78Analysis of the Cell Vertex Finite Volume Method for the Cauchy-Riemann Equations-第2页
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78Analysis of the Cell Vertex Finite Volume Method for the Cauchy-Riemann Equations-2
1394；¨K.W.MORTON,MARTINSTYNES；Usingtheelementaryidenti；δx(bc)=μx(bc)=；(μxb)(δxc)+(δxb)(μxc),；(μxb)(μxc)+(δxb)(δxc)；andtheirδx,μyanalogues,w；Bc(w,GPw)=；?1N?1N????i=1j=1；N?1N?1；1????
1394¨K.W.MORTON,MARTINSTYNES,ANDENDRESULIUsingtheelementaryidentitiesδx(bc)=μx(bc)=(μxb)(δxc)+(δxb)(μxc),1(μxb)(μxc)+(δxb)(δxc)4andtheirδx,μyanalogues,wecanrewrite(3.2)asBc(w,GPw)=h?1N?1N????i=1j=1N?1N?11????hGij(μxμywij)(δxδywij)[μxδy(a1)ij+δxμy(a2)ij]+4i=1j=1hGij(μxμywij)2[δxμy(a1)ij+μxδy(a2)ij]+?1N?1N????i=1j=11hGij(μxμywij)(δxμywij)[μxμy(a1)ij+δxδy(a2)ij]41hGij(μxμywij)(μxδywij)[δxδy(a1)ij+μxμy(a2)ij]4+(3.3)Wede?ne?1N?1N????i=1j=1≡S1+S2+S3+S4.G(x,y)=e?(κ1xi?1+κ2yj?1),for(x,y)∈Kij,whereκl,l=1,2,arepositiveconstantswhichwillbechosenappropriatelyinthecourseoftheproof.FirstweboundS1frombelow.Observingthat(3.4)itfollowsthatS1(3.5)≥N?1N?11????2ij?hG(μxwij)2|Aij|2i=1j=1N?1N?21????2i,j+1hG(μxwij)2|Ai,j+1|,?2i=1j=1(μxμywij)2≤1[(μxwij)2+(μxwi,j?1)2],2withasimilarboundintermsof(μywij)2,where(3.6)Aij=1(δxμy(a1)ij+μxδy(a2)ij).hNotingthatGij≥Gi,j+1andseparatingoutthetermwithj=N?1fromthe?rstdoublesummation,(3.7)(3.8)S1≥N?1N?21????2ij?hG(μxwij)2(|Aij|+|Ai,j+1|)2i=1j=1N?11??2i,N?1hG(μxwi,N?1)2|Ai,N?1|.?2i=1CELLCVERTEXFINITEVOLUMEMETHOD1395Now(μxbij)(δxbij)=(1/2)δxb2weusethisidentityandsumbypartstogetS3(3.9)where1Bij=h(μxμy(a1)ij+δxδy(a2)ij).4LetuswriteGijBij?Gi+1,jBi+1,j=(Gij?Gi+1,j)Bi+1,j+Gij(Bij?Bi+1,j).Recallingthede?nitionofGij,itfollowsthat1κ1hGij,2ˉprovidedh≤1/κ1.Inaddition,sincea1,a2∈C1(?),Gij?Gi+1,j≥Bij=h(a1)ij+O(h2)=h(a1)i?1,j+O(h2).Thus,for0&h≤h0,whereh0=h0(??a),wehaveBij≥1hα1.2=N?2N?11????(μywij)2(GijBij?Gi+1,jBi+1,j)2i=1j=1N?11??(μywN?1,j)2GN?1,jBN?1,j,+2j=1Similarly,for0&h≤h0,whereh0=h0(??a)(withapossibleadjustmentoftheprevioush0),|Bij?Bi+1,j|≤2h2?????a??L∞(?).Consequently,GijBij?Gi+1,jBi+1,j≥h2GijReturningtoS3,wededucethatS3≥????N?2N?11????2ij1κ1α1?2?????hG(μywij)2a??L∞(?)2i=1j=14N?11??hGN?1,j(μywN?1,j)2α1.+4j=1????1κ1α1?2?????a??L∞(?).4Analogously,S4≥????N?1N?21????2ij1κ2α2?2?????hG(μxwij)2a??L∞(?)2i=1j=14N?11??hGi,N?1(μxwi,N?1)2α2.+4i=11396¨K.W.MORTON,MARTINSTYNES,ANDENDRESULINowcombiningthelowerboundsforS1andS4weobtain1S1+S42≥?2N?1N????i=1j=1hG(μxwij)2ij2??11κ2α2??????a??L∞(?)?|Aij+Ai,j+1|84??N?11??hGi,N?1(μxwi,N?1)2(α2?h|Ai,N?1|).+4i=1Notingthat|Ai,j(±1)|≤2?????a??L∞(?)for0&h≤h0,whereh0=h0(??a)(withapossibleadjustmentoftheprevioush0),itfollowsthatα2?h|Ai,N?1|≥Choosingκ2suchthatκ2≥??8??1+2?????a??L∞(?),α21α2.2itfollowsthatfor0&h≤h0,whereh0dependsonlyon??a,?2N?1N?1N??????112ij2S1+S4≥hG(μxwij)+α2hGi,N?1(μxwi,N?1)2.28i=1j=1i=1Similarly,choosingκ1suchthatκ1≥??8??1+2?????a??L∞(?)α1andusingthealternativeboundforS1,wehavethat?1N?1N?2N??????112ij2S1+S3≥hG(μywij)+α1hGN?1,j(μywN?1,j)2.28i=1j=1j=1Finally,S1+S3+S4≥?1N?2N????i=1j=1hG(μywij)+2ij2?2N?1N????i=1j=1h2Gij(μxwij)2N?1N?1????11N?1,j2+α1hG(μywN?1,j)+α2hGi,N?1(μxwi,N?1)2,88j=1i=1a),andκi,i=1,2,arechosenasindicatedabove.Insertingthisprovidedh≤h0(??lowerboundinto(3.3)andrecallingthatdueto(3.1)thetermS2=0,wededucethat???1?2N?2NN?1N????????h2(μywij)2+h2(μxwij)2?Bc(w,GPhw)≥C2?(3.10)+?1?α18i=1j=1N?1??j=1i=1j=1N?1??i=1?hGN?1,j(μywN?1,j)2+α2hGi,N?1(μxwi,N?1)2?CELLCVERTEXFINITEVOLUMEMETHOD1397forallw∈Uhandforallh≤h0(??a).Tocompletetheproofofthelemmaweboundfrombelowtheright-handsideofthisinequalityintermsof??w??l2(?h).Thisiseasilyaccomplishedbyde?ning??1μwij=wdxdy,hKijandnotingthat,forw∈Uh,??w??2l2(?h)=?1N?1N????i=1j=1h2(μwij)2,μwij=andμwij=Since(μwij)2≤andw=0on??,itfollowsthatC2?1N?1N????i=1j=11(μxwij+μxwi,j?1),21(μywij+μywi?1,j).211(μxwij)2+(μxwi,j?1)2,22h|μwij|≤C222?2N?1N????i=1j=1h(μxwij)+C2h22N?1??i=1h(μxwi,N?1)2.Similarly,C2?1N?1N????i=1j=1h|μwij|≤C222?1N?2N????i=1j=1h(μywij)+C2h22N?1??j=1h(μywN?1,j)2.Substitutingthesumofthesetwoinequalitiesinto(3.10),wededucethedesireda).coercivityofthebilinearformBc(?,?)forallw∈Uhandforallh≤h0(??Wenotethatcondition(3.1)wasnecessaryinordertoremovethetermS2thatcontainedthesecond-di?erenceδxδy;thistermcannotbeabsorbedintoanyofthepositivetermsinthelowerboundonBc(w,GPhw).Lemma3.2.Forallw∈Uhandallλ∈Mh,λ≥0,Bc(w,λPh??Ih(??aw))=|λ1/2??Ih(??aw)|2l2(?h).Proof.Thisisimmediatefrom(2.1).Lemma3.3.AssumethatthereexistpositiveconstantsC2andC5suchthatGij≥C2,|Gij?Gi?1,j|≤C5h,and|Gij?Gi,j?1|≤C5hforalliandj.Thenthere1398¨K.W.MORTON,MARTINSTYNES,ANDENDRESULIexistpositiveconstantsC6=C6(C2,C5)andh1=h1(C2,C5),suchthat??1NN????1hh2(?μy(wx)i?1,j)2Bd(w,GPw)≥C2ε?8i=1j=1?NN?1????+h2(?μx(wy)i,j?1)2??C6ε|w|2l2(?h)i=1j=1??ε?hGi,N?1|μxwi,N?1|2+8hi=1N?1??N?1??j=1?hGN?1,j|μywN?1,i|2?forallw∈Uhandallh≤h1.Proof.Wegivedetailsonlyfortheμ?yterms,postponingtheanalogouscontributionfromtheμ?xtermsof(2.3)untillaterintheproof.Thuswewrite,foranyw∈Uh,Bd(w,GPw)=?εh?1N?1N????i=1j=1N?1??j=1hGij(μxμywij)[?μy(wx)ij?μ?y(wx)i?1,j]+(wyterms)(3.11)=εh??N?1??i=2μ?y(wx)i?1,j[Gijμxμywij?Gi?1,jμxμywi?1,j]???μ?y(wx)N?1,jGN?1,jμxμywN?1,j+μ?y(wx)0,jG1,jμxμyw1,jNowfor1≤i≤Nwehavethat+(wyterms).?y(wx)i?1,j+(Gij?Gi?1,j)μxμywi?1,j.Gijμxμywij?Gi?1,jμxμywi?1,j=hGijμTherefore,using|Gij?Gi?1,j|≤C5htogetherwiththearithmetic-geometricmeaninequality,wegetμ?y(wx)i?1,j(Gijμxμywij?Gi?1,jμxμywi?1,j)????μy(wx)i?1,j)2?C5|μ?y(wx)i?1,jμxμywi?1,j|≥hGij(???1??2μy(wx)i?1,j)2?C5(Gij)?1(μxμywi?1,j)2.≥hGij(?2=h(?μy(wx)N?1,j)2?μ?y(wx)N?1,jμxμywN,jh1(?μy(wx)N?1,j)2?(μywN?1,j)2,≥28hhN?1,jG(?μy(wx)N?1,j)221?GN?1,j(μywN?1,j)2.8h(3.12)Ontheotherhand,?μ?y(wx)N?1,jμxμywN?1,jandtherefore?μ?y(wx)N?1,jGN?1,jμxμywN?1,j(3.13)≥包含各类专业文献、应用写作文书、文学作品欣赏、生活休闲娱乐、外语学习资料、高等教育、专业论文、78Analysis 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78Analysis of the Cell Vertex Finite Volume Method for the Cauchy-Riemann Equations-4
1404¨K.W.MORTON,MARTINST；Proof.(ofTheorem3.7)Letu；WebeginbyconsideringT1.F；ij≡χij(1)+χ(2),1≤i,j≤N?1；For2≤i≤N?1and1≤j≤N?1,asi；?2r?1|χijh|u|Hr(Tij),(1)；whereTij=(xi?2,xi+1)×(yj；??1/2
1404¨K.W.MORTON,MARTINSTYNES,ANDENDRESULIProof.(ofTheorem3.7)Letuslabelthethreetermsontheright-handsideoftheinequalityinTheorem3.6byT1,T2andT3.WebeginbyconsideringT1.Forthesakeofnotationalsimplicity,wede?ne,asintheproofofTheorem3.6,??yj??yj111μy(uIχij=[?ux(xi,y)dy?μ?y(uIux(xi?1,y)dy]x)ij?x)i?1,j+hhyj?1hyj?1????1xi1xi1IIμx(uy)ij?uy(x,yj)dx?μ?x(uy)i,j?1+uy(x,yj?1)dx]+[?hhxi?1hxi?1ij≡χij(1)+χ(2),1≤i,j≤N?1.For2≤i≤N?1and1≤j≤N?1,asimpleapplicationoftheBramble-Hilbertlemmayields?2r?1|χijh|u|Hr(Tij),(1)|≤Ch2&r≤min(s,4),whereTij=(xi?2,xi+1)×(yj?1,yj).Consequently,for2≤i≤N?1and1≤j≤N?1,??1/2N?1N?1????2??h2|χij≤Chr?2|u|Hr(?),2&r≤min(s,4).(1)|i=2j=1Nowletusconsiderthecasewheni=1and1≤j≤N?1;recallingthede?nitionofμ?y(uIx)0,jandappealingtotheBramble-Hilbertlemma,wededucethat???1/2?1/2N?1N?1????j2?212t?2???h2|χ1|≤Chh|u|2Ht((x0,x2)×(yj?1,yj)(1)hj=1j=1≤Cht?2|u|Ht(?0),2&t≤min(s,3),where?0=(x0,x2)×(y0,yN?1).Combiningthesetwoboundsweget??1/2N?1N?1????2??h2|χij≤C(hr?2|u|Hr(?)+ht?2|u|Ht(?0)),(1)|i=1j=1with2&r≤min(s,4)and2&t≤min(t,3).ExploitingtheboundarylayerestimatestatedinProposition3.1,|u|Ht(?0)≤Ch1/2??u??Ht+1(?h).Thus,?ε??1N?1N????i=1j=1?1/22?h2|χij(1)|≤Cε(hr?1|u|Hr+1(?)+ht?2h1/2??u??Ht+1(?h)),with1&r≤min(s,3),2&t≤min(s,3).Similarly,?1/2?N?1N?1????2??h2|χij≤Cε(hr?1|u|Hr+1(?)+ht?2h1/2??u??Ht+1(?h)),ε(2)|i=1j=1CELLCVERTEXFINITEVOLUMEMETHOD1405with1&r≤min(s,3),2&t≤min(s,3).Thus,recallingfromthestatementofTheorem3.5thatε≤Ch,itfollowsthatT1≤C1(εhr?1|u|Hr+1(?)+ε1/2ht?1??u??Ht+1(?h)),for1&r≤min(s,3),2&t≤min(s,3).??=??weobtaintheboundTermT2isestimatedusingProposition3.1withdT2≤C2hr?1|u|Hr(?h),1&r≤min(s,3).Finally,thetermT3canbeboundedusingastandardinterpolationerrorestimatetoobtainT3≤C3hr|u|Hr(?h),1&r≤min(s,2)=2.CombiningtheboundsonT1,T2andT3yieldsthedesirederrorestimate.4.ConclusionsInthispaperwehavebeenconcernedwiththestabilityandtheconvergenceofacell-vertex?nitevolumemethodforlinearellipticconvection-dominateddi?usionequationsintheplane.Usingacombinationoftechniquesfromthetheoryof?nitedi?erenceand?niteelementmethodsweprovedthattheschemeisstable,uniformlyasthedi?usioncoe?cienttendstozero,andsecond-orderconvergent.Inadditiontotheerrorboundinthemesh-dependentl2-norm,Theorem3.7implies1(?),thederivativeoftheglobalerrorinthestream-that,providedu∈H4(?)∩H0√wisedirectionisO(h3/2),aslongash≥2.Theresultspresentedheremaybeextendedtotensor-productnon-uniformmeshes.References1.P.BallandandE.S¨uli,Analysisofthecellvertex?nitevolumemethodforhyperbolicequationswithvariablecoe?cients,SIAMJ.Numer.Anal.34,No.3,June1997.2.P.I.Crumpton,J.A.MackenzieandK.W.Morton,CellvertexalgorithmsforthecompressibleNavier-Stokesequations,JournalofComputationalPhysics,109(.MR94e:760813.A.Jameson,Accelerationoftransonicpotential?owcalculationsonarbitrarymeshesbythemultiplegridmethod,AIAAPaper79,p..4.H.Keller,Anew?nitedi?erenceschemeforparabolicproblems,In:NumericalSolutionofPartialDi?erentialEquationsII,SYNSPADE1970(Ed.,B.Hubbard,)AcademicPress,.MR43:28665.K.W.Morton,NumericalSolutionofConvection-Di?usionProblems,AppliedMathematicsandMathematicalComputation,12,ChapmanandHall,London,1996.6.K.W.Morton,P.I.CrumptonandJ.A.Mackenzie,Cellvertexmethodsforinviscidandviscous?ows,ComputersFluids,22(.7.J.A.MackenzieandK.W.Morton,Finitevolumesolutionsofconvection-di?usiontestproblems,MathematicsofComputation,60(0.MR93d:760658.K.W.MortonandM.F.Paisley,A?nitevolumeschemewithshock?ttingforthesteadyEulerequations,JournalofComputationalPhysics,80(3.9.K.W.MortonandM.Stynes,Ananalysisofthecellvertexmethod,MathematicalModellingandNumericalAnalysis,28(4.MR95h:6507210.K.W.MortonandE.S¨uli,Finitevolumemethodsandtheiranalysis,IMAJournalofNumericalAnalysis,11(0.MR93e:6514511.R.H.Ni,AmultiplegridmethodforsolvingtheEulerequations,AIAAJ.20(C1571.12.L.A.OganesianandL.A.Ruhovec,Variational-di?erencemethodsforthesolutionofellipticequations,Publ.oftheArmenianAcademyofSciences,Yerevan,1979.(InRussian).1406¨K.W.MORTON,MARTINSTYNES,ANDENDRESULI13.A.Preissmann,Propagationdesintumescencesdanslescanauxetrivieras,PaperpresentedattheFirstCongressoftheFrenchAssociationforComputation,heldatGrenoble,France,1961.14.H.G.Roos,M.StynesandL.Tobiska,NumericalMethodsforSingularlyPerturbedDi?er-entialEquations,SpringerComputationalMathematics,24,Springer-Verlag,1996.15.E.S¨uli,Finitevolumemethodsondistortedpartitions:stability,accuracy,adaptivity,Tech-nicalReportNA89/6,OxfordUniversityComputingLaboratory,1989.16.E.S¨uli,Theaccuracyof?nitevolumemethodsondistortedpartitions,MathematicsofFiniteElementsandApplicationsVII(J.R.Whiteman,ed.)AcademicPress,.MR92i:6517117.E.S¨uli,Theaccuracyofcellvertex?nitevolumemethodsonquadrilateralmeshes,Mathe-maticsofComputation,59(2.MR93a:6515818.H.A.Thomas,HydraulicsofFloodMovementsinRivers,CarnegieInstituteofTechnology,Pittsburgh,Pennsylvania,1937.19.B.Wendro?,Oncentereddi?erenceequationsforhyperbolicsystems,J.Soc.Indust.Appl.Math.8(5.MR22:7259OxfordUniversityComputingLaboratory,WolfsonBuilding,ParksRoad,OxfordOX13QD,UnitedKingdomE-mailaddress:Bill.Morton@comlab.ox.ac.ukDepartmentofMathematics,UniversityCollege,Cork,IrelandE-mailaddress:STMT8007@iruccvax.ucc.ieOxfordUniversityComputingLaboratory,WolfsonBuilding,ParksRoad,OxfordOX13QD,UnitedKingdomE-mailaddress:Endre.Suli@comlab.ox.ac.uk包含各类专业文献、应用写作文书、文学作品欣赏、生活休闲娱乐、外语学习资料、高等教育、专业论文、78Analysis of the Cell Vertex Finite Volume Method for the Cauchy-Riemann Equations等内容。　Analysis of the Cell Vertex Finite Volume Method for the Cauchy-Rie..
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Analysis of the Cell Vertex Finite Volume Method for the Cauchy-Riemann Equations
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3秒自动关闭窗口Abstract:Abstract: We present a solution algorithm for a second-order accurate discrete form of the inhomogeneous Cauchy-Riemann equations. The algorithm is comparable in speed and storage requirements with fast Poisson solvers. Error estimates for the discrete approximation of sufficiently smooth solutions of the pro numerical results indicate that second-order accuracy obtains even for solutions which do not have the required smoothness.Journal Name: Mathematics of ComputationPublication Date: 1979Loading PreviewA fast Cauchy-Riemann solver51 PagesSign upBefore we can start your download,please take a moment to join our communityof 25,090,045 academic researchers.&&Connect&&Connect&&Sign up with emailBy signing up, you agree to our&Download PDFs forover 6.6 Million papers Share your paperswith other researchersSee analytics on yourprofile & papersFollow other peoplein your field
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