A Numerical Method for Steady State Free Boundary

ANumericalMethodforSteadyStateFreeBoundaryProblemsZhiminZhangDepartmentofMathematicsTexasTechUniversity,Lubbock,TX79409IvoBabuskaInstituteforPhysicalScienceandTechnologyUniversityofMaryland,CollegePark,MD20742AbstractThisistherstinaseriesofpapersinwhichwediscussanumericalmethodtosolveacertainclassofsteadystatefreeboundaryproblems.ThemethodconstructsasequenceofsolutionsfortheLaplaceequationindierentdomains.WeshowinthispaperthatthesequenceconvergestothesolutionofgivenfreeboundaryproblemundertheassumptionthatwecansolvetheLaplaceequationinaxeddomainexactly.Unlikethetraditionalapproach,thenewmethodtakesadvantageoftheanalyticnatureofthefreeboundarytoachieveanexponentialrateofconvergence.TheniteelementapproximationoftheLaplaceequationateachiteration,relatednumericalaspects,andnumericalimplementationwillbeconsideredinforthcomingpapers.Keywords.freeboundary,theLaplaceequation,minimizationsequence,variationalformulation,aprioriestimate,exponentialrateofconvergence.AMSsubjectclassications.65N30.1IntroductionInthispaperweshallconsideramodelfreeboundary(steadystate)problemgovernedbytheLaplaceequation.Sincethefreeboundaryhastobesolvedaspartsofthesolution,methodsfor(xed)boundaryvalueproblemscannotbeapplieddirectly.Intheliterature,therearemanymethodstosolvefreeboundaryproblems.See[5],[6],[7]andthereferencetherein.Amongthem,thevariationalinequalitymethodisprobablythemostecientThisworkwaspartiallysupportedbytheNationalScienceFoundationgrantCCR-88-20279whiletheauthorwasatTheUniversityofMarylandatCollegePark.1one.Unfortunately,onlycertaintypesoffreeboundaryproblemshavevariationalinequalityformulationsorsuitablevariationalformulationsfornumericalmethods.See[11]forarecentworkinthisdirection.Weconsiderhereamodelproblemwhichhasnopropervariationalformulationfordirectnumericalimplementation.ThisproblemwasdiscussedbyAckerusingastabilizedtrialfree-boundarymethod.AckerprovedthattheconvergencerateofhismethodisO(1=n),wherenisthenumberofiterations.Thereaderisreferredto[1]fordetails.alsosee[2]forasocalled\xedmesh-variationalapproach.Ourmethoddiscretizesthefreeboundarybyparameterizedanalyticcurves.Thisallowsthebestutilizationoftheanalyticnatureofthefreeboundary.WeconstructasequenceofsolutionsfortheLaplaceequationindierentdomainsbyiterativelyminimizingacertainfunctional.WeassumethatateachiterationwecansolvetheLaplaceequationexactly.ThenweareabletoachieveanexponentialconvergencerateO(en)insomeSobolevnorm,wherenisthenumberofparameters.Thesignicanceofouranalysisisthatwerelateapproximationerrorwiththeobjectivefunctionintheminimizationprocess.Thenweareabletoestablishareliableaposteriorierrorestimatesincetheobjectivefunctionispracticallycomputable.Wewilladdressthisinaforthcomingpaperinwhichwediscussthenumericalimplementationofourmethod.Hereistheoutlineofthepaper.WeintroducethemodelproblemandsomenotationsinSection2.TheadmissiblesetoftheproblemisdenedinSection3anditsbasicpropertiesarediscussedinSection3andSection5.InSection4weconstructaminimizationprocessandshowitsconvergence.AnaprioriestimatewhichrelateserrorestimatewiththeobjectivefunctionisestablishedinSection6.WeproveanexponentialrateofconvergenceforadiscretizedminimizationsequenceinSection7andgivesomeconclusionremarksinSection8.2TheModelProblemandNotationsLet:=f(x;(x));x2Rg,whereisaperiodicLipschitzcontinuousfunction,andletQ0beaconstant.Considerthefreeboundaryproblem(FF)Findacurve:=f(x;(x));x2Rgwith(x)(x)andafunctionusuchthatu=0in;u=1on;u=0;@u=Qon;whereisthestripregionboundedbyand,[e-mail](xed)boundaryvalueproblem,partoftheboundaryinProblem(FF)isun-knownandhastobefoundaspartofthesolution.Wecallthispartoftheboundary\freeboundary.Wealsonoticethattherearetwoboundaryconditionsonthefreeboundary.Thesolutionpair(;u)hashydrodynamicalmeaning:u(x;y)isthestreamfunctionoftheirrotationalowofanidealincompressibleuid,withoutsources,overthebottom.describesthewavegeometry.TheconstantQcharacterizestheforceeldwhichactsonthemovinguid.Forthemathematicalnatureofthisproblem,wequotethefollowingtheoremfrom[1](Theorem1):2Theorem2.1Thefree-boundaryproblem(FF)isuniquelysolvedbyanananalyticcurve:=f(x;(x));x2Rg.Fromtheperiodicitycondition,itispossibletoconsideraprobleminaboundeddomaininsteadofaninniteone.Thiscanbedonebycuttingooneperiodoftheinnitedomaininxdirection.Withoutlossofgeneralityweassumethattheperiodis1andthecutisfrom0to1.Inordertofurthersimplifytheproblem,weassumethat(x)issymmetricintoeachperiodwithrespecttoitsmiddlepoint,andhencetheboundaryconditiononthetwoverticalboundariesis@xu=0.Therefore,equivalenttotheproblem(FF),wehavethefollowing:(P)Findananalyticcurve:=f(x;(x));x2[0;1]gwith(x)(x)andafunctionusuchthat(u=0in;@xu=0onx=0;1;u=1on;u=0;@u=Qon;whereisanitedomainboundedbylinesx=0,x=1andcurves:=f(x;(x));x2[0;1]g;:=f(x;(x));x2[0;1];(0)=(1);0(0)=0(1)=0g:Notehere,anequivalentconditionfor@u=Qonisjruj=Q,sinceu=0onandu=1on.Obviously,Theorem2.1isalsoapplicabletoProblem(P).Next,wewritetheproblem(P)i