2005a), ‘On the convergence of a general class of

ONTHECONVERGENCEOFAGENERALCLASSOFFINITEVOLUMEMETHODSHOLGERWENDLANDAbstract.Inthispaperweinvestigatenumericalmethodsforsolvinghyperbolicconservationlawsbasedonnitevolumesandoptimalrecovery.ThesemethodscanforexamplebeappliedincertainENOschemes.Theirapproximationpropertiesdependinparticularonthereconstructionfromcellaverages.Hence,thispaperisdevotedtoproveconvergenceresultsforsuchreconstructionprocessesfromcellaverages.Keywords.optimalrecovery,nitevolumes,positivedenitekernels,approximationordersAMSsubjectclassications.65M2065M1541A301.Introduction.Finitevolumemethodsarewell-establishedtoolsforsolvinghyperbolicconservationlawsoftheform@@tu+dX`=1@@x`f`(u)=0(1.1)numerically.Here,u:Rd[0;1)!Rnisthevector-valuedsolutioncontainingthequantitytobeconservedwhilef`:Rn!Rndenotetheso-calleduxfunctions.Fordiscretizinginspace,nitevolumemethodsusecellaverageinformation.Tobemoreprecise,foraxedtimesuchcellaveragesareemployedtoreconstructtheunknownfunctionuapproximately.Foragoodreconstructioninregionswherethesolutionof(1.1)isknownorexpectedtobesmoothahigherorderreconstructionschemeisdesirable.Hence,suchhighorderschemescurrentlyformamajorresearchdirectioninthetheoryofnitevolumes.Thersthigherorderreconstructionschemesemployed,werebasedonpolyno-mialsandsueredfromthetypicalbehaviorofmultivariatepolynomials,suchasoscillationandill-conditioning.Inaseriesofpapers[4,9,10,11],Sonarproposedtoemployoptimalrecoverybasedonconditionallypositivedenitekernelsinstead.Hisnumericalexamplesindi-catethattheserecoveryprocessesindeedleadtohigherorderschemes.Nonetheless,uptonowtherehasnomathematicalproofbeengivenforthisobservation.In[11],heconcludedwith\[...]nearlynothingisknownaboutapproximationordersinthecaseofrecoveryfromcellaveragedata.[...]Atthemoment,however,wearefacedwiththefactthatimportanttheoreticalresultsaremissinginthisareaofresearch.\Itisthegoalofthispapertollthistheoreticalgapandtoshowthattherecoveryprocesscanleadtoarbitraryhighorders,providedthetargetfunctionuissucientlysmoothandthecorrect(conditionally)positivedenitekernelisemployed.However,sinceouranalysisisbaseduponapproximationpropertiesofpolynomi-als,ourproofwillneedslightlylargerstencilsthanthoseproposedbySonar.Ontheotherhand,sincethe\correctselectionofstencilsisstillunderinvestigation,ourresultsmightalsocontributetothisproblem.Moreover,theresultswewillachievearenotrestrictedto(conditionally)positivedenitekernelsatall.Onthecontrary,theywillworkforeveryinterpolatoryandInstitutfurNumerischeundAngewandteMathematik,UniversitatGottingen,Lotzestr.16-18,D-37083Gottingen,[e-mail],ourresultsareestablishedforanarbitraryspacedimension.Thispaperisorganizedasfollows.Intherestofthesectionwewillintroducesomegeneralnotationswewillneedtostateourconvergenceresults.ThenextsectionisdevotedtoashortreviewonnitevolumeandENO(essentiallynonoscillatory)schemes.Thethirdsectiondescribeshowsuchschemescanbederivedusingoptimalrecovery.Thefourthsectionisthemainsectionwhereweprovideourerroranalysis.Inthenalsectionwetakeaspeciallookatthin-platesplineapproximation,whichisoneofthemostpopularreconstructionmethodsinthiscontext.FornumericalexampleswereferthereadertothepreviouslymentionedpapersbySonar.WewillestablishourerrorestimatesusingavarietyofSobolevspaces,whichwewanttointroducenow.LetRdbeadomain.Fork2N0,and1p1,wedenetheSobolevspacesWkp()toconsistofalluwithdistributionalderivativesDu2Lp(),jjk.Associatedwiththesespacesarethe(semi-)normsjujWkp()=0@Xjj=kkDukpLp()1A1=pandkukWkp()=0@XjjkkDukpLp()1A1=p:Thecasep=1isdenedintheobviousway:jujWkp()=supjj=kkDukL1()andkukWk1()=supjjkkDukL1()WewillalsobedealingwithfractionalorderSobolevspaces.Let1p1,k2N0,and0s1.WedenethefractionalorderSobolevspacesWk+sp()tobealluforwhichthefollowing(semi-)normsarenite:jujWk+sp():=0@Xjj=kZZjDu(x)Du(y)jpkxykd+ps2dxdy1A1=p;kukWk+sp():=kukpWkp()+jujpWk+sp()1=p:2.FiniteVolumeandENOSchemes.Finitevolumeschemesintroduceweaksolutionsto(1.1)inthefollowingsense.IfVRdisanarbitrarycompact,smallre-gion,calledthecontrolvolume,thenuhastosatisfytheweakformoftheconservationlaw(1.1)intheformddtZVu(x;t)dx=Z@VdX`=1f`(u(x;t))`(x)dS;(2.1)where(x)[e-mail](1.1)oftendirectlyresultsfromthephysicalconservationlawandistheninacertainsenseevenmorenaturalthan(1.1).Toconvert(2.1)intoanumericalprocedure,theregionRdofinterestissubdividedintonon-overlappingsubregionsTh=fVjg,i.e.=N[j=1Vj;ConvergenceofFiniteVolumeMethods3wheretheVjaresimpliceshavingsizeO(h).Then,(2.1)canobviouslyberewrittenusingthecellaveragesj(u)(t):=uj(t)=1jVjjZVju(x;t)dx;1jN:Moreover,ifNjdenotesthesetoftheneighboringsimplicestothesimplexVj2Th,wehave:ddtj(u)(t)=1jVjjXV2NjZ@V\@VjdX`=1f`(u)(V)`dS;where(V)denotestheouterunitnormalvectortotheboundaryface@V\@VjofV.IftheuxisreplacedbyanumericaluxfunctionoranapproximateRiemannsolverH:RnRnRd!Rn,satisfyingH(u;u;)=dX`=1f`(u)`;andiftheintegrationontheboundaryhyperplane@V\@Vjisreplacedby