{"id":"https://openalex.org/W1975968248","doi":"https://doi.org/10.1137/s0036142997331669","title":"Finite Element Approximation of the Cahn--Hilliard Equation with Degenerate Mobility","display_name":"Finite Element Approximation of the Cahn--Hilliard Equation with Degenerate Mobility","publication_year":1999,"publication_date":"1999-01-01","ids":{"openalex":"https://openalex.org/W1975968248","doi":"https://doi.org/10.1137/s0036142997331669","mag":"1975968248"},"language":"en","primary_location":{"id":"doi:10.1137/s0036142997331669","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036142997331669","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":"https://durham-repository.worktribe.com/output/1599393","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5071664437","display_name":"John W. Barrett","orcid":"https://orcid.org/0000-0002-4048-1692"},"institutions":[{"id":"https://openalex.org/I47508984","display_name":"Imperial College London","ror":"https://ror.org/041kmwe10","country_code":"GB","type":"education","lineage":["https://openalex.org/I47508984"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"John W. Barrett","raw_affiliation_strings":["Imperial College London,#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Imperial College London,#TAB#","institution_ids":["https://openalex.org/I47508984"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5070982393","display_name":"James F. Blowey","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"James F. Blowey","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5071758656","display_name":"Harald Garcke","orcid":"https://orcid.org/0000-0001-5235-2073"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Harald Garcke","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.7933,"has_fulltext":false,"cited_by_count":252,"citation_normalized_percentile":{"value":0.83280336,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":98,"max":100},"biblio":{"volume":"37","issue":"1","first_page":"286","last_page":"318"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9993000030517578,"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"}},"topics":[{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9993000030517578,"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"}},{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9983999729156494,"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9886000156402588,"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":0.8458474278450012},{"id":"https://openalex.org/keywords/degenerate-energy-levels","display_name":"Degenerate energy levels","score":0.7292566299438477},{"id":"https://openalex.org/keywords/cahn\u2013hilliard-equation","display_name":"Cahn\u2013Hilliard equation","score":0.6887534856796265},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.5755069255828857},{"id":"https://openalex.org/keywords/homogeneous","display_name":"Homogeneous","score":0.5650969743728638},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5584734678268433},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5581590533256531},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.555653989315033},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.4770207405090332},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.47073256969451904},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.45769578218460083},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.4537392854690552},{"id":"https://openalex.org/keywords/energy","display_name":"Energy (signal processing)","score":0.4286792278289795},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.3503933846950531},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.2268490493297577},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.21745839715003967},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.11366546154022217},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.08044543862342834}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8458474278450012},{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.7292566299438477},{"id":"https://openalex.org/C24822716","wikidata":"https://www.wikidata.org/wiki/Q5017440","display_name":"Cahn\u2013Hilliard equation","level":3,"score":0.6887534856796265},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.5755069255828857},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.5650969743728638},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5584734678268433},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5581590533256531},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.555653989315033},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.4770207405090332},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.47073256969451904},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.45769578218460083},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.4537392854690552},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.4286792278289795},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.3503933846950531},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2268490493297577},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.21745839715003967},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.11366546154022217},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.08044543862342834},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","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/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1137/s0036142997331669","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036142997331669","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:10.1.1.23.1474","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.1474","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.iam.uni-bonn.de/~harald/paper/dmob5.ps.gz","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.37.2579","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.2579","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://maths.dur.ac.uk/~dma0jfb/Post/97-07.ps.Z","raw_type":"text"},{"id":"pmh:oai:durham-repository.worktribe.com:1599393","is_oa":true,"landing_page_url":"https://durham-repository.worktribe.com/output/1599393","pdf_url":null,"source":{"id":"https://openalex.org/S4306400188","display_name":"Durham Research Online (Durham University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I190082696","host_organization_name":"Durham University","host_organization_lineage":["https://openalex.org/I190082696"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Journal Article"},{"id":"pmh:oai:spiral.imperial.ac.uk:10044/1/1104","is_oa":false,"landing_page_url":"http://hdl.handle.net/10044/1/1104","pdf_url":null,"source":{"id":"https://openalex.org/S4306401396","display_name":"Spiral (Imperial College London)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I47508984","host_organization_name":"Imperial College London","host_organization_lineage":["https://openalex.org/I47508984"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"318","raw_type":"Journal Article"}],"best_oa_location":{"id":"pmh:oai:durham-repository.worktribe.com:1599393","is_oa":true,"landing_page_url":"https://durham-repository.worktribe.com/output/1599393","pdf_url":null,"source":{"id":"https://openalex.org/S4306400188","display_name":"Durham Research Online (Durham University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I190082696","host_organization_name":"Durham University","host_organization_lineage":["https://openalex.org/I190082696"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Journal Article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/7","score":0.8700000047683716,"display_name":"Affordable and clean energy"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W1539990887","https://openalex.org/W1965753687","https://openalex.org/W1970380607","https://openalex.org/W1979743485","https://openalex.org/W1984314510","https://openalex.org/W1985642229","https://openalex.org/W1998579166","https://openalex.org/W1998903284","https://openalex.org/W2003173073","https://openalex.org/W2008209583","https://openalex.org/W2011723842","https://openalex.org/W2017385247","https://openalex.org/W2019569173","https://openalex.org/W2046528357","https://openalex.org/W2058436955","https://openalex.org/W2069739223","https://openalex.org/W2071643566","https://openalex.org/W2078279585","https://openalex.org/W2079950371","https://openalex.org/W2083629653","https://openalex.org/W2106465111","https://openalex.org/W2138026128","https://openalex.org/W2144225312","https://openalex.org/W2569713119","https://openalex.org/W4254216534","https://openalex.org/W4296816584"],"related_works":["https://openalex.org/W2353039109","https://openalex.org/W2967558384","https://openalex.org/W2973230771","https://openalex.org/W4365141110","https://openalex.org/W2946071405","https://openalex.org/W2008209583","https://openalex.org/W4318242465","https://openalex.org/W1168660807","https://openalex.org/W4385920456","https://openalex.org/W4247443486"],"abstract_inverted_index":{"We":[0,73],"consider":[1],"a":[2,31,37,89],"fully":[3],"practical":[4],"finite":[5],"element":[6],"approximation":[7,78],"of":[8],"the":[9,66],"Cahn--Hilliard":[10],"equation":[11],"with":[12],"degenerate":[13],"mobility":[14,33],"$$":[15,26],"\\textstyle":[16],"\\frac{\\partial":[17],"u}{\\partial":[18],"t}=":[19],"\\del":[20,22],".(\\,b(u)\\,":[21],"(-\\gamma\\lap":[23],"u+\\Psi'(u)))":[24],",":[25],"where":[27],"$b(\\cdot)\\geq":[28],"0$":[29],"is":[30,36,71],"diffusional":[32],"and":[34,47],"$\\Psi(\\cdot)$":[35],"homogeneous":[38,91],"free":[39,92],"energy.":[40,93],"In":[41],"addition":[42],"to":[43,80,85,88],"showing":[44],"well":[45],"posedness":[46],"stability":[48],"bounds":[49],"for":[50,64],"our":[51,77],"approximation,":[52],"we":[53],"prove":[54],"convergence":[55],"in":[56,83],"one":[57],"space":[58],"dimension.":[59],"Furthermore,":[60],"an":[61],"iterative":[62],"scheme":[63],"solving":[65],"resulting":[67],"nonlinear":[68],"discrete":[69],"system":[70],"analyzed.":[72],"also":[74],"discuss":[75],"how":[76],"has":[79],"be":[81,86],"modified":[82],"order":[84],"applicable":[87],"logarithmic":[90],"Finally,":[94],"some":[95],"numerical":[96],"experiments":[97],"are":[98],"presented.":[99]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":21},{"year":2024,"cited_by_count":17},{"year":2023,"cited_by_count":19},{"year":2022,"cited_by_count":16},{"year":2021,"cited_by_count":13},{"year":2020,"cited_by_count":15},{"year":2019,"cited_by_count":17},{"year":2018,"cited_by_count":15},{"year":2017,"cited_by_count":10},{"year":2016,"cited_by_count":7},{"year":2015,"cited_by_count":6},{"year":2014,"cited_by_count":12},{"year":2013,"cited_by_count":13},{"year":2012,"cited_by_count":6}],"updated_date":"2026-07-27T08:26:11.824852","created_date":"2025-10-10T00:00:00"}