{"id":"https://openalex.org/W7163875392","doi":"https://doi.org/10.48550/arxiv.2606.06719","title":"High-order conforming finite elements for the Cahn-Hilliard equation: Relative-energy stability and energy defects","display_name":"High-order conforming finite elements for the Cahn-Hilliard equation: Relative-energy stability and energy defects","publication_year":2026,"publication_date":"2026-06-04","ids":{"openalex":"https://openalex.org/W7163875392","doi":"https://doi.org/10.48550/arxiv.2606.06719"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2606.06719","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.06719","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2606.06719","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5138129025","display_name":"Aaron Brunk","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Brunk, Aaron","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5019190389","display_name":"Marvin Fritz","orcid":"https://orcid.org/0000-0002-8360-7371"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Fritz, Marvin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"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":0.7128000259399414,"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":0.7128000259399414,"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":0.13600000739097595,"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":0.052799999713897705,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.7319999933242798},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.7171000242233276},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5577999949455261},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5453000068664551},{"id":"https://openalex.org/keywords/a-priori-and-a-posteriori","display_name":"A priori and a posteriori","score":0.5067999958992004},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.5013999938964844},{"id":"https://openalex.org/keywords/fourier-transform","display_name":"Fourier transform","score":0.4575999975204468},{"id":"https://openalex.org/keywords/energy","display_name":"Energy (signal processing)","score":0.4487000107765198},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.39480000734329224}],"concepts":[{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.7319999933242798},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.7171000242233276},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.65829998254776},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5892999768257141},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5577999949455261},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5453000068664551},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.5067999958992004},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.5013999938964844},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4699000120162964},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.4575999975204468},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.4487000107765198},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.39480000734329224},{"id":"https://openalex.org/C176321772","wikidata":"https://www.wikidata.org/wiki/Q1430640","display_name":"Numerical stability","level":3,"score":0.3865000009536743},{"id":"https://openalex.org/C2777423268","wikidata":"https://www.wikidata.org/wiki/Q3663116","display_name":"Energy balance","level":2,"score":0.35190001130104065},{"id":"https://openalex.org/C57733114","wikidata":"https://www.wikidata.org/wiki/Q1006032","display_name":"Discrete Fourier transform (general)","level":5,"score":0.3508000075817108},{"id":"https://openalex.org/C203024314","wikidata":"https://www.wikidata.org/wiki/Q1365258","display_name":"Fourier analysis","level":3,"score":0.3465999960899353},{"id":"https://openalex.org/C92244383","wikidata":"https://www.wikidata.org/wiki/Q428273","display_name":"Discontinuous Galerkin method","level":3,"score":0.3375000059604645},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.31119999289512634},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.2992999851703644},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.2971000075340271},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.29510000348091125},{"id":"https://openalex.org/C181330731","wikidata":"https://www.wikidata.org/wiki/Q2068418","display_name":"Finite difference","level":2,"score":0.2930000126361847},{"id":"https://openalex.org/C55689738","wikidata":"https://www.wikidata.org/wiki/Q15963867","display_name":"Discrete time and continuous time","level":2,"score":0.29109999537467957},{"id":"https://openalex.org/C207864730","wikidata":"https://www.wikidata.org/wiki/Q179467","display_name":"Fourier series","level":2,"score":0.28630000352859497},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.2736999988555908}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2606.06719","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.06719","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2606.06719","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.06719","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/7","display_name":"Affordable and clean energy","score":0.7627063989639282}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"We":[0,103],"study":[1],"a":[2,34,99,106,111,124,161],"semidiscrete":[3,105],"single-field":[4],"Galerkin":[5],"approximation":[6,141],"of":[7,37,98],"the":[8,38,55,60,64,95,130,139,151,155],"Cahn-Hilliard":[9,57],"equation":[10],"using":[11,118],"high-order":[12],"conforming":[13],"finite":[14,21,90],"element":[15],"spaces.":[16],"More":[17],"specifically,":[18],"globally":[19],"$C^1$":[20,89],"elements":[22,149],"with":[23,145,160],"$H^2$-conforming":[24],"trial":[25],"spaces,":[26,78,82],"including":[27],"Argyris,":[28],"Bell,":[29],"and":[30,41,123,147,153],"Bogner-Fox-Schmit":[31],"elements,":[32],"allow":[33],"direct":[35],"discretization":[36],"fourth-order":[39],"formulation":[40],"preserve":[42],"mass":[43],"exactly.":[44],"The":[45,114,134],"main":[46],"structural":[47],"result":[48],"is":[49,84,116],"an":[50,67,119],"exact":[51],"energy":[52,70,122],"balance":[53,65],"for":[54,75,87,129],"physical":[56],"energy.":[58],"Besides":[59],"expected":[61,140],"discrete":[62,100,125,132],"dissipation,":[63],"contains":[66],"explicitly":[68],"computable":[69],"defect.":[71],"This":[72],"defect":[73,156],"vanishes":[74],"Laplacian-invariant":[76],"periodic":[77],"such":[79],"as":[80],"Fourier":[81,162],"but":[83],"generally":[85],"nonzero":[86],"classical":[88],"elements.":[91],"It":[92],"therefore":[93],"quantifies":[94],"precise":[96],"loss":[97],"gradient-flow":[101],"structure.":[102],"prove":[104],"priori":[107],"error":[108],"estimates":[109],"by":[110,158],"relative-energy":[112],"argument.":[113],"estimate":[115],"closed":[117],"augmented":[120],"relative":[121],"elliptic":[126],"reconstruction":[127],"bound":[128],"inverse":[131],"Laplacian.":[133],"resulting":[135],"convergence":[136],"rates":[137,152],"match":[138],"orders.":[142],"Numerical":[143],"experiments":[144],"Bell":[146],"Argyris":[148],"confirm":[150],"demonstrate":[154],"mechanism":[157],"comparison":[159],"reference":[163],"discretization.":[164]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2026-06-09T00:00:00"}
