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A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods

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Niveau: Supérieur, Doctorat, Bac+8
A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods Emmanuel Creuse?, Serge Nicaise† October 7, 2009 Abstract We consider some (anisotropic and piecewise constant) diffusion problems in do- mains of R2, approximated by a discontinuous Galerkin method with polynomials of any degree. We propose an a posteriori error estimator based on gradient recovery by averaging. It is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability. The lower bound is also established. Moreover these additional terms are negligible when the recovered gradient is superconvergent. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests. Key Words A posteriori estimator, Discontinuous Galerkin finite elements. AMS (MOS) subject classification 65N30; 65N15, 65N50, 1 Introduction Among other methods, the finite element method is one of the more popular that is com- monly used in the numerical realization of different problems appearing in engineering applications, like the Laplace equation, the Lame system, the Stokes system, the Maxwell system, etc.... (see [7, 8, 23]). More recently discontinuous Galerkin finite element methods become very attractive since they present some advantages, like flexibility, adaptivity, etc... In our days a vast literature exists on the subject, we refer to [3, 10] and the references cited there.

  • constant ?

  • approximated problem

  • wise constant

  • discontinuous galerkin

  • diffusion coefficients approximated

  • †universite de valenciennes et du hainaut cambresis

  • positive integer

  • galerkin method


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AposteriorierrorestimatorbasedongradientrecoverybyaveragingfordiscontinuousGalerkinmethodsEmmanuelCreuse´,SergeNicaiseOctober7,2009AbstractWeconsidersome(anisotropicandpiecewiseconstant)diffusionproblemsindo-mainsofR2,approximatedbyadiscontinuousGalerkinmethodwithpolynomialsofanydegree.Weproposeanaposteriorierrorestimatorbasedongradientrecoverybyaveraging.Itisshownthatthisestimatorgivesrisetoanupperboundwheretheconstantisoneuptosomeadditionaltermsthatguaranteereliability.Thelowerboundisalsoestablished.Moreovertheseadditionaltermsarenegligiblewhentherecoveredgradientissuperconvergent.Thereliabilityandefficiencyoftheproposedestimatorisconfirmedbysomenumericaltests.KeyWordsAposterioriestimator,DiscontinuousGalerkinfiniteelements.AMS(MOS)subjectclassification65N30;65N15,65N50,1IntroductionAmongothermethods,thefiniteelementmethodisoneofthemorepopularthatiscom-monlyusedinthenumericalrealizationofdifferentproblemsappearinginengineeringapplications,liketheLaplaceequation,theLame´system,theStokessystem,theMaxwellsystem,etc....(see[7,8,23]).MorerecentlydiscontinuousGalerkinfiniteelementmethodsbecomeveryattractivesincetheypresentsomeadvantages,likeflexibility,adaptivity,etc...Inourdaysavastliteratureexistsonthesubject,wereferto[3,10]andthereferencescitedthere.Adaptivetechniquesbasedonaposteriorierrorestimatorshavebecomeindispens-abletoolsforsuchmethods.ForcontinuousGalerkinfiniteelementmethods,therenowUniversite´desSciencesetTechnologiesdeLille,LaboratoirePaulPainleve´UMR8524,EPISIMPAF-INRIALilleNordEurope,UFRdeMathe´matiquesPuresetApplique´es,Cite´Scientifique,59655Villeneuved’AscqCedexemail:creuse@math.univ-lille1.frUniversite´deValenciennesetduHainautCambre´sis,LAMAV,FRCNRS2956,InstitutdesSci-encesetTechniquesdeValenciennes,F-59313-ValenciennesCedex9France,email:Serge.Nicaise@univ-valenciennes.fr1
existsavastamountofliteratureonaposteriorierrorestimationforproblemsinmechanicsorelectromagnetismandobtaininglocallydefinedaposteriorierrorestimates.Werefertothemonographs[2,4,24,31]foragoodoverviewonthistopic.Ontheotherhandasimilartheoryfordiscontinuousmethodsislessdeveloped,letusquote[5,12,17,18,19,27,30].Usuallyupperandlowerboundsareprovedinordertoguaranteethereliabilityandtheefficiencyoftheproposedestimator.Mostoftheexistingapproachesinvolveconstantsdependingontheshaperegularityoftheelementsand/orofthejumpsinthecoefficients;butthesedependencesareoftennotgiven.Onlyafewnumberofapproachesgivesrisetoestimateswithexplicitconstants,letusquote[2,6,20,22,25,26,16]forcontinuousmethods.Fordiscontinuousmethods,wemaycitetherecentpapers[1,21,9,14,15].Ourgoalisthereforetoconsidersecondorderellipticoperatorswithdiscontinuousdiffusioncoefficientsintwo-dimensionaldomainswithmixedboundaryconditionsandadiscontinuousGalerkinmethodwithpolynomialsofanydegree.Inspiredfromthepaper[16],whichtreatsthecaseofcontinuousdiffusioncoefficientsapproximatedbyacontinuousGalerkinmethod,wefurtherderiveanaposterioriestimatorwithanexplicitconstantintheupperbound(moreprecisely1)uptosomeadditionaltermsthatareusuallysupercon-vergentandsomeoscillatingterms.Theapproach,calledgradientrecoverybyaveraging[16]isbasedontheconstructionofaZienkiewicz/Zhuestimator,namelythedifferenceinanappropriatenormofarhuhGuh,whererhuhisthebrokengradientofuhandGuhisaH(div)-conformingapproximationofthisvariable.Herespecialattentionhastobepaidduetotheassumptionthatamaybediscontinuous.MoreoverthenonconformingpartoftheerrorismanagedusingaHelmholtzdecompositionoftheerrorandastandardOswaldinterpolationoperator[19,1].Furthermoreusingstandardinverseinequalities,weshowthatourestimatorislocallyefficient.Twointerestsofthisapproacharefirstthesimplic-ityoftheconstructionofGuh,andsecondlyitssuperconvergenceproperty(validatedbynumericaltests).Thescheduleofthepaperisasfollows:Werecallinsection2thediffusionproblem,itsnumericalapproximationandanappropriateHelmholtzdecompositionoftheerror.Section3isdevotedtotheintroductionoftheestimatorbasedongradientaveragingandtheproofsoftheupperandlowerbounds.Theupperbounddirectlyfollowsfromtheconstructionoftheestimatorandsomeresultsfrom[16],whilethelowerboundrequirestheuseofsomeinverseinequalitiesandaspecialconstructionofGuh.Finallyinsection4somenumericaltestsarepresentedthatconfirmthereliabilityandefficiencyofourestimatorandthesuperconvergenceofGuhtoaru.Letusfinishthisintroductionwithsomenotationusedintheremainderofthepaper:OnD,theL2(D)-normwillbedenotedbyk∙kD.InthecaseD=Ω,wewilldroptheindexΩ.Theusualnormandsemi-normofHs(D)(s0)aredenotedbyk∙ks,Dand|∙|s,D,respectively.Finally,thenotationa.bandabmeanstheexistenceofpositiveconstantsC1andC2,whichareindependentofthemeshsize,ofthequantitiesaandbunderconsiderationandofthecoefficientsoftheoperatorssuchthata.C2bandC1b.a.C2b,respectively.Inotherwords,theconstantsmaydependontheaspectratioofthemeshaswellasthepolynomialdegreel(seebelow).2