High Order Discontinuous Galerkin Method

ÉCOLEPOLYTECHNIQUEFÉDÉRALEDELAUSANNEHighOrderDiscontinuousGalerkinMethodSemesterProjectofBenjaminStammDirectedby:Prof.A.QuarteroniDr.E.BurmanSectionofMathematics,EPFL,LausanneJune24,20041CONTENTS2Contents1Motivation32Outline33TheTransport-ReactionProblem33.1UniquenessandexistenceofthevariationalformulationinL2()..43.2Stability.................................44ThepureTransportProblem54.1Stability.................................55DiscontinuousGalerkinMethod65.1Notations................................65.2Themethod...............................65.3ConvergenceAnalysis.........................106NumericalResults166.1Thecode.................................166.2TestProblem..............................166.2.1h-refinement..........................176.2.2N-refinement..........................186.3Conclusionsofthenumericaltests...................187Conclusion198IntroductiontoLegendre’spolynomials199Introductiontonumericalintegration2010AppendixA211MOTIVATION31MotivationStandardcontinuousGalerkin-basedfiniteelementmethodshavepoorstabilityprop-ertieswhenappliedtotransport-dominatedflowproblems,soexcessivenumericalsta-bilizationisneeded.Incontrast,theDiscontinuousGalerkinmethodisknowntohavegoodstabilitypropertieswhenappliedtofirstorderhyperbolicproblems.2OutlineInthissemesterprojectwewillconsidertheDiscontinuousGalerkinmethod.Insection3,thetransport-reactionproblemispresented.Itmentionsalsothehy-pothesisunderwhichuniquenessandexistenceofthevariationalformulationinL2isguaranteed.Astabilityresultfollows.Insection4,thepuretransportproblemispresentedaswellasastabilityresultforthisproblem.Section5introducestheDiscontinuousGalerkinmethodfollowedbyaconver-genceanalysis.Themainresultofthisprojectistheproofoftheconvergencetheorem.Section6dealswithnumericalresults.IntheframeworkofthissemesterprojectaMatlabcodeisdevelopedforthecomputationofanumericalapproximation.Themethodisappliedtoasimpletestcasewithknownsolution.Finally,theresultsareanalysed.Section7istheconclusionoftheDiscontinuousGalerkinmethod.Insections8and9wegivearudimentaryintroductiontoorthogonalpolynomialsandnumericalintegration.3TheTransport-ReactionProblemInthissection,thetransport-reactionproblemisstudiedwithnonconstantcoefficients.Thefollowingproblemisconsidered:Findu:!Rsuchthat:u+u=fin(1)u=gon(2)whereu=
rudenotesthederivativeinthe-direction.isdefinedby=fx2@:n(x)
0g,wheren(x)istheoutwardnormalunitvectoratthepointx.Analogously+isdefinedby+=fx2@:n(x)
0g.isavectorfieldsuchthat2[W1;1()]d,and2L1(),f2H1(),g2H12().LetbeW=fw2L2():
rw2L2()gL2()withnormkuk2W;=kuk2L2()+k
ruk2L2WisaHilbertspace.3THETRANSPORT-REACTIONPROBLEM43.1UniquenessandexistenceofthevariationalformulationinL2()Multiplying(1)byasmoothtestfunctionv:!Randintegratingonthedomainleadsto:Findu2W(K)Z(u+
ru)v=Zfv8v2L2()LetVbethespaceV=fw2W:wj=ggW(3)Thebilinearforma:WL2()!RandthelinearformF:L2()!Raredefinedbya(u;v)=Z(u+
ru)v8u2W;8v2L2()F(v)=Zvf8v2L2()Then,thevariationalformulationinL2()is:Findu2Vsuchthat:a(u;v)=F(v)8v2L2()(4)Itcanbeshownthatunderthehypothesisthatthereexistsaconstant0suchthat(x)12r
(x)00a.e.in(5)theconditionsoftheNeˇcasTheoremaresatisfied.Thatimpliesthatthereexistsauniquesolutionof(4).Inaddition,condition(5)isalsonecessaryforuniquenessandexistence.Formoredetailssee[1].3.2StabilityInthissection,astabilityresultfortheTransport-ReactionProblemonthewholedo-mainisdeveloped.Lemma3.1(stabilityforthetransport-reactionproblem)Ifitexistsaconstant1suchthat(x)108x2,thenthefollowingstabilityresultisgiven:1kuk2L2()+Z+j
nju211kfk2L2()+Zj
njg2PROOF.Letustakeequation(1),multiplyitbyuandintegrateover.Weget(u;u)+(u;u)=(f;u)Then,thefollowingintegrationbypartsisused(u;u)=(u;u)+(
nu;u)@=(u;u)+(j
nju;u)+(j
njg;g)4THEPURETRANSPORTPROBLEM5sothat(u;u)+12(j
nju;u)+=(f;u)+12(j
njg;g)Now,theCauchy-Schwarzinequalityisappliedto(f;u).Thisleadsto2(u;u)+(j
nju;u)+=2kfkkfk+(j
njg;g)FinallytheYounginequalitywith=121isapplied.Then1kuk2L2()+Z+j
nju211kfk2L2()+Zj
njg24ThepureTransportProblemInthissection,thepureTransportProblemisconsidered:Findu:!Rsuchthatu=finRd(6)u=gon(7)Thenthevariationalformulationofthisproblemis:Findu2Wsuchthat:(u;v)=(f;v)8v2L2()(8)u=gon4.1StabilityLemma4.1(stabilityforthepureTransportProblem)Inthecaseofthepuretrans-portproblem,i.e.if0andunderthehypothesisthatthereexistsavectorfunction2L1(K)
dsuchthatitexistsaconstant1whichsatisfies~
~10then,thefollowingstabilityresultisgiven1ke~
~xuk2L2()+(j
nje~
~xu;e~
~xu)+11ke~
~xfk2L2()+(j
nje~
~xg;e~
~xg)PROOF.Letbe~u(~x)=e~
~xu(~x)andnotethatproblem(6)isequivalenttothefol-lowingproblem:Find~u:!Rsuchthat~
r~u+~u=e~
~xfin~u=e~
~xgonwhere=~
~.Thisisatransport-reactionproblemandthankstothehypothesis,Lemma(3.1)canbeapplied,sothatwegettheresult.5DISCONTINUOUSGALERKINMETHOD65DiscontinuousGalerkinMethod5.1NotationsLetusfirstintroducesomenotations.ConsideranelementK.Kcanbearbitraryasimplexoraparallelepiped,then@Kissplitinto@K=fx2@K:n(x)
0g@K+=fx2@K:n(x)
0gandwehavethat