{"id":"https://openalex.org/W2048343906","doi":"https://doi.org/10.1090/mcom3052","title":"Convergence analysis of a fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equation","display_name":"Convergence analysis of a fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equation","publication_year":2015,"publication_date":"2015-03-11","ids":{"openalex":"https://openalex.org/W2048343906","doi":"https://doi.org/10.1090/mcom3052","mag":"2048343906"},"language":"en","primary_location":{"id":"doi:10.1090/mcom3052","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom3052","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","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/A5100693236","display_name":"Wenbin Chen","orcid":"https://orcid.org/0000-0001-8305-0764"},"institutions":[{"id":"https://openalex.org/I24943067","display_name":"Fudan University","ror":"https://ror.org/013q1eq08","country_code":"CN","type":"education","lineage":["https://openalex.org/I24943067"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wenbin Chen","raw_affiliation_strings":["Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, People\u00e2\u0080\u0099s Republic of China 200433","Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, People's Republic of China 200433#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, People\u00e2\u0080\u0099s Republic of China 200433","institution_ids":["https://openalex.org/I24943067"]},{"raw_affiliation_string":"Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, People's Republic of China 200433#TAB#","institution_ids":["https://openalex.org/I24943067"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100390949","display_name":"Yuan Liu","orcid":"https://orcid.org/0000-0003-3973-4207"},"institutions":[{"id":"https://openalex.org/I24943067","display_name":"Fudan University","ror":"https://ror.org/013q1eq08","country_code":"CN","type":"education","lineage":["https://openalex.org/I24943067"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yuan Liu","raw_affiliation_strings":["School of Mathematical Sciences, Fudan University, Shanghai, People\u00e2\u0080\u0099s Republic of China 200433","School of Mathematical Sciences, Fudan University, Shanghai, People's Republic of China 200433#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Fudan University, Shanghai, People\u00e2\u0080\u0099s Republic of China 200433","institution_ids":["https://openalex.org/I24943067"]},{"raw_affiliation_string":"School of Mathematical Sciences, Fudan University, Shanghai, People's Republic of China 200433#TAB#","institution_ids":["https://openalex.org/I24943067"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100417045","display_name":"Cheng Wang","orcid":"https://orcid.org/0000-0003-4220-8080"},"institutions":[{"id":"https://openalex.org/I100633361","display_name":"University of Massachusetts Dartmouth","ror":"https://ror.org/00fzmm222","country_code":"US","type":"education","lineage":["https://openalex.org/I100633361"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Cheng Wang","raw_affiliation_strings":["Department of Mathematics, University of Massachusetts, North Dartmouth, Massachusetts 02747"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Massachusetts, North Dartmouth, Massachusetts 02747","institution_ids":["https://openalex.org/I100633361"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5028411348","display_name":"Steven M. Wise","orcid":"https://orcid.org/0000-0003-3824-2075"},"institutions":[{"id":"https://openalex.org/I75027704","display_name":"University of Tennessee at Knoxville","ror":"https://ror.org/020f3ap87","country_code":"US","type":"education","lineage":["https://openalex.org/I75027704"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Steven Wise","raw_affiliation_strings":["Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996","Department of Mathematics; University of Tennessee; Knoxville Tennessee 37996"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996","institution_ids":["https://openalex.org/I75027704"]},{"raw_affiliation_string":"Department of Mathematics; University of Tennessee; Knoxville Tennessee 37996","institution_ids":["https://openalex.org/I75027704"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.6386,"has_fulltext":false,"cited_by_count":128,"citation_normalized_percentile":{"value":0.82717978,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"85","issue":"301","first_page":"2231","last_page":"2257"},"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/T12141","display_name":"Fluid Dynamics and Thin Films","score":0.9901000261306763,"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/T10591","display_name":"Theoretical and Computational Physics","score":0.9800999760627747,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8595508933067322},{"id":"https://openalex.org/keywords/cahn\u2013hilliard-equation","display_name":"Cahn\u2013Hilliard equation","score":0.7332949638366699},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6155510544776917},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4960068166255951},{"id":"https://openalex.org/keywords/balanced-flow","display_name":"Balanced flow","score":0.46659162640571594},{"id":"https://openalex.org/keywords/finite-difference","display_name":"Finite difference","score":0.45509928464889526},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.44596871733665466},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.428529292345047},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4220839738845825},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.13350817561149597}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8595508933067322},{"id":"https://openalex.org/C24822716","wikidata":"https://www.wikidata.org/wiki/Q5017440","display_name":"Cahn\u2013Hilliard equation","level":3,"score":0.7332949638366699},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6155510544776917},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4960068166255951},{"id":"https://openalex.org/C167879884","wikidata":"https://www.wikidata.org/wiki/Q727568","display_name":"Balanced flow","level":2,"score":0.46659162640571594},{"id":"https://openalex.org/C181330731","wikidata":"https://www.wikidata.org/wiki/Q2068418","display_name":"Finite difference","level":2,"score":0.45509928464889526},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.44596871733665466},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.428529292345047},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4220839738845825},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.13350817561149597},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C119599485","wikidata":"https://www.wikidata.org/wiki/Q43035","display_name":"Electrical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","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/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/mcom3052","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom3052","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1545671193","display_name":"Collaborative Research: Stable and Efficient Convexity-Splitting Schemes for Bistable Gradient PDEs","funder_award_id":"1115390","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G1711889295","display_name":null,"funder_award_id":"11331004","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G179464631","display_name":null,"funder_award_id":"91130004","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2091062436","display_name":"Collaborative Research: Stable and Efficient Convexity-splitting Schemes for Bistable Gradient PDEs","funder_award_id":"1115420","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G2170284883","display_name":null,"funder_award_id":"201406100085","funder_id":"https://openalex.org/F4320322725","funder_display_name":"China Scholarship Council"},{"id":"https://openalex.org/G2638463404","display_name":null,"funder_award_id":"91130004","funder_id":"https://openalex.org/F4320322725","funder_display_name":"China Scholarship Council"},{"id":"https://openalex.org/G2700745455","display_name":null,"funder_award_id":"1115420","funder_id":"https://openalex.org/F4320322725","funder_display_name":"China Scholarship Council"},{"id":"https://openalex.org/G306313590","display_name":null,"funder_award_id":"1115390","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3185265237","display_name":null,"funder_award_id":"1115420","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3948081026","display_name":null,"funder_award_id":"11331004","funder_id":"https://openalex.org/F4320322725","funder_display_name":"China Scholarship Council"},{"id":"https://openalex.org/G47597050","display_name":null,"funder_award_id":"1418692","funder_id":"https://openalex.org/F4320322725","funder_display_name":"China Scholarship Council"},{"id":"https://openalex.org/G4832448216","display_name":null,"funder_award_id":"11331004","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5102833212","display_name":null,"funder_award_id":"91130004","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5217067652","display_name":null,"funder_award_id":"11171077","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G53304062","display_name":null,"funder_award_id":"1418689","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5740527225","display_name":null,"funder_award_id":"11171077","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5909355770","display_name":null,"funder_award_id":"1418692","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5933115280","display_name":null,"funder_award_id":"1418689","funder_id":"https://openalex.org/F4320322725","funder_display_name":"China Scholarship Council"},{"id":"https://openalex.org/G6480509492","display_name":null,"funder_award_id":"201406100085","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6857168495","display_name":"Highly efficient and accurate numerical schemes  for nonlinear gradient flows with energy stability","funder_award_id":"1418689","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7956358305","display_name":null,"funder_award_id":"1115390","funder_id":"https://openalex.org/F4320322725","funder_display_name":"China Scholarship Council"},{"id":"https://openalex.org/G8072635636","display_name":null,"funder_award_id":"11171077","funder_id":"https://openalex.org/F4320322725","funder_display_name":"China Scholarship Council"},{"id":"https://openalex.org/G846989147","display_name":"Efficient, Adaptive, and Convergent Numerical Methods for Phase Field and Phase Field Crystal Equations with Applications","funder_award_id":"1418692","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8950462385","display_name":null,"funder_award_id":"201406100085","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320322725","display_name":"China Scholarship Council","ror":"https://ror.org/04atp4p48"},{"id":"https://openalex.org/F4320337380","display_name":"Division of Mathematical Sciences","ror":"https://ror.org/051fftw81"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":52,"referenced_works":["https://openalex.org/W1550687986","https://openalex.org/W1566292686","https://openalex.org/W1593414945","https://openalex.org/W1963865657","https://openalex.org/W1966158133","https://openalex.org/W1969100313","https://openalex.org/W1985007325","https://openalex.org/W1987241994","https://openalex.org/W1989224938","https://openalex.org/W1990486429","https://openalex.org/W1991621869","https://openalex.org/W1992445851","https://openalex.org/W1999107555","https://openalex.org/W2003173073","https://openalex.org/W2010718945","https://openalex.org/W2014292079","https://openalex.org/W2018728680","https://openalex.org/W2027822086","https://openalex.org/W2028413297","https://openalex.org/W2029449803","https://openalex.org/W2029966914","https://openalex.org/W2030350543","https://openalex.org/W2031651409","https://openalex.org/W2044562696","https://openalex.org/W2046528357","https://openalex.org/W2050638715","https://openalex.org/W2050969603","https://openalex.org/W2053384289","https://openalex.org/W2068934186","https://openalex.org/W2077001143","https://openalex.org/W2077028260","https://openalex.org/W2079028266","https://openalex.org/W2079994134","https://openalex.org/W2083994446","https://openalex.org/W2086238793","https://openalex.org/W2093560293","https://openalex.org/W2112149397","https://openalex.org/W2128204029","https://openalex.org/W2130457461","https://openalex.org/W2151917717","https://openalex.org/W2167347425","https://openalex.org/W2168653054","https://openalex.org/W2328649268","https://openalex.org/W2521393946","https://openalex.org/W2963577202","https://openalex.org/W3098605801","https://openalex.org/W3148050609","https://openalex.org/W4229576011","https://openalex.org/W4231829751","https://openalex.org/W4248764030","https://openalex.org/W4256064871","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W57680709","https://openalex.org/W3203278930","https://openalex.org/W4302608385","https://openalex.org/W4297571134","https://openalex.org/W2983063598","https://openalex.org/W2963867164","https://openalex.org/W4297969557","https://openalex.org/W1529867338","https://openalex.org/W2230799146","https://openalex.org/W2992451487"],"abstract_inverted_index":{"We":[0],"present":[1],"an":[2,6,356],"error":[3,168,285],"analysis":[4],"for":[5,15,287],"unconditionally":[7],"energy":[8],"stable,":[9],"fully":[10],"discrete":[11,179],"finite":[12,429],"difference":[13,430],"scheme":[14],"the":[16,25,39,63,173,288,296,309,312,333,353,411,415,421],"Cahn-Hilliard-Hele-Shaw":[17],"equation,":[18],"a":[19,178,301,428,433],"modified":[20],"Cahn-Hilliard":[21],"equation":[22],"coupled":[23],"with":[24,352],"Darcy":[26],"flow":[27],"law.":[28],"The":[29],"scheme,":[30],"proposed":[31],"by":[32,332],"S.":[33],"M.":[34],"Wise,":[35],"is":[36,306,327,350],"based":[37],"on":[38],"idea":[40],"of":[41,62,355,410,414,432],"convex":[42],"splitting.":[43],"In":[44],"this":[45],"paper,":[46],"we":[47,176,426],"rigorously":[48],"prove":[49],"first":[50],"order":[51,57],"convergence":[52,58,305,424],"in":[53,59,300,308,328],"time":[54,313],"and":[55,423],"second":[56],"space.":[60],"Instead":[61],"(discrete)":[64],"<inline-formula":[65,180,315,336,357],"content-type=\"math/mathml\">":[66,181,316,337,358],"<mml:math":[67,182,317,338,359],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[68,183,318,339,360],"alttext=\"upper":[69,184,361],"L":[70,84,93,185,208,362],"Subscript":[71,85,94,107,186,200,209,222,363,376],"s":[72,95,187,210,364],"Superscript":[73,87,96,109,188,202,211,224,365,378],"normal":[74,189],"infinity":[75,190],"Baseline":[76,89,98,111,191,204,213,226,367,380],"left-parenthesis":[77,99,192,214,368],"0":[78,100,193,215,369],"comma":[79,101,194,216,370],"upper":[80,83,92,102,105,195,198,207,217,220,371,374],"T":[81,103,196,218,372],"semicolon":[82,104,197,219,373],"h":[86,108,201,223,377],"2":[88,97,110,212,366],"right-parenthesis":[90,205],"intersection":[91,206],"H":[106,199,221,375],"right-parenthesis\">":[112,227,381],"<mml:semantics>":[113,228,320,341,382],"<mml:mrow>":[114,229,383],"<mml:msubsup>":[115,129,139,150,230,244,254,265,384,395],"<mml:mi>L</mml:mi>":[116,130,140,231,255,385],"<mml:mi>s</mml:mi>":[117,141,232,256,321,386],"<mml:mi":[118,233],"mathvariant=\"normal\">":[119,234],"\u221e":[120,235],"</mml:mi>":[121,236],"</mml:msubsup>":[122,133,143,154,237,248,258,269,388,399],"<mml:mo":[123,134,144,155,238,249,259,270,389,400],"stretchy=\"false\">(</mml:mo>":[124,145,239,260,390],"<mml:mn>0</mml:mn>":[125,146,240,261,391],"<mml:mo>,</mml:mo>":[126,147,241,262,392],"<mml:mi>T</mml:mi>":[127,148,242,263,393],"<mml:mo>;</mml:mo>":[128,149,243,264,394],"<mml:mi>h</mml:mi>":[131,152,246,267,342,397],"<mml:mn>2</mml:mn>":[132,142,153,257,387],"stretchy=\"false\">)</mml:mo>":[135,156,250,271,401],"<mml:mo>":[136,251],"\u2229":[137,252],"</mml:mo>":[138,253],"<mml:mi>H</mml:mi>":[151,245,266,396],"</mml:mrow>":[157,272,402],"<mml:annotation":[158,273,322,343,403],"encoding=\"application/x-tex\">L_s^\\infty":[159,274],"(0,T;L_h^2)":[160],"\\cap":[161,277],"L_s^2":[162,278],"(0,T;":[163,275,279],"H_h^2)</mml:annotation>":[164],"</mml:semantics>":[165,282,324,345,406],"</mml:math>":[166,283,325,346,407],"</inline-formula>":[167,284,326,347,408],"estimate,":[169],"which":[170,291],"would":[171],"represent":[172],"typical":[174],"approach,":[175],"provide":[177],"1":[203],"3":[225,379],"<mml:mn>1</mml:mn>":[247],"<mml:mn>3</mml:mn>":[268,398],"H_h^1)":[276],"H_h^3":[280],")</mml:annotation>":[281],"estimate":[286],"phase":[289,416],"variable,":[290],"allows":[292],"us":[293],"to":[294],"treat":[295],"nonlinear":[297],"convection":[298],"term":[299],"straightforward":[302],"way.":[303],"Our":[304],"unconditional":[307],"sense":[310],"that":[311],"step":[314],"alttext=\"s\">":[319],"encoding=\"application/x-tex\">s</mml:annotation>":[323],"no":[329],"way":[330],"constrained":[331],"mesh":[334],"spacing":[335],"alttext=\"h\">":[340],"encoding=\"application/x-tex\">h</mml:annotation>":[344],".":[348],"This":[349],"accomplished":[351],"help":[354],"encoding=\"application/x-tex\">L_s^2":[404],"(0,T;H_h^3)</mml:annotation>":[405],"bound":[409],"numerical":[412],"approximation":[413],"variable.":[417],"To":[418],"facilitate":[419],"both":[420],"stability":[422],"analyses,":[425],"establish":[427],"analog":[431],"Gagliardo-Nirenberg":[434],"type":[435],"inequality.":[436]},"counts_by_year":[{"year":2026,"cited_by_count":12},{"year":2025,"cited_by_count":14},{"year":2024,"cited_by_count":10},{"year":2023,"cited_by_count":20},{"year":2022,"cited_by_count":13},{"year":2021,"cited_by_count":28},{"year":2020,"cited_by_count":9},{"year":2019,"cited_by_count":9},{"year":2018,"cited_by_count":8},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":2}],"updated_date":"2026-08-09T07:27:16.801131","created_date":"2025-10-10T00:00:00"}
