barles1990-Convergence-of-Approximation-Schemes-Fo

Proceedlngaofthe29thConferenceonDeclrlonandControlHonolulu.HawallDecember1990TP-11=4:40ConvergenceofApproximationSchemesForFullyNonlinearSecondOrderEquations*G.BarlesandP.E.SouganidisBrownUniversityProvidence,R.I.02912,USACEREMADEDivisionofAppliedMathematicsUniversityofParis-DauphineParis,FRANCEAbstractWestudytheconvergenceofawideclassofapproxima-tionschemestotheviscositysolutionoffullynonlinearsecond-orderellipticorparabolic,possiblydegenerate,partialdifferen-tialequations.I.IntroductionInthisnoteweshowtheconvergenceofawideclassofap-proximationschemestothesolutionoffullynonlinearsecond-orderellipticorparabolic,possiblydegenerate,pde's.Roughlyspeaking,weprovethatanymonotone,stableandconsistentschemeconverges(tothecorrectsolution)providedthatthereexistsacomparisonprincipleforthelimitingequation.Wethengiveseveralexamplesofconcreteschemes,wheretheresultap-plies.TheformulationoftheschemesfollowsalongthelinesofSouganidis[14],wheretheanalogousproblemwasstudiedforfirst-ordereuqations,butnotinthepresentgenerality.TheconvergenceresultisbaseduponexploringabasicideaofBar-lesandPerthame[2,3]regardingpassagetothelimitsinfullynon-linear,second-order,ellipticpde's,withonlyLestimates.Thismethodreliesonthenotionofviscositysolutions,intro-ducedbyCrandallandLions[6]forfirst-orderproblemsandextendedtosecond-orderbyLions[12].Ourapproachispurelyanalyticanddoesnotrelyonanyconvexityorconcavityorsmoothnessassumptions;wearethusabletopresentcompletelynewresultsconcerningconvergenceofnumericalschemes...etc.TheseresultsincludeasspecialcasesmostoftheresultsofBardiandFalcone[l],Capuzzo-DolcettaandFalcone[5],Fal-cone(71,Kushner[ll]andMenaldi[13]...whodealwithconvexorconcaveproblems.Inthisnotewechosetopresentasimplecaseinordertopresentationaswellasthemainideassimple.Formoredetails,werefertoBarlesandSouganidis[4].*G.BarleswassupportedbyAFOSRcontract86-0315.P.E.Sougani-diswaspartiallysupportedbyNSFgrantsDMS-8657464andDMS-8801208,AROcontractDAAL03-90-G-0012,DARPAcontractF49620-88-C-0129andtheSloanFoundation.CH2917-3/90/0000-2347$1.OO@1990IEEETheorem:Assume(2)-(5),uoEBUC(R)andFEBUC(Sn).Thenup4UlocallyuniformlyonEnx[O,T],asthemeshllPllofthepartitionPgoestozero.023472.TheMainResultWeconsidertheCauchyproblemuttF(D2u)=OinRx(o,T)(1){u(z,D)=~(z)onR.HereUandFarecontinuousfunctionsoftheirarguments,D2udenotesthehessianmatrixofUwithrespecttox,andFisassumedtobeelliptic,ie.F(M)5F(N)ifM2N(2)forallM,NESn(thespaceofnxnsymmetricmatrices).Itiswellknown(cq9,10]),thatifFisuniformlycontinuousanduoEBUC(Rn),then(1)hasauniqueviscositysolu-tioninBUC(Rx[O,T]),whereBUC(D)denotesthespaceofbounded,uniformlycontinuousfunctionsdefinedonD.Wenowconstructageneralschemethatissupposedtoap-proximate(1).Tothisend,forp0letS(p):BUC(E).+BUC(R)besuchthatS(p)u2S(p)vifU2w,S(p)(utk)=S(p)utk(kER),(3)(4)and4-s(p)'-+F(D*~)asp-+ofordl4Ecw(5).PGivensuchanSandapartitionP=(0=totlRnx[O,T].+RbytN=T}of[O,T],wedefineup:S(t-LI)uM(.,~~-I)(-),iftE(ti-1,tiIUO(*),ift=0.(6)up(*,t)=andpayoffEuo(XT).Souganidis[8]).Example2:Numericalapproximations.Forsimplicityassumen=1anduotobe1-periodic,whichinturnyieldsthatUis1-periodicinz.Tocomputethesolution,weconsideragridinspaceandtimeofmeshsizeAzandAtrespectively,anddevotebyMandKtheintegerssuchthatMAX=1andKAt=T.Finally,wewriteU:forthequantityu(jAz,kAt).OneofthesimplestschemesapproximatingUisgivenby(Fordetailssee[4]andFlemingand3.ProofoftheTheoremBeforewebeginwiththeproof,werecallthedefinitionofviscositysolution.Tothisend,werecallthenotionsofupperandlowersemicontinuousenvelopesofafunctionW.Theseare:w*(z)=limsupw(y)andtu&)=liminfw(y).y-2Y-*ZDefinition.AboundedfunctionU:Rx[O,T]-+Risa(viscosity)subsolution(resp.supersolution)of(l),ifforall4ECmandall(z,t)ERx(O,T]suchthatU*-4(resp.U.-4)hasalocalmu(resp.min)at(x,t),wehave4t(z,t)tF(D24(.,t))50#t(z,t)tF(D24(z,t))20.)(7)(8)(respThefunctionUissaidtobea(viscosity)solutionof(l),ifitisbothsub-andsupersolutionof(1).Wecontinuenowwiththeproofofthetheorem.Itisimmediate,inviewofourassumptions,thatthereexistsaconstantCsuchthatIup(z,t)l5Cforall(z,t)ERx[O,T]andallpartitionsP.WethendefineUandup*.Inviewoftheirdefinitionwehaveup,5UinRnx[O,T].Ontheotherhand,wewillshowthatUisasubsolutionandup,isasupersolutionof(1).Therefore,bythestrongcomparisonpropertyof(1)(cf.[I),weobtainU$5U5uptinEnx[O,T],whereUEBUC(Rx[O,T]).Hence,U=up*=up*,thustheresult.HereweonlyshowthatU;isasubsolutionof(1).Indeed,let(z,t)ERx(O,T]beastrictlocalmaxofU-4,forsmoothfunction4.Then(cf[9,10])thereexistpartitionsP,,andpoints(rPn,tpn)ERx(0,T]suchthatup-4attainsalocalTherefore(3)and(4)yieldmaxat(zpn,tpn)and(zpn,tpn)+(z,t).Buttp,E(ti,-l,tn].4(zp,rtpn)Is(tp,-tin-l)~(.,~in-l)(zpn),Dividingbyt,,-tin-l,lettingIIPnll+0andusing(5),weconclude.ExamplesExample1:Stochasticdifferentialgames.SupposethatF(M)=mumint-trace-a(y,z))M)YEYZEZ[:IwhereY,2arecompactsetsanda=aa',anddefinewhere9is,ameanzerorandomvector,withcomponentsin-dependentandtakingvaluesfl.Finally,Egdenotestheex-pectedvalueofg.Thefunctionupdefinedby(6)andtheSaboveconvergesasllPll+0tothelowervalueofthestochasticdifferential