1 Bounds for Linear--Functional Outputs of Coerciv

1BoundsforLinear{FunctionalOutputsofCoercivePartialDierentialEquations:LocalIndicatorsandAdaptiveRenementJ.PeraireaandA.T.PaterabaDepartmentofAeronauticsandAstronautics,MassachusettsInstituteofTechnology,Cambridge,MA02139,USAbDepartmentofMechanicalEngineering,MassachusettsInstituteofTechnology,Cambridge,MA02139,USAWepresentaframeworkfortheecientcalculationoflowerandupperboundstoout-putswhicharelinearfunctionalsofthesolutionstosymmetricornonsymmetricsecond{ordercoercivepartialdierentialequations.ThemethodisbasedupontheconstructionofanaugmentedLagrangian,inwhichtheobjectiveisaquadraticenergyreformulationofthedesiredoutput,andtheconstraintsaretheniteelementequilibriumconditionsandinterelementcontinuityrequirements;theboundsarethenderivedbyevokingthedualmax-minproblemforappropriatelychosencandidateLagrangemultipliers.Theboundcomputationcomprisestwocomponents:severalglobalcalculationsonarelativelycoarse\workingnite{elementtriangulationTHconsistingofKHelementsTH;and2KHin-dependentTH{localcalculationsonarelativelyne\truthnite{elementtriangulationTh.Inthispaperwefocusonthreenewdevelopments.First,weintroduceamodiedenergyobjective,andhencemodiedLagrangian,thatpermitsbothmoretransparentinterpretationandmorereadygeneralization.Second,wedemonstratethattheboundgap|thedierencebetweentheupperandlowerboundsforthedesiredoutput|canberepresentedasthesumofpositivecontributions|localindicators|associatedwiththeelementsTHofTH.Third,basedontheselocalbound{gaperrorindicators,wedevelopadaptivestrategiesbywhichtoreducetheboundgap|andhenceimproveourvalidatedpredictionfortheoutputofinterest|throughoptimalrenementofTH.Theresultingmethodisappliedtoanillustrativeprobleminlinearelasticity.1.INTRODUCTIONTheeldofaposteriorierrorestimationandadaptivemeshrenementnowhasalonghistoryinniteelementanalysis.Thetwogoalsoftheserelatedpursuitsare,rst,inex-pensiveconrmationoftheaccuracyofaparticularniteelementsolution,andsecond,ecientimprovementoftheniteelementsolutionbyoptimaladaptivemeshrenement.Wenowmakethesenotionsmoreprecise.Wedenotetheexactsolutiontoourcoercivepartialdierentialequationbyu(say,thedisplacement),andtheniteelementapproximationtouassociatedwithatriangu-2lationTHbyuH.TheniteelementerroristhusgivenbyeuuH;wedenotethe(pseudo)metricinwhichwewishtomeasuretheniteelementerrorbyE(v).Aposterioriproceduresprovideanestimate,E(TH),fortheE{metricoftheniteelementerror,E(e);thisestimateE(TH)istypicallyexpressedasthesumofpositivecontributionsETHasso-ciatedwiththeKHelementsTHofthetriangulationTH.TheelementalcontributionsETHareinterpretedaslocalindicatorsforthepurposeofsubsequentmeshrenementstrategies.Thegeneralapproachestoaposteriorierrorestimationmaybecategorizedas\ex-plicitor\implicit[3].Explicittechniquesaretypicallybasedonresidualevaluationandaprioriapproximationandstabilityresults[4,18,7]:theadvantageiscomputationaleciency;thedisadvantageisthepresenceofconstantsthatcannotbepreciselyevalu-ated.Implicittechniques[11,6,2]arebasedonthesolutionofKHresidual{forcedTH{localindependentsubproblems:theadvantageismoreprecisequantication;thedisadvantageisincreasedcomplexityandcomputationaleort.Althoughbothexplicitandimplicitmethodstypicallyprovideerrorboundsinthesensethat(roughly)thereexistsacon-stantCEindependentofTHandusuchthatE(e)CEETH[6,18],theconstantCEistypicallynotknown.Inwhatfollows,weshallreservetheterm\boundforestimatorsforwhichCEisknowntohighaccuracy.Mostearlyworkonaposteriorierrorestimationfocused,rst,onsymmetricproblems,andsecond,onthenaturalenergymeasureoftheerror,inwhichE(v)ischosentobethenorminducedbythesymmetricbilinearformassociatedwiththeweakformulationoftheproblem.Inthiscase,veryeectiveexplicitandimplicittechniques[4,6,18]canbedevelopedthatrequireessentiallynoregularityassumptionsandcontainonlymini-malunknownapproximation(andperhapscoercivity)contributionstoCE.Furthermore,implicitprocedurescanbedeveloped[11,2]thatproviderigorousboundsfortheerror;theunknowncontributionstoCEarereducedtothe(typicallyverysmall)inaccuraciesincurredinthesolutionoftheTH{localsubproblems.Manyoftheseexplicitandimplicitmethodscanbereadilygeneralizedtononsymmetricproblems[18]andmoregeneralerrormetrics;incontrast,theearlierboundprocedures[11,2]aredevelopedonlyforcoercivesymmetricproblems,andarefundamentallyrestrictedtothenaturalenergynorm.Therehasrecentlybeengreatlyincreasedinterestintheextensionofaposteriories-timationtechniquestoerrormetricsmoredirectlyrelevanttoengineeringanalysis.Inparticular,thequantity(orquantities)ofinterestinengineeringstudiesisnottheeldvariableu,ortheerrorintheenergynorm,butrathertheoutput|thesystemperfor-mancemetric{thatreectsthespecicgoalsandobjectivesofthedesignoroptimizationexercise.Tobemoreprecise,wedenotethisengineeringoutputofinterestbys(say,theforceoverpartoftheboundary);wefurtherassumethatsmaybeexpressedass=lO(u),wherelO(v)isa(preferablybounded)anefunctional.Theniteelementapproxima-tiontos,sH,isthengivenbysH=lO(uH).Itisclearthat,inordertomeasuretheerrorinthelinear{functionaloutputs,weshouldchoose