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1、ᜧắ ᦻLocalDiscontinuousGalerkinMethodForNon-FickianDiffusionInViscoelasticPolymersWangDexinSchoolofMathematicalSciences,ShandongNormalUniversityJinan,Shandong,250014ABSTRACTManypracticalphenomenons,suchastheeffectofhumidityonthinpolymerfilms,thepenetrationanddiffusionofliquidflowinpolymermater
2、ial,aregovernedbynon-FickianDiffusioninviscoelasticpolymers.Numerousexperimentshaverecognizedthatthesephenomenonscharacterizedbyaconstantvelocityspreadingofthepenetratewithawell-definedsharpfront,whichisgenericallycallednon-Fickianbehaviorornon-Fickianflow.Forindustrialapplications,theprimaryinteres
3、tsinthemathematicalmodelofthenon-Fickiandiffusionaretheconcentrationwhichshowstheextentandlocationofthepenetrableliquid,thefluxwhichindicatesthequantityofthepenetrableliquidthroughsomeareaoftheporousmedia,andtheviscoelasticstressinthepolymerfilm.Wewanttoproposeanumericalmethodwhichcanapproximatesimu
4、ltaneouslytheunknownfunction(theconcentration),theviscoelasticstressinthepolymerfilmandtheadjointvector-function(theflux).Inthispaper,wesimulatenumericallythefollowingnonlinearnon-FickianD3万方数据ᜧắ ᦻiffusionproblem(a)ut-V-(D(u)Vu)=f+(K(u)t),(x,t)GQxI,(b)Tt+y(u)t=^Vu,(x,t)GQxI,(c)u(x,t)=0,onFdxI
5、,(d)(D(u)Vu+K(u)t)•n=gNonrnxI,(e)u(x,0)=uo(x),a(x,0)=ao(x),xGQ,byalocaldiscontinuousGalerkinfiniteelementmethod.TheoreticalanalysisinthisthesisindicatesthatthismethodinheritstheadvantagesoflocaldiscontinuousGalerkinmethod,suchasapproximatingsimultaneouslytheconcentrationu,theviscoelasticstresst,andt
6、hefluxfunctiong=D(u)Vu+K(u)twithhighaccuracy.AndfromthefluxgandstresstwecaneasilyobtainVu,whichdescribesthewell-definedsharpfront.Weusee-inequalityandGronwallinequalitytoanalysisthelocaldiscontinuousfiniteelementmethod.Andweprovethestabilityoflocaldiscontinuousfiniteelementschemeundertheratioofgrids
7、(K/h
