A Posteriori Error Analysis and Adaptive Methods f

APosterioriErrorAnalysisandAdaptiveMethodsforPartialDifferentialEquationsZhimingChen∗Abstract.Theadaptivefiniteelementmethodbasedonaposteriorierrorestimatesprovidesasystematicwaytorefineorcoarsenthemeshesaccordingtothelocalaposte-riorierrorestimatorontheelements.Oneoftheremarkablepropertiesofthemethodisthatforappropriatelydesignedadaptivefiniteelementprocedures,themeshesandtheassociatednumericalcomplexityarequasi-optimalinthesensethatthefiniteelementdiscretizationerrorisproportionaltoN−1/2intermsoftheenergynorm,whereNisthenumberofdegreesoffreedomoftheunderlyingmesh.Thepurposeofthispaperistoreportsomeoftherecentadvancesintheaposteriorierroranalysisandadaptivefiniteelementmethodsforpartialdifferentialequations.EmphaseswillbepaidonanadaptiveperfectlymatchedlayertechniqueforscatteringproblemsandasharpL1aposteriorierroranalysisfornonlinearconvection-diffusionproblems.MathematicsSubjectClassification(2000).Primary65N15;Secondary65N30.Keywords.Aposteriorierrorestimates,adaptivity,quasi-optimality1.IntroductionTheaimoftheadaptivefiniteelementmethod(AFEM)forsolvingpartialdiffer-entialequationsistofindthefiniteelementsolutionandthecorrespondingmeshwithleastpossiblenumberofelementswithrespecttodiscreteerrors.Thetasktofindthemeshwiththedesiredpropertyishighlynontrivialbecausethesolutionisaprioriunknown.Thebasicideaoftheseminalwork[3]istofindthedesiredmeshundertheprincipleoferrorequidistribution,thatis,thediscretizationerrorshouldbeapproximatelyequaloneachelement.Theerrorontheelementswhichisalsounknowncan,however,canbeestimatedbyaposteriorierrorestimates.TodayAFEMbasedonaposteriorierrorestimatesattractsincreasinginterestsandbecomesoneofthecentralthemesofscientificcomputation.ThepurposeofthispaperistoreportsomeoftherecentadvancesintheaposteriorierroranalysisandAFEMforpartialdifferentialequations.Aposteriorierrorestimatesarecomputablequantitiesintermsofthediscretesolutionanddata,whichprovideinformationforadaptivemeshrefinement(andcoarsening),errorcontrol,andequidistributionofthecomputationaleffort.We∗TheauthorisgratefultothesupportofChinaNationalBasicResearchProgramunderthegrant2005CB321701andtheChinaNSFunderthegrant1002510and10428105.2ZhimingChennowdescribebrieflythebasicideaofAFEMusingtheexampleofsolvingthePossionequationonapolygonaldomainΩinR2−Δu=finΩ,u=0on∂Ω.(1.1)HerethesourcefunctionfisassumedtobeinL2(Ω).Itiswell-knownthatthesolutionoftheproblem(1.1)maybesingularduetothereentrantcornersofthedomaininwhichcasethestandardfiniteelementmethodswithuniformmeshesarenotefficient.LetMhbearegulartriangulationofthedomainΩandBhbethecollectionofallinter-elementsidesofMh.DenotebyuhthepiecewiselinearconformingfiniteelementsolutionoverMh.Foranyinter-elementsidee∈Bh,letΩebethecollectionoftwoelementssharingeanddefinethelocalerrorindicatorηeasη2e:=K∈ΩehKf2L2(K)+h1/2eJe2L2(e),wherehK:=diam(K),he:=diam(e),andJe:=[[∇uh]]e·νstandsforthejumpoffluxacrosssideewhichisindependentoftheorientationoftheunitnormalνtoe.Thefollowingaposteriorierrorestimateiswell-known[2]u−uh2H1(Ω)≤Ce∈Bhη2e.Thatηereallyindicatestheerrorisexplainedbythefollowinglowerbound[39].η2e≤CK∈Ωeu−uh2L2(K)+CK∈ΩehK(f−fK)2L2(K),wherefK=1|K|Kfdx.Basedonthelocalerrorindicator,theusualadaptivealgorithmsolvingtheellipticproblem(1.1)readsasfollowsSolve→Estimate→Refine.Theimportantconvergenceproperty,whichguaranteestheiterativelooptermi-natesinfinitestepsstartingfromaninitialcoarsemesh,isprovedin[23,30].Itisalsoobserved(cf.e.g.[30])thatforappropriatelydesignedadaptivefiniteelementprocedures,themeshesandtheassociatednumericalcomplexityarequasi-optimalinthesensethat∇(u−uh)L2(Ω)≈CN−1/2(1.2)isvalidasymptotically,whereNisthenumberofelementsoftheunderlyingfiniteelementmesh.Sincethenonlinearapproximationtheory[5]indicatesthatN−1/2isthehighestattainableconvergenceorderforapproximatingfunctionsinH1(Ω)intwospacedimensionsoverameshwithNelements,oneconcludesthatAFEMisanoptimaldiscretizationmethodforsolvingtheellipticproblem(1.1).AdaptiveFiniteElementMethods3Insection2weconsidertouseAFEMtosolvetheHelmholtz-typescatteringproblemswithperfectlyconductingboundaryΔu+k2u=0inR2\¯D,(1.3a)∂u∂n=−gonΓD,(1.3b)√r∂u∂r−iku→0asr=|x|→∞.(1.3c)HereD⊂R2isaboundeddomainwithLipschitzboundaryΓD,g∈H−1/2(ΓD)isdeterminedbytheincomingwave,andnistheunitouternormaltoΓD.Weas-sumethewavenumberk∈Risaconstant.Westudyanadaptiveperfectlymatchedlayer(APML)techniquetodealwiththeSommerfeldradiationcondition(1.3c)inwhichthePMLparameterssuchasthethicknessofthelayerandthefictitiousmediumpropertyaredeterminedthroughsharpaposteriorierrorestimates.TheAPMLtechniquecombinedwithAFEMprovidesacompletenumericalmethodforsolvingthescatteringproblemintheframeworkoffiniteelementwhichhasthenicepropertythatthetotalcomputationalcostsareinsensitivetothethicknessofthePMLabsorbinglayers.Thequasi-optimalityofunderlyingFEMmeshesisalsoobserved.ThingsbecomemuchmorecomplicatedwhenapplyingAFEMtosolvetime-dependentpartialdifferentialequations.Oneimportantquestionisifoneshouldusetheadaptivetimemarching(ATM)methodinwhichvariabletimestepsizes(butconstantateachtimestep)andvariablespac