Numerical solution of parabolic integrodifferentia

NUMERICALSOLUTIONOFPARABOLICINTEGRO-DIFFERENTIALEQUATIONSBYTHEDISCONTINUOUSGALERKINMETHODSTIGLARSSON,VIDARTHOMEE,ANDLARSB.WAHLBINAbstract.Thenumericalsolutionofaparabolicequationwithmemoryisconsidered.Theequationisrstdiscretizedintimebymeansofthediscontin-uousGalerkinmethodwithpiecewiseconstantorpiecewiselinearapproximat-ingfunctions.Theanalysispresentedallowsvariabletimestepswhich,aswillbeshown,canthenecientlybeselectedtomatchsingularitiesinthesolutioninducedbysingularitiesinthekernelofthememorytermorbynonsmoothinitialdata.Thecombinationwithniteelementdiscretizationinspaceisalsostudied.1.IntroductionLetHbeaseparableHilbertspaceandassumethatAisalinear,selfadjoint,positivedenite,notnecessarilyboundedoperator,withcompactinverse,denedinD(A)H,andthat,for0stT,B(t;s)isalinearoperatorinHwithD(B(t;s))D(A).Considertheinitialvalueproblemut+Au+Zt0B(t;s)u(s)ds=f;fort2(0;T];withu(0)=u0;(1.1)wheref=f(t),u=u(t),ut=du=dt.Settingkvkp=kAp=2vk=(Apv;v)1=2,wherekkisthenormand(;)thetheinnerproductinH,weassumethroughoutthepaperthattheoperatorAdominatesB(t;s)inthesensethat,forsome2(0;1],j(B(t;s)v;w)jC(ts)1kvkpkwkq;p=0;1;2;p+q=2:(1.2)For01,(1.2)reectsaweaklysingularbehaviorofB(t;s).When=1weshallsometimesassumethatanappropriatenumberofderivativesofB(t;s)existandarealsodominatedbyA;inthiscasewerefertoB(t;s)asa\smoothkernel.Intheapplicationsthatwehaveinmind,eitherAisanellipticsecondorderdierentialoperatorinaboundeddomainRdwithhomogeneousDirichletboundaryconditions,andB(t;s)isasecondorderdierentialoperator,orelseAandB(t;s)arediscreteanalogsofsuchoperators,arisingfromaniteelementdiscretizationinthespatialvariables.Ourabstractframeworkmakesitpossibletotreatthesecasessimultaneously.Inthedierentialoperatorscase,(1.2)amounts1991MathematicsSubjectClassication.65M60,65R20,45L10.Keywordsandphrases.Parabolicintegro-dierentialequation,weaklysingularkernel,discon-tinuousGalerkin,variabletimestep,niteelement,errorestimate,Gronwalllemma.ThersttwoauthorswerepartlysupportedbytheSwedishResearchCouncilforEngineeringSciences(TFR).ThethirdauthorthankstheNationalScienceFoundation,USA,fornancialsupportandalsoChalmersUniversityofTechnologyandGoteborgUniversityfortheirhospitalityduringtheSpringof1995.PublishedinMath.Comp.67(1998),45{71.12STIGLARSSON,VIDARTHOMEE,ANDLARSB.WAHLBINtoellipticregularity,plusaboundforthecoecientsofB(t;s).Theproblemconsideredmay,e.g.,bethoughtofasamodelproblemoccurringinthetheoryofheatconductioninmaterialswithmemory,cf.[3].Equationswithweaklysingularkernelsoccurin[7],[9],[10].Forotherreferences,see,e.g.,[16].Weshallconsidertheapproximatesolutionof(1.1)bymeansofthediscon-tinuousGalerkinmethod(cf.[4],[5]),whichweshalldenebelowinthepresentcontext.WhenAandB(t;s)aredierentialoperators,weshallconsideralsothediscretizationinspacebyniteelements,whichwillthendeneafullydiscretemethodfor(1.1)inthiscase.Forearlierworkondiscretizationintimeorspace,orboth,ofequationssuchas(1.1),see,e.g.,[1],[2],[8],[11],[12],[13],[14],[15],[16],[17].AsweshallseeinSection5,aweaklysingularkernelinthememorytermtypicallyleadstoasingularityinthesolution(withrespecttotime),asdononsmoothinitialdata.ItishenceofinterestthatthediscontinuousGalerkinmethod,ascomparedtostandardnitedierencemethodsintime,facilitatestheanalysisofvariabletimestepsand,also,acceptslowerregularityofsolutions.TheseadvantagesofthediscontinuousGalerkinmethodarewellknowninthecontextofstandardparabolicproblems,cf.[4],[5].However,wepointoutthatouranalysisinthispaperdoesnottakeintoaccountnumericalapproximationofcertainintegralsoccurring.Todeneourtimesteppingmethod,let0=t0t1tnTbeapartitionoftheinterval[0;T],anddeneIn=(tn1;tn),kn=tntn1.FurtherletVN=VNq,fortN2(0;T],denotethesetofscalarfunctionson[0;tN],which,forn=1;:::;N,reducetopolynomialsofdegreelessthanqonInwithq=1or2.WeshallworkwithfunctionsinWNVND(A1=2);inthedierentialoperatorapplicationsthesearefunctionsof(x;t)2[0;tN],whichareeitherpiecewiseconstantorpiecewiselinearintime,notnecessarilycontinuousatthenodesofthepartition.LettingA(v;w)andB(t;s;v;w)denotethenaturalbilinearformsonD(A1=2)generatedby(Av;w)and(B(t;s)v;w),respectively,weset,forpiecewisesmoothfunctionsV;W,with[V]n=V+nVn,Vn=limt!tnV(t)denotingjumpterms,GN(V;W)=NXn=1ZIn(Vt(t);W(t))+A(V(t);W(t))+Zt0B(t;s;V(s);W(t))dsdt+N1Xn=1([V]n;W+n)+(V+0;W+0):(1.3)ForB(t;s)0werecognizethebilinearformusedintheanalysisofthediscontin-uousGalerkinmethodforaparabolicdierentialequation.MultiplicationinHof(1.1)byXandintegrationover(0;tN)showthattheexactsolutionsatisesGN(u;X)=(u0;X+0)+ZtN0(f(t);X(t))dt;8X2WN:ThenumericalapproximationU2WNisnowdenedbyGN(U;X)=(u0;X+0)+ZtN0(f(t);X(t))dt;8X2WN:(1.4)NUMERICALSOLUTIONOFINTEGRO-DIFFERENTIALEQUATIONS3Wenotethatthisisatimesteppingscheme,whichdeterminesUsuccessivelyonInforn=1;:::;N,whenitisknownon[0;tn1),fromZIn(Ut;X)+A(U;X)+Zttn1B(;s;U(s);X)dsdt+(U+n1;X+n1)=(Un1;X+n1)+ZIn(f;X)dtZInZtn10B(;s;U(s);X)dsdt;8X2WN;whereU0=u0.TheuniquenessofUfollowsbyGronwall’slemmaprovidedthatk=maxnknissmallenough