Residual Velocities in Steady Free Boundary Value

arXiv:0709.1734v1[math.NA]11Sep2007ResidualVelocitiesinSteadyFreeBoundaryValueProblemsofVectorLaplacianTypeWanChen∗BrianWetton†February1,2008AbstractThispaperdescribesatechniquetodeterminethelinearwell-posednessofageneralclassofvectorellipticproblemsthatincludeasteadyinterface,tobedeterminedaspartoftheproblem,thatsep-aratestwosubdomains.TheinterfacesatisfiesmixedDirichletandNeumannconditions.Weconsider“2+2”models,meaningtwoin-dependentvariablesrespectivelyoneachsubdomain.ThegoverningequationsaretakentobevectorLaplacian,tobeabletomakeana-lyticprogress.Theinterfaceconditionscanbeclassifiedintofourlargecategories,andweconcentrateontheonewithmostphysicalinterest.Thewell-posednesscriteriainthiscaseareparticularlyclear.Inmanyphysicalcases,themovementoftheinterfaceintime-dependentsitua-tionscanbereducedtoanormalmotionproportionaltotheresidualinoneofthesteadystateinterfaceconditions(theellipticinteriorprob-lemsandtheotherinterfaceconditionsaresatisfiedateachtime).Ifonlythesteadystateisofinterest,onecanconsiderusingotherresid-ualsforthenormalvelocity.Ouranalysiscanbeextendedtogiveinsightintochoosingresidualvelocitiesthathavesuperiornumericalproperties.Hence,inthesecondpart,wediscussaniterativemethodtosolvefreeboundaryproblems.Theadvantagesofthecorrectlycho-sen,non-physicalresidualvelocitiesaredemonstratedinanumericalexample,basedonasimplifiedmodeloftwo-phaseflowwithphasechangeinporousmedia.∗DepartmentofMathematics,UBC,[e-mail]†correspondingauthor,DepartmentofMathematics,UBC,[e-mail].1Keywords:Well-Posedness,FreeBoundaryProblem,ResidualVe-locity.1IntroductionFreeboundaryproblems(FBPs)havemotivatedseveralstudiesinthepastduetotheirvastapplicationsinfluidflow,phasechangemodelsandotherfields.Someclassicalfreeboundaryproblemsare:thedamseepageproblem[6],incompressibletwo-phaseflow[5,14](i.e.fallingdropletsorrisingbubbles[12],theAlt-Caffarelliproblem[2],theclassicalStefanproblem[6],etc.Fromamathematicalpointofview,FBPsareboundaryvalueproblemswithanunknownboundary.Themotion(unsteadycase)orposition(steadycase)oftheboundaryhastobedeterminedtogetherwiththesolutionofthegivenpartialdifferentialequationsonone(freesurface)orbothsides(interface)ofthefreeboundary.Thecouplingofthefreeboundarytotheinteriorisalwaysnonlinear[13],andthusFBPsareoftennoteasytosolve.Thecommonstructureofalltheexamplesaboveisthatatsteadystate,theyallhavesecondorderellipticgoverningequationsformunknownsononesideandnunknownsontheothersideofthefreeboundary.Wedenotethissituationasan“m+n”problem.Theremustthenbem+n+1Dirichlet-Neumannconditionsattheinterface.Forfourthorderproblemssuchasbiharmonicequation,wewouldneed2m+2n+1conditionsattheinterfacetodeterminean“m+n”problem.Bythisgeneralization,manymorecomplicatedproblemscanbeformulated.However,fourthorderproblemsandsecondorderellipticsystemsotherthanvectorLaplacianarenotconsideredfurtherinthispresentstudy.Tosolvefreeboundaryproblemsnumericallytherearethreemainkindsofmethods:capturingmethods,fronttrackingmethodsandlevelsetmethods.CapturingmethodsarebasedonEulerianfor-mulationandtheproblemsarereformulatedandsolvedinthewholedomain.Theinterfacelocationisrecoveredfromthediscretesolution.Inthesemethods,theinterfaceconditionsarenotspecifiedexplicitly.AclassicalexampleistheEnthalpymethod[6].Thealternativeistodiscretizetheinterfaceexplicitly[16][18]orviaalevelsetapproach[11].Inboththesecases,theinterfaceconditions(andinterfaceve-locityfortime-dependentproblems)areimplementedexplicitly.Thiscanbedonebyconsideringthedomainsasdisjointanddiscretizing2theequationsandinterfaceconditionsdirectly[8]orbydiscretizingtheentiredomainandmodifyingthediscretizationneartheinterface[10].Thelatterapproachcombinedwithmodernlevelsettechniquescanalsobeconsideredacapturingmethod.Inafewcases,steadystatesolutionscanbeobtaineddirectlyusingshapeoptimization[13,9].Anotherapproachtoreachsteadystatesolutionsistosolvethetransienttime-dependentproblemtolongtime.Intheclassofproblemsweconsider,thenormalmotionoftheinterfaceisdrivenbythedifferenceofsolutionsoneithersideoftheinterface.Forexample,solidificationboundariesaredrivenbyaStefancondition[6].Itshouldbemadeclearthatwearenotconsideringheretheproblemofdendriticgrowth[1,4]whichinourframeworkwouldbeill-posed,butisregularizedbyhigherorder(curvature)terms.Weconsideronlywell-posedproblemswhichdonotneedregularization.Inthephysicalexampleofsection3,theinterfaceismovedbythenetmassfluxattheinterface.Thosearetypicalconditionsthatgivenormalvelocityofeachpointontheinterface.Intheseexamplesandothersofphysicalinterest,thenormalinterfacevelocityisgivenbytheresidualinoneofthesteadystateinterfaceconditions.Inthispaper,weapplytheresidualvelocitymethod(atrackingtechnique)of[7]:1.GivenaninitialinterfaceΓandtrackingpoints{x:x∈Γ},solvenumericallyafixedboundaryvalueproblemsatisfyingallofthesteadyinterfaceconditionsbutone.2.SubstitutethesolutionsintotheunsatisfiedconditionandfindtheresidualR(x).3.Iftheresidualislargerthanatolerance,explicitlyevolvetheinterfaceusingtheresidualasanormalvelocityofthetrackingpoints:Vn=R(x)4.Repeatthe