A least square extrapolation method for the a post

ALeastSquareExtrapolationMethodfortheAPosterioriErrorEstimateoftheIncompressibleNavierStokesProblemM.Garbey,andW.ShyyDepartmentofComputerScienceUniversityofHoustonHouston,TX,77204,USA:PartialDifferentialEquations,LeastSquareMethod,Richardsonextrapolation,aposteriorierrorestimate.AbstractAposteriorierrorestimatorsarefundamentaltoolsforprovidingcondenceinthenumericalcomputationofPDEs.Todate,themaintheoriesofaposterioriestimatorshavebeendevelopedlargelyintheniteelementframework,foreitherlinearellipticoperatorsornon-linearPDEsintheabsenceofdisparatelengthscales.Ontheotherhand,thereisastronginterestinusinggridrenementcombinedwithRichardsonextrapolationtoproduceCFDsolutionswithimprovedaccuracyand,therefore,aposteriorierrorestimates.Butinpractice,theeffectiveorderofanumericalmethodoftendependsonspacelocationandisnotuniform,renderingtheRichardsonextrapolationmethodunreliable.Wehaverecentlyintroduced[Garbey13thinternationalconferenceondomaindecompositionandGarbey&ShyyJCP2003]anewmethodwhichestimatestheorderofconvergenceofacomputationasthesolutionofaleastsquareminimizationproblemontheresidual.Thismethod,calledleastsquareextrapolation,introducesaframeworkfacilitatingmulti-levelextrapolation,improvesaccuracyandprovidesaposteriorierrorestimate.Thismethodcanaccommodatedifferentgridarrangements.ThegoalofthispaperistoinvestigatethepowerandlimitsofthismethodviaincompressibleNavierStokesowcomputations.DepartmentofComputerScience,UniversityofHouston,Houston,TX77204,USADepartmentofMechanicalandAerospaceEngineering,UniversityofFlorida,Gainesville,FL32611,USA1ALeastSquareExtrapolationMethodfortheAPosterioriErrorEstimateoftheIncompressibleNavierStokesProblemM.Garbey,andW.ShyyAbstractAposteriorierrorestimatorsarefundamentaltoolsforprovidingcondenceinthenumericalcomputationofPDEs.Todate,themaintheoriesofaposterioriestimatorshavebeendevelopedlargelyintheniteelementframework,foreitherlinearellipticoperatorsornon-linearPDEsintheabsenceofdisparatelengthscales.Ontheotherhand,thereisastronginterestinusinggridrenementcombinedwithRichardsonextrapolationtoproduceCFDsolutionswithimprovedaccuracyand,therefore,aposteriorierrorestimates.Butinpractice,theeffectiveorderofanumericalmethodoftendependsonspacelocationandisnotuniform,renderingtheRichardsonextrapolationmethodunreliable.Wehaverecentlyintroduced[Garbey13thinternationalconferenceondomaindecompositionandGarbey&ShyyJCP2003]anewmethodwhichestimatestheorderofconvergenceofacomputationasthesolutionofaleastsquareminimizationproblemontheresidual.Thismethod,calledleastsquareextrapolation,introducesaframeworkfacilitatingmulti-levelextrapolation,improvesaccuracyandprovidesaposteriorierrorestimate.Thismethodcanaccommodatedifferentgridarrangements.ThegoalofthispaperistoinvestigatethepowerandlimitsofthismethodviaincompressibleNavierStokesowcomputations.IndexTermsPartialDifferentialEquations,LeastSquareMethod,Richardsonextrapolation,aposteriorierrorestimate.I.INTRODUCTIONANDMOTIVATIONRichardsonextrapolation(RE)isasimple,elegantandgeneralmathematicalideathatworksfornumericalquadraturewiththeRombergmethodorODEintegrationsthathavesmoothenoughsolutionwiththeBulirsch-Stoermethod.ItsuseinComputationalFluidDynamics(CFD)[3],[4],[10],[12],[13],[16],[17],[18],[21]islimitedbythefactthatmeshesmightnotbeneenoughtosatisfyaccuratelytheaprioriconvergenceestimatesthatareonlyasymptoticinnature.FurthermoretheorderofconvergenceofaCFDcodeisoftenspacedependentandeventuallyparameters,suchastheReynoldsnumber,dependent.TocopewiththeselimitationsofRE,wehaveintroducedrecently[8],[9]theso-calledLeastSquareExtrapolationmethod(LSE)thatisbasedontheideaofndingautomaticallytheorderofamethodasthesolutionofaleastsquareminimizationproblemontheresidual.OurLSEmethodisbasedonthepost-processingofdataproducedbyexistingPDEcodes.Themethodhasbeendescribedindetailedin[9].Fromapracticalpointofview,wehaveusedatwodimensionalturningpointproblem[14]exhibitingasharptransitionlayeraswellasanitedifferenceapproximationofthecavityowproblemin!formulation[15]toshowthatourmethodismorereliablethanREwhiletheimplementationisstillfairlyeasyandthenumericalprocedureinexpensive.OurobjectiveistouseanyPDEorCFDsolvers,independentoftheirinnerworkingalgorithmandprocedures,providedthattheycanoffertheinformationincludingtheresidualofthenumericalapproximation,stabilityestimates,andvaryinggridresolutionsandnumericalsolutions,toaccomplishthefollowinggoals:(i)aposterioriestimatesofPDEsthataremorereliableandrobustthanstraightforwardRichardsonextrapolation-basedmethodswithlowcostinadditionalCPUtime,(ii)asolutionwithimprovedaccuracy,(iii)arithmeticefciencyofthePDEmultilevelsolutionprocedurebyprovidingagoodstartingpointforiterativesolvers[6],and(iv)adynamicsolutionvericationsoftware.DepartmentofComputerScience,UniversityofHouston,Houston,TX77204,USADepartmentofMechanicalandAerospaceEngineering,UniversityofFlorida,Gainesville,FL32611,USA2Fromtheappliedmathematicspointofview,aposterioriestimatesh