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5063675442","display_name":"Paola F. Antonietti","orcid":"https://orcid.org/0000-0002-2138-3878"},"institutions":[{"id":"https://openalex.org/I4210137033","display_name":"Istituto di Matematica Applicata e Tecnologie Informatiche","ror":"https://ror.org/03m0n3c07","country_code":"IT","type":"education","lineage":["https://openalex.org/I4210137033","https://openalex.org/I4210155236"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"P. F. Antonietti","raw_affiliation_strings":["Istituto di Matematica Applicata e Tecnologie Informatiche \"E. Magenes\" -CNR, Via Ferrata 1, I-27100 Pavia, Italy"],"affiliations":[{"raw_affiliation_string":"Istituto di Matematica Applicata e Tecnologie Informatiche \"E. Magenes\" -CNR, Via Ferrata 1, I-27100 Pavia, Italy","institution_ids":["https://openalex.org/I4210137033"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5062969935","display_name":"L. Beir\u00e3o da Veiga","orcid":"https://orcid.org/0000-0001-5895-469X"},"institutions":[{"id":"https://openalex.org/I66752286","display_name":"University of Milano-Bicocca","ror":"https://ror.org/01ynf4891","country_code":"IT","type":"education","lineage":["https://openalex.org/I66752286"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"L. Beira\u0342o da Veiga","raw_affiliation_strings":["Dipartimento di Matematica e Applicazioni, Universit\u00e0 di Milano-Bicocca, Via Cozzi 53, I-20125 Milano, Italy"],"affiliations":[{"raw_affiliation_string":"Dipartimento di Matematica e Applicazioni, Universit\u00e0 di Milano-Bicocca, Via Cozzi 53, I-20125 Milano, Italy","institution_ids":["https://openalex.org/I66752286"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5028302493","display_name":"Simone Scacchi","orcid":"https://orcid.org/0000-0001-6011-784X"},"institutions":[{"id":"https://openalex.org/I4210137033","display_name":"Istituto di Matematica Applicata e Tecnologie Informatiche","ror":"https://ror.org/03m0n3c07","country_code":"IT","type":"education","lineage":["https://openalex.org/I4210137033","https://openalex.org/I4210155236"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"S. Scacchi","raw_affiliation_strings":["Istituto di Matematica Applicata e Tecnologie Informatiche \"E. Magenes\" -CNR, Via Ferrata 1, I-27100 Pavia, Italy"],"affiliations":[{"raw_affiliation_string":"Istituto di Matematica Applicata e Tecnologie Informatiche \"E. Magenes\" -CNR, Via Ferrata 1, I-27100 Pavia, Italy","institution_ids":["https://openalex.org/I4210137033"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5079733575","display_name":"Marco Verani","orcid":"https://orcid.org/0000-0001-8015-4947"},"institutions":[{"id":"https://openalex.org/I189158943","display_name":"University of Milan","ror":"https://ror.org/00wjc7c48","country_code":"IT","type":"education","lineage":["https://openalex.org/I189158943"]},{"id":"https://openalex.org/I93860229","display_name":"Politecnico di Milano","ror":"https://ror.org/01nffqt88","country_code":"IT","type":"education","lineage":["https://openalex.org/I93860229"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"M. Verani","raw_affiliation_strings":["MOX-Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy,","Dipartimento di Matematica, Universit\u00e0 degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy"],"affiliations":[{"raw_affiliation_string":"MOX-Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy,","institution_ids":["https://openalex.org/I93860229"]},{"raw_affiliation_string":"Dipartimento di Matematica, Universit\u00e0 degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy","institution_ids":["https://openalex.org/I189158943"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5079733575"],"corresponding_institution_ids":["https://openalex.org/I189158943","https://openalex.org/I93860229"],"apc_list":null,"apc_paid":null,"fwci":29.6238,"has_fulltext":false,"cited_by_count":244,"citation_normalized_percentile":{"value":0.99740267,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"54","issue":"1","first_page":"34","last_page":"56"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9987000226974487,"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":0.9987000226974487,"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":0.9983000159263611,"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/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9980000257492065,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/polygon-mesh","display_name":"Polygon mesh","score":0.915294349193573},{"id":"https://openalex.org/keywords/cahn\u2013hilliard-equation","display_name":"Cahn\u2013Hilliard equation","score":0.7461388111114502},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7146006226539612},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.6007896065711975},{"id":"https://openalex.org/keywords/degrees-of-freedom","display_name":"Degrees of freedom (physics and chemistry)","score":0.5765168070793152},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5467097759246826},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.5219273567199707},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.512628972530365},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.499434232711792},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4913610517978668},{"id":"https://openalex.org/keywords/scheme","display_name":"Scheme (mathematics)","score":0.46114829182624817},{"id":"https://openalex.org/keywords/element","display_name":"Element (criminal law)","score":0.41614076495170593},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2833741307258606},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.26486554741859436},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.23784899711608887},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.1836761236190796},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.16142138838768005}],"concepts":[{"id":"https://openalex.org/C31487907","wikidata":"https://www.wikidata.org/wiki/Q1154597","display_name":"Polygon mesh","level":2,"score":0.915294349193573},{"id":"https://openalex.org/C24822716","wikidata":"https://www.wikidata.org/wiki/Q5017440","display_name":"Cahn\u2013Hilliard equation","level":3,"score":0.7461388111114502},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7146006226539612},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.6007896065711975},{"id":"https://openalex.org/C208081375","wikidata":"https://www.wikidata.org/wiki/Q274502","display_name":"Degrees of freedom (physics and chemistry)","level":2,"score":0.5765168070793152},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5467097759246826},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.5219273567199707},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.512628972530365},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.499434232711792},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4913610517978668},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.46114829182624817},{"id":"https://openalex.org/C200288055","wikidata":"https://www.wikidata.org/wiki/Q2621792","display_name":"Element (criminal law)","level":2,"score":0.41614076495170593},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2833741307258606},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.26486554741859436},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.23784899711608887},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.1836761236190796},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.16142138838768005},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","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/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"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/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","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":4,"locations":[{"id":"doi:10.1137/15m1008117","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1008117","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:air.unimi.it:2434/391198","is_oa":true,"landing_page_url":"http://hdl.handle.net/2434/391198","pdf_url":null,"source":{"id":"https://openalex.org/S4306400516","display_name":"Archivio Istituzionale della Ricerca (Universita Degli Studi Di Milano)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I189158943","host_organization_name":"University of Milan","host_organization_lineage":["https://openalex.org/I189158943"],"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":"","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:boa.unimib.it:10281/138475","is_oa":false,"landing_page_url":"http://hdl.handle.net/10281/138475","pdf_url":null,"source":{"id":"https://openalex.org/S4306401259","display_name":"BOA (University of Milano-Bicocca)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I66752286","host_organization_name":"University of Milano-Bicocca","host_organization_lineage":["https://openalex.org/I66752286"],"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":"info:eu-repo/semantics/article"},{"id":"pmh:oai:re.public.polimi.it:11311/1000808","is_oa":true,"landing_page_url":"http://epubs.siam.org/doi/pdf/10.1137/15M1008117","pdf_url":null,"source":{"id":"https://openalex.org/S4306400312","display_name":"Virtual Community of Pathological Anatomy (University of Castilla La Mancha)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79189158","host_organization_name":"University of Castilla-La Mancha","host_organization_lineage":["https://openalex.org/I79189158"],"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":"","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:air.unimi.it:2434/391198","is_oa":true,"landing_page_url":"http://hdl.handle.net/2434/391198","pdf_url":null,"source":{"id":"https://openalex.org/S4306400516","display_name":"Archivio Istituzionale della Ricerca (Universita Degli Studi Di Milano)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I189158943","host_organization_name":"University of Milan","host_organization_lineage":["https://openalex.org/I189158943"],"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":"","raw_type":"info:eu-repo/semantics/article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/16","display_name":"Peace, Justice and strong institutions","score":0.8100000023841858}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":48,"referenced_works":["https://openalex.org/W1512153219","https://openalex.org/W1863866458","https://openalex.org/W1963803848","https://openalex.org/W1986187395","https://openalex.org/W1987723981","https://openalex.org/W1989512090","https://openalex.org/W1990258816","https://openalex.org/W1990518285","https://openalex.org/W1992923859","https://openalex.org/W1996774337","https://openalex.org/W1997748595","https://openalex.org/W2003173073","https://openalex.org/W2006000775","https://openalex.org/W2014931158","https://openalex.org/W2032257836","https://openalex.org/W2038887300","https://openalex.org/W2039254108","https://openalex.org/W2039955529","https://openalex.org/W2044562696","https://openalex.org/W2045686441","https://openalex.org/W2046528357","https://openalex.org/W2047049848","https://openalex.org/W2053384289","https://openalex.org/W2057105966","https://openalex.org/W2058585715","https://openalex.org/W2061524394","https://openalex.org/W2064807134","https://openalex.org/W2065302471","https://openalex.org/W2074439496","https://openalex.org/W2077028260","https://openalex.org/W2090496377","https://openalex.org/W2095195575","https://openalex.org/W2097181817","https://openalex.org/W2108609294","https://openalex.org/W2126328932","https://openalex.org/W2130763970","https://openalex.org/W2157252460","https://openalex.org/W2158598889","https://openalex.org/W2161530217","https://openalex.org/W2168914934","https://openalex.org/W2169464027","https://openalex.org/W2569713119","https://openalex.org/W2883313970","https://openalex.org/W2963396286","https://openalex.org/W2970692887","https://openalex.org/W3022780580","https://openalex.org/W3101743426","https://openalex.org/W3106057804"],"related_works":["https://openalex.org/W2990970471","https://openalex.org/W3113339583","https://openalex.org/W2555456828","https://openalex.org/W4327546956","https://openalex.org/W2027096693","https://openalex.org/W2361126117","https://openalex.org/W124014347","https://openalex.org/W1987597427","https://openalex.org/W1990508873","https://openalex.org/W2915984213"],"abstract_inverted_index":{"In":[0],"this":[1],"paper":[2],"we":[3,79],"develop":[4],"an":[5],"evolution":[6],"of":[7,12,18,28,36,41,43,48,52,69,83,91,99],"the":[8,16,19,26,53,57,81,84,89,92],"$C^1$":[9],"virtual":[10],"elements":[11],"minimal":[13],"degree":[14],"for":[15],"approximation":[17],"Cahn--Hilliard":[20],"equation.":[21],"The":[22],"proposed":[23],"method":[24,59],"has":[25],"advantage":[27],"being":[29],"conforming":[30],"in":[31],"$H^2$":[32],"and":[33,76,87],"making":[34],"use":[35,68],"a":[37,74,97],"very":[38],"simple":[39],"set":[40,98],"degrees":[42,47],"freedom,":[44],"namely,":[45],"3":[46],"freedom":[49],"per":[50],"vertex":[51],"mesh.":[54],"Moreover,":[55],"although":[56],"present":[58],"is":[60],"new":[61],"also":[62],"on":[63],"triangles,":[64],"it":[65],"can":[66],"make":[67],"general":[70],"polygonal":[71],"meshes.":[72],"As":[73],"theoretical":[75],"practical":[77],"support,":[78],"prove":[80],"convergence":[82],"semidiscrete":[85],"scheme":[86,95],"investigate":[88],"performance":[90],"fully":[93],"discrete":[94],"through":[96],"numerical":[100],"tests.":[101]},"counts_by_year":[{"year":2026,"cited_by_count":5},{"year":2025,"cited_by_count":16},{"year":2024,"cited_by_count":25},{"year":2023,"cited_by_count":20},{"year":2022,"cited_by_count":37},{"year":2021,"cited_by_count":39},{"year":2020,"cited_by_count":19},{"year":2019,"cited_by_count":17},{"year":2018,"cited_by_count":30},{"year":2017,"cited_by_count":20},{"year":2016,"cited_by_count":15},{"year":2014,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}