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1、AppliedMathematicsandComputation110(2000)239±250www.elsevier.nl/locate/amcAposteriorierrorestimatorfor®nitevolumemethodsAbdellatifAgouzal*,FabienneOudinU.M.R.5585±Equiped'AnalyseNumeriqueLyonSaint-EtienneUniversiteLyon1,Ba^t.101,69622VilleurbanneCedex,FranceAbstractInthispaper,®r
2、stwediscussatechniquetocompare®nitevolumemethodandsomewell-known®niteelementmethods,namelythedualmixedmethodsandnoncon-formingprimalmethods,forellipticequations.Thesebothequivalencesareexploitedtogiveusaposteriorierrorestimatorfor®nitevolumemethods.Thisestimatorisexplicitlygiven,ea
3、sytocomputeandasymptoticallyexactwithoutanyregularityofthesolutioninunstructuredgrids.Ó2000ElsevierScienceInc.Allrightsreserved.Keywords:Finitevolume;Mixed®niteelement;Nonconforming®niteelement;Aposterioriestimator1.IntroductionInrecentyearsconsiderableinteresthasbeenshowninthedeve
4、lopmentofcomputableaposteriorierrorestimates.Itiswellknownthatapriorierrorestimatescangiveconvergencerateonmeshsize,butcannotprovideactualerrorbounds.Ontheotherhand,aposteriorierrorestimatesattempttoprovidesuchinformation.Alltheworkinaposteriorierrorestimatorisdoneinthecaseof®nitee
5、lementmethods(see[6]andreferencestherein).Inthispaper,weproposeanewaposteriorierrorestimatorfor®nitevolumemethods.Thefactthat®nitevolumemethodsisequivalenttoslightnoncon-formingmethodsplaysanessentialroleinouranalysis.*Correspondingauthor.E-mail:agouzal@iris.univ-lyon1.fr.0096-3003
6、/00/$-seefrontmatterÓ2000ElsevierScienceInc.Allrightsreserved.PII:S0096-3003(99)00118-6240A.Agouzal,F.Oudin/Appl.Math.Comput.110(2000)239±2502.Problemandnotation2LetXdenoteaboundeddomainofR,which,forthesakeofsimplicity,wesupposetobeaconvexpolygon.Weconsiderthesecondorderboundaryval
7、ueproblem:ÿDu‡ruˆfinX;…1†uˆ0onCˆoX;wheref2L2…X†andr2L1…X†withrP0:Now,weconsideraregularsequenceofdecomposition…Th†hofXontotriangles.LetEhdenotethesetofedgesoftrianglesin…Th†h.Weseto0oEhˆfge2EhjeoX;EhˆEhnEh;…2†and121oH0…Th†ˆs2L…X†;8T2Th;sjT2H…T†=8e2Eh;ZZ…3†0sdrˆ0;8e2Eh;eˆT1T2;…s
8、jT1ÿsjT2†drˆ0:eeLetWhbethe
