A posteriori error estimates for finite element ap

APOSTERIORIERRORESTIMATESFORFINITEELEMENTAPPROXIMATIONSOFTHECAHN-HILLIARDEQUATIONANDTHEHELE-SHAWFLOWXIAOBINGFENGANDHAIJUNWUyAbstract.ThispaperdevelopsaposteriorierrorestimatesofresidualtypeforconformingandmixedniteelementapproximationsofthefourthorderCahn-Hilliardequationut+

u1f(u)
=0.Itisshownthattheaposteriorierrorboundsdependson1onlyinsomelowpolynomialorder,insteadofexponentialorder.Usingtheseaposteriorierrorestimates,wecon-structanadaptivealgorithmforcomputingthesolutionoftheCahn-Hilliardequationanditssharpinterfacelimit,theHele-Shawow.Numericalexperimentsarepresentedtoshowtherobustnessandeectivenessofthenewerrorestimatorsandtheproposedadaptivealgorithm.Keywords.Cahn-Hilliardequation,Hele-Shawow,phasetransition,conformingelements,mixedniteelementmethods,aposteriorierrorestimates,adaptivityAMSsubjectclassications.65M60,65M12,65M15,53A101.Introduction.Inthispaperwederiveaposteriorierrorestimatesandde-velopanadaptivealgorithmbasedontheerrorestimatesforconformingandmixedniteelementapproximationsofthefollowingCahn-HilliardequationanditssharpinterfacelimitknownastheHele-Shawow[2,37]ut+

u1f(u)
=0inT:=(0;T);(1.1)@u@n=@@n
u1f(u)
=0in@T:=@(0;T);(1.2)u=u0inf0g;(1.3)whereRN(N=2;3)isaboundeddomainwithC2boundary@oraconvexpolygonaldomain.T0isaxedconstant,andfisthederivativeofasmoothdoubleequalwellpotentialtakingitsglobalminimumvalue0atu=1.Awellknownexampleoffisf(u):=F0(u)andF(u)=14(u21)2:Forthenotationbrevity,weshallsuppressthesuper-indexonuthroughoutthispaperexceptinSection5.Theequation(1.1)wasoriginallyintroducedbyCahnandHilliard[11]todescribethecomplicatedphaseseparationandcoarseningphenomenainameltedalloythatisquenchedtoatemperatureatwhichonlytwodierentconcentrationphasescanexiststably.TheCahn-Hilliardhasbeenwidelyacceptedasagood(conservative)modeltodescribethephaseseparationandcoarseningphenomenainameltedalloy.ThefunctionurepresentstheconcentrationofoneofthetwometalliccomponentsDepartmentofMathematics,TheUniversityofTennessee,Knoxville,TN37996,U.S.A.([e-mail]).TheworkofthisauthorispartiallysupportedbytheNSFgrantDMS-0410266.yDepartmentofMathematics,NanjingUniversity,Jiangsu,210093,PRChina.([e-mail]).TheworkofthisauthorispartiallysupportedbytheChinaNSFgrant10401016,bytheChinaNationalBasicResearchProgramgrant2005CB321701,andbytheNaturalScienceFoundationofJiangsuProvinceunderthegrantBK2006511.1arXiv:0708.2116v1[math.NA]15Aug20072X.FengandH.Wuofthealloy.Theparameterisan\interactionlength,whichissmallcomparedtothecharacteristicdimensionsonthelaboratoryscale.Cahn-Hilliardequation(1.1)isaspecialcaseofamorecomplicatedphaseeldmodelforsolidicationofapurematerial[10,29,33].Forthephysicalbackground,derivation,anddiscussionoftheCahn-Hilliardequationandrelatedequations,wereferto[4,2,7,11,13,20,35,36]andthereferencestherein.ItshouldbenotedthattheCahn-Hilliardequation(1.1)canalsoberegardedastheH1-gradientowfortheenergyfunctional[28](1.4)J(u):=Zh12jruj2+12F(u)idx:Inadditiontoitsapplicationinphasetransition,theCahn-Hilliardequation(1.1)hasalsobeenextensivelystudiedinthepastduetoitsconnectiontothefollowingfreeboundaryproblem,knownastheHele-ShawproblemandtheMullins-Sekerkaproblem
w=0innt;t2[0;T];(1.5)@w@n=0on@;t2[0;T];(1.6)w=ont;t2[0;T];(1.7)V=12h@w@nitont;t2[0;T];(1.8)0=00whent=0:(1.9)Here=Z11rF(s)2ds:andVare,respectively,themeancurvatureandthenormalvelocityoftheinterfacet,nistheunitoutwardnormaltoeither@ort,[@w@n]t:=@w+@n@w@n,andw+andwarerespectivelytherestrictionofwin+tandt,theexteriorandinterioroftin.Undercertainassumptionontheinitialdatumu0,itwasrstformallyprovedbyPego[37]that,as&0,thefunctionw:=
u+1f(u),knownasthechemicalpotential,tendstow,which,togetherwithafreeboundary:=[0tT(tftg)solves(1.5)-(1.9).Alsou!1intforallt2[0;T],as&0.TherigorousjusticationofthislimitwascarriedoutbyAlikakos,BatesandChenin[2]undertheassumptionthattheaboveHele-Shaw(Mullins-Sekerka)problemhasaclassicalsolution.Later,Chen[13]formulatedaweaksolutiontotheHele-Shaw(Mullins-Sekerka)problemandshowed,usinganenergymethod,thatthesolutionof(1.1)-(1.3)approaches,as&0,toaweaksolutionoftheHele-Shaw(Mullins-Sekerka)problem.OneofaconsequencesoftheconnectionbetweentheCahn-HilliardequationandtheHele-Shawowisthatforsmallthesolutionto(1.1)-(1.3)equals1inthetwobulkregionsofwhichisseparatedbyathinlayer(calleddiuseinterface)ofwidthO().Asexpected,thesolutionhasasharpmovingfrontoverthetransitionlayer.AnothermotivationfordevelopingecientadaptivenumericalmethodsfortheCahn-Hilliardequationisitsapplicationsfarbeyonditsoriginalroleinphasetransi-tion.TheCahn-Hilliardequationisindeedafundamentalequationandanessentialbuildingblockinthephaseeldtheoryformovinginterfaceproblems(cf.[31]),itAdaptivemethodsfortheCahn-Hilliardequation3isoftencombinedwithotherfundamentalequationsofmathematicalphysicssuchastheNavier-Stokesequation(cf.[22,30,34]andthereferencestherein)tobeusedasdiuseinterfacemodelsfordescribingvariousinterfacedynamics,suchasowoftwo-phaseuids,fromvariousapplications.Theprimarynumericalchallengeforsol