Finite element and difference approximation of som

FiniteElementandDierenceApproximationofSomeLinearStochasticPartialDierentialEquationsE.J.Allen,S.J.Novosel,Z.ZhangTexasTechUniversityLubbock,TX79409AbstractDierenceandniteelementmethodsaredescribed,analyzed,andtestedfornu-mericalsolutionoflinearparabolicandellipticSPDEsdrivenbywhitenoise.Weakandintegralformulationsofthestochasticpartialdierentialequationsareapproximated,respectively,byniteelementanddierencemethods.Thewhitenoiseprocessesareapproximatedbypiecewiseconstantrandomprocessestofacilitateconvergenceproofsfortheniteelementmethod.Erroranalysesofthetwonumericalmethodsyieldesti-matesofconvergencerates.Computationalexperimentsindicatethatthetwonumer-icalmethodshavesimilaraccuracybuttheniteelementmethodiscomputationallymoreecientthanthedierencemethod.KEYWORDS:Stochasticpartialdierentialequations,niteelements,nitedierences.1IntroductionStochasticinitial-valueproblemsandstochasticboundary-valueproblemsareofincreas-inginterestastheirtheoreticalfoundationisdevelopedandnewapplicationsarediscov-ered[7,8,13,15].Manyinterestingnumericalmethodsforapproximatingstochasticinitial-valueproblemshaverecentlybeendeveloped,analyzed,andtested.(See,e.g.,[1,8,11,14].)However,numericalapproximationofstochasticboundary-valueproblemshasnotbeenre-searchedasthoroughly[2,3,6,12].Inthepresentinvestigation,niteelementanddier-encemethodsaredescribed,analyzed,andcomparedfornumericalsolutionofellipticandparabolicstochasticdierentialequationsdrivenbywhitenoise.1Dierenceandniteelementmethodsarestudiedforapproximatesolutionoflinearellipticstochasticequationsoftheform8:u(x)+bu(x)=g(x)+_W(x);0x1u(0)=u(1)=0(1.1)where_W(x)denoteswhitenoiseandforlinearparabolicstochasticequationsoftheform8:@u@t(t;x)@2u@x2(t;x)+bu(t;x)=@2W@t@x(t;x)+g(t;x);t0u(0;x)=u0(x);0x1u(t;0)=u(t;1)=0;t0(1.2)where@2W@t@xdenotesthemixedsecond-orderderivativeoftheBrowniansheetandbisaconstant.Anapplicationof(1.2)occurs,forexample,inamodelinneurophysiology[15].Thenumericalproceduresareappliedtoweakandintegralformulationsoftheaboveequations.TheweakandintegralformulationswereshownbyBuckdahnandPardoux[4]tobeequivalent.Theweakformulationof(1.1)hastheformZ10u(x)(x)dx+Z10bu(x)(x)dx=Z10g(x)(x)dx+Z10(x)dW(x)(1.3)for2C2(0;1)\C0[0;1]andtheintegralformof(1.1)hastheformu(x)+Z10k(x;y)bu(y)dy=Z10k(x;y)g(y)dy+Z10k(x;y)dW(y)(1.4)wherek(x;y)=x^yxyisGreen’sfunctionassociatedwiththeellipticequationv(x)=(x),v(0)=v(1)=0sothatv(x)=R10k(x;y)(y)dy.Theweakformulationof(1.2)hastheformZ10u(t;x)(x)dxZt0Z10u(s;x)d2dx2(x)dxds+Zt0Z10bu(s;x)(x)dxds(1.5)=Z10u0(x)(x)dx+Zt0Z10(x)dW(s;x)+Zt0Z10g(s;x)(x)dxdsfor2C2[0;1]\C0[0;1]andtheintegralformulationof(1.2)hastheformu(t;x)+Zt0Z10Gts(x;y)bu(s;y)dyds=Z10Gt(x;y)u0(y)dy(1.6)+Zt0Z10Gts(x;y)dW(s;y)+Zt0Z10Gts(x;y)g(s;y)dyds2whereGt(x;y)=21Xn=1sinnxsinnye(n)2tisthefundamentalsolutionofvt(t;x)vxx(t;x)=0;v(0;x)=(x);v(t;0)=v(t;1)=0;sothatv(t;x)=Z10Gt(x;y)(y)dy.Inthepresentinvestigation,itisassumedthatthevalueofbin(1.1)issucientlysmallsothat2=b2Z10Z10k2(x;y)dxdy1:Similarly,itisassumedthatthevalueofbin(1.2)issucientlysmallsothat~2=b2ZT0Z10Zt0Z10G2ts(x;y)dydsdxdt1:ThefollowingresultisprovedbyBuckdahnandPardouxconcerningexistenceanduniquenessofsolutionsto(1.3)and(1.4)and(1.5)and(1.6)andisreproducedhereforconvenience.Theorem1.1Letg2L2(0;1),then(1.4)possessesauniquesolutionwhichisa.s.con-tinuouson[0;1].Letu02C0[0;1]andg2L2(loc)((0;1)(0;1)),then(1.6)hasauniquesolutionu2C((0;1)[0;1])a.s.Furthermore,(1.3)and(1.4)areequivalentand(1.5)and(1.6)areequivalent.Inthisinvestigation,niteelementproceduresaredeveloped,analyzed,andtestedfornumericalsolutionoftheweakformulations(1.3)and(1.5).Dierencemethodsarede-veloped,analyzed,andtestedfornumericalsolutionoftheintegralformulations(1.4)and(1.6).Twodierentnumericalmethodsareusefulforcomparisonpurposesasexactsolutionsgenerallyarenotavailablefortheseproblems.Intheconvergenceproofsofthesenumericalmethods,inparticulartheniteelementmethod,satisfactionofcertainregularityconditionsonthesolutionsto(1.4)and(1.6)fa-cilitatetheanalyses.Indeed,iftheseregularityconditionsaremet,thenstandardanalysistechniquesintheniteelementmethodcanbeapplied.Unfortunately,asdescribedbelow,3therequiredconditionsarenotsatisedfortheaboveproblems.Toovercomethisdiculty,thewhitenoiseprocessesarerstapproximatedbypiecewiseconstantrandomprocesses.(Theauthorsareunawareofpreviousformulationandapplicationofsuchwhitenoiseap-proximations.)Theseapproximationsaresubstitutedforthewhitenoiseprocessesin(1.4)and(1.6).Itisshownthatthesolutionstothenew\simplerproblemsconvergetotheactualsolutionsof(1.4)and(1.6)asthewhitenoiseapproximationsbecomener.Itisthenshownthatcertainregularityconditionsaremetbythesolutionstothesimplerproblems.Finally,itisprovedthatniteelementanddierenceapproximationsconvergetothesolu-tionsofthesimplerproblemsandhencetothesolutionsoftheoriginalproblems.Inthenalsection,computationalresultsaredescribedandcomparedforthetwonumericalmethods.Inthenextsection,theapproximatewhitenoiseprocessesar