A posteriori error estimation of residual type for anisotropic diffusion–convection–reaction problems

Thomas Apel

Description

Niveau: Supérieur, Licence, Bac+2
A posteriori error estimation of residual type for anisotropic diffusion–convection–reaction problems Thomas Apel? Serge Nicaise† May 22, 2009 Abstract: This paper presents an a posteriori residual error estimator for diffusion– convection–reaction problems with anisotropic diffusion, approximated by a SUPG finite element method on isotropic or anisotropic meshes in Rd, d = 2 or 3. The equivalence between the energy norm of the error and the residual error estimator is proved. Numerical tests confirm the theoretical results. Key words: anisotropic diffusion, SUPG, a posteriori error estimate. AMS subject classification: 65N30, 65N15 1 Introduction This paper is devoted to the singularly perturbed diffusion–convection–reaction problem with special focus on anisotropic diffusion: for f ? L2(?) and g ? L2(?N), let u be the solution of ? ? ? ?div (A?u) + b · ?u+ cu = f in ?, u = 0 on ?D, A?u · n = g on ?N , (1) where the matrix A and the functions b and c satisfy assumptions (A1) to (A6) below, and ? ? Rd, d = 2 or 3, is a bounded domain with a polygonal (d = 2) or polyhedral (d = 3) boundary ?. This boundary is divided into two parts ?D and ?N , where Dirichlet and Neumann boundary conditions are imposed, respectively.

• a?u ·


• error estimator


• refined anisotropic meshes


• †universite de valenciennes et du hainaut cambresis


• lower bounds


• within boundary layers


• ?v ?

Sujets

• Saint Nicaise

Informations

• Publié par: profil-vieg-2012
• Nombre de lectures: 14
• Langue: English

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Aposteriorierrorestimationofresidualtypefor anisotropicdiffusion–convection–reactionproblems ThomasApel ∗ SergeNicaise † May22,2009 Abstract:Thispaperpresentsanaposterioriresidualerrorestimatorfordiffusion– convection–reactionproblemswithanisotropicdiffusion,approximatedbyaSUPGfinite elementmethodonisotropicoranisotropicmeshesinR d ,d=2or3.Theequivalence betweentheenergynormoftheerrorandtheresidualerrorestimatorisproved.Numerical testsconfirmthetheoreticalresults. Keywords:anisotropicdiffusion,SUPG,aposteriorierrorestimate. AMSsubjectclassification:65N30,65N15 1Introduction Thispaperisdevotedtothesingularlyperturbeddiffusion–convection–reaction problemwithspecialfocusonanisotropicdiffusion:forf∈L 2 (Ω)andg∈L 2 (Γ N ), letubethesolutionof   −div(Aru)+b∙ru+cu=finΩ, (1)u=0onΓ D ,  Aru∙n=gonΓ N , wherethematrixAandthefunctionsbandcsatisfyassumptions(A1)to(A6) below,andΩ⊂R d ,d=2or3,isaboundeddomainwithapolygonal(d=2)or polyhedral(d=3)boundaryΓ.ThisboundaryisdividedintotwopartsΓ D and Γ N ,whereDirichletandNeumannboundaryconditionsareimposed,respectively. ∗ Institutfu¨rMathematikundBauinformatik,Universita¨tderBundeswehrMu¨nchen,D–85577 Neubiberg,Germany,[e-mail] † Universit´edeValenciennesetduHainautCambre´sis,LAMAV,ISTV,F–59313-Valenciennes Cedex9,France,[e-mail] 1

WeareparticularlyinterestedinthecasewhenAbecomessmallinsomedirec- tion,forinstancethecases  µ ε0 ¶ ε00 A=(d=2),orA=  010  (d=3), 10 100 ε>0.Inthecasewhenεissmallwithrespecttobandc,theproblemissingularly perturbedandthesolutionmaygeneratesharpboundaryorinteriorlayers,where thesolutionofthelimitproblem(correspondingtoε=0)isnotsmoothordoes notsatisfytheboundarycondition.Letusquote[17,18,19]fortheapriorierror analysisintwodimensions.Itisshownthatanisotropicfiniteelementsmustbeused inordertoachieveconvergenceuniformintheperturbationparameterε. Thereisavastamountofliteratureonaposteriorierrorestimation.Forsin- gularlyperturbedproblemswithconvectionwecite[2,8,11,16,21,25,26],where anisotropicfiniteelementmesheswereconsideredin[8,16,21]only.Ananisotropic diffusiontensorisconsideredonlyin[7]. Inthispaperwecombineallthoseingredientsandderivearesidualtypeerror estimator.Weprovethereliabilityandefficiencyofthiserrorestimatorwherethe dependenceonεistraced.Thelowerboundmainlydependsonthelocalmesh PecletnumberPe T :=h min,A,T kA −1/2 bk ∞,T ,thereforetheefficiencyisachievedif Pe T ≤cwhichisalwayssatisfiedintheabsenceofconvection.Thereliabilityis basedontheintroductionofanalignmentmeasureasitwasdonebyKunert[12,13]. Thequantityisoftheorderoneifthemeshiswelladaptedtotheproblem,seethe discussioninSubsection3.3. Letusmentionthat,toourknowledge,noapproachisknownthatleadstotwo- sidesestimatesonanisotropicmesheswithoutanyassumptiononthemesh.The classicalresultsassummarizedin[1,24]areobtainedforisotropicmeshesonly. Thedualweightedresidualmethod,see[5]foranoverview,isappliedin[8,9]on anisotropicmeshes,butthereisnoestimatefrombelow.Themorerecentapproach in[20]isnotyetanalyzedforanisotropicmeshesandtwodifferenterrorestimators areusedfortheupperandlowerbounds.Letusfinallymentiontheapproachby Picasso[21]whoconsidersanisotropicmeshesandprovesreliabilityforanestimator thatdependsonr(u−u h )whereruisreplacedinpracticebyarecoveredgradient r R u.Wenotethatwecancontrolinthesamewaythealignmentmeasure. Inthispaperwedevelopanestimatorofresidualtypeforproblemswithconvec- tion,reactionandanisotropicdiffusion.Forthediscretizationweusetheh-versionof thestreamlineupwindPetrov–Galerkinmethod(SUPG).Withoutthestabilization term,themethodreducestoastandardGalerkinmethodandproducesnon-physical oscillations.Wenotethatourerrorestimatorworksaswellinthiscase. 2