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COMPUTERMETHODSINAPPLIEDMECHANICSANDENGINEERING45(1984)28.5-312NORTH-HOLLANDFINITEELEMENTMETHODSFORLINEARHYPERBOLICPROBLEMSClaesJOHNSONDepartmentofMathematics,ChalmersUniversityofTechnology,S-41296Giireborg,Sweden..UnoNAVERTFlygdivisionenSaab-Scania.S-58266Linkiiping,SwedenJuhaniPITKARANTADepartmentofMathema~cs,HelsinkiUniz)ers~tyofTechnology,SF-02150Esbo15,~i~lan~~Received6December1982Revisedmanuscriptreceived15March1983Wegiveasurveyofsomerecentworkbytheauthorsonfiniteelementmethodsforconvection-diffusionproblemsandfirst-orderlinearhyperbolicproblems.0.IntroductionThepurposeofthispaperistogiveasurveyofsomerecentworkonfiniteelementmethodsforconvection~iffusionproblemsandfirst-orderlinearhyperbolicproblemsbytheauthors[9-13,171.Thepresentpaperisanelaborationof[lo].Weshallfirstconsiderastationaryscalarlinearconvection-dominatedconvection-diffusionproblemoftheform-shuip*Vu+cru=finft,(0.1)u=gonr,where0isaboundeddomaininRNwithboundaryr,p=(&,...,&)andaaresmoothlyvaryingcoefficientswithIpI-1andEOisasmallconstant.Thesolutionuofthisproblemisingeneralnotgloballysmoothevenforsmoothdatafandg[20];ingeneraluwillvaryrapidlyinalayerofwidthO(E)attheoutflowboundaryr+={xET:n(x)*/3(x)30)wheren(x)istheoutwardunitnormaltoratxEf.Moreover,inthelimitcaseE=0wheretheboundarydatagareonlyprescribedontheinflowboundaryr...={xEr:n(x)*/3(x)0}uwillbediscontinuousacrossthecharacteristiccurve(streamline)x(s)givenbydxfds=p(x),x(O)=x0Er_,ife.g.gisdiscontinuousatx0.IfE0thensuchadiscontinuityisspreadoutoveralayeraroundthecharacteristicx(s)ofwidthO(~E).A‘classical’probleminnumericalanalysisistoconstruct,usingameshwithmeshlengthhnotorientedtofollowthecharacteristics,afinitedifferenceorfiniteelementmethodfor(0.1)that(i)ishigher-orderaccurateand(ii)hasgoodstabilitypropertieswithoutrequiringhtobesmallerthanE.Conventionalschemesfor(0.1)fallintooneofthefollowingtwocategories00457825/84/$3.00@1984,ElsevierSciencePublishersB.V.(North-Holland)2XhC.Johnsonetal.,FEMforhyperbolicproblemseachonesatisfyingonlyoneoftheaboveconditions.Thefirstclassofmethodsconsistsofformallyhigher-orderaccuratemethodssuchase.g.thestandardGalerkinmethod(seebelow)orfinitedifferencemethodsbasedonusingcentereddifferenceapproximationsfortheconvectivetermp.Vu.ThesemethodswillproduceseverelyoscillatingapproximatesolutionsunlesshEortheexactsolutionhappenstobegloballysmooth.Intheotherclasswefindtheclassicalmonotoneupwindschemesobtainedbyaddinganartificialdiffusion(orviscosity)termoftheform-hAu.Thesemethodssatisfy(ii)andproducenon-oscillatingapproximatesolutionsbutareonlyfirst-orderaccuratebeingbasedonsolvingamodifiedproblem.Notethatwedonotassumethatthemeshmatchesthecharacteristicsof(0.1)sothatinparticularthesamemeshmaybeusedfordifferentflowfieldsp.Ifthemeshisorientedtofollowthecharacteristics,wegetmethodswithqualitativelydifferentpropertieswhichmaybeconsideredtobevariantsofthemethodofcharacteristics.WeshallinSection1considerafiniteelementmethodfor(O.l),thestreamlinedijfusionmethodintroducedbyHughesandBrooks[2,5,61whichsatisfiesboththeconditions(i)and(ii)statedabove.ThismethodisaPetrov-GalerkinmodificationofthestandardGalerkinmethodwhereartificialdiffusioninthestreamlinedirectionisintroducedbymodifyingthetestfunctionsfromutou+hp*Vv.Themathematicalanalysisofthismethodwasstartedin[9]andwascarriedfurtherin[171.Themainresultsin[17]concerningthestreamlinediffusionmethodfor(0.1)areroughlyspeakingthefollowingifFhandpiecewisepolynomialsofdegreekareused.Hereuhdenotesthefiniteelementsolution.(A)Globalerrorestimates.(0.2)wherelllwllln=~/rllVwllL*cn,+-\/QP*WlL2u2~+II42V2~andH”(R)forsapositiveintegerdenotestheusualSobolevspacewithnorm11.IIHyRj.WenotethatthisisanimprovementofthecorrespondingestimateforthestandardGalerkinmethodwhichreads(0.3)NoticeinparticularthepresenceofthetermX&II/~.V(u-~~~~~~~~~~~ontheleft-handsideof(0.2),whichmeansthatthestreamlinediffusionmethodhasanimprovedstabilityforthestreamlinederivative/3*VascomparedtothestandardGalerkinmethod.(B)Localizationresults.Theseresultsstatethateffectsarepropagatedinthediscreteproblemapproximatelyasinthecontinuousproblem,i.e.approximatelyalongthecharac-teristics.Morepreciselyitisprovedthattheinfluenceofasourceinthediscreteproblemdecayswiththedistancedtothesourcelikeexp(-Cd/h)_inanydirectionwithapositivecomponentintheupwinddirection-pandlikeexp(-Cd/d\/)indirectionsorthogonaltothestreamlines(crosswinddirections).Alternatively,theseresultscanbephrasedaslocalerrorestimatesoftheformlllu-UhlllR'~Chk+“2tllulI~~+1cn,,,+llfIIw,+kbd(0.3)C.Johnsonetal.,FEMforhyperbolicprobietns287wherea’CW,thedistance_fromapointxEa’to0\0”isO(hlogl/h)indirectionsnotorthogonaltopandO(ghlogl/h)incrosswinddirections.Moreover0”satisfiesthefollowingcondition:iffE0”thenallpointsx=2-$3withs0belongingtoflalsobelongtoKY’,i.e.,allpointsupstreamapointin0”alsobelongtoa”,orinotherwords,a“doesnotallow‘upstreamcut-off’(seeFig.1).NotethatthestandardGalerkinmethodfor(0.1)doesnotallowsuchlocalestimates;infactinthiscaseeffectsmaypropagateincrosswindorevenupwinddirectionswithlittledamping.Prev