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Iterative Solvers for Hybridized Finite Element Methods

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Title:
Iterative Solvers for Hybridized Finite Element Methods
Creator:
Tan, Shuguang
Place of Publication:
[Gainesville, Fla.]
Publisher:
University of Florida
Publication Date:
Language:
english
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1 online resource (100 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Gopalakrishnan, Jayadeep
Committee Members:
Ungor, Alper
Pilyugin, Sergei
Hager, William W.
Park, Trevor H.
Graduation Date:
8/8/2009

Subjects

Subjects / Keywords:
Approximation ( jstor )
Boundary value problems ( jstor )
Data smoothing ( jstor )
Error rates ( jstor )
Estimation methods ( jstor )
Finite element method ( jstor )
Galerkin methods ( jstor )
Lagrange multipliers ( jstor )
Mathematics ( jstor )
Multigrid methods ( jstor )
Mathematics -- Dissertations, Academic -- UF
algorithm, condition, differential, discontinuous, discretization, element, equation, finite, galerkin, hybridized, iterative, lagrange, matrix, mesh, method, mixed, multigrid, numerical, partial, triangulation
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Electronic Thesis or Dissertation
born-digital ( sobekcm )
Mathematics thesis, Ph.D.

Notes

Abstract:
We consider the application of a variable V-cycle multigrid algorithm for the hybridized mixed method for second order elliptic boundary value problems. Our algorithm differs from previous works on multigrid for the mixed method in that it is targeted at efficiently solving the matrix system for the Lagrange multiplier of the method. Since the mixed method is best implemented by first solving for the Lagrange multiplier and recovering the remaining unknowns locally, our algorithm is more useful in practice. The critical ingredient in the algorithm is a suitable intergrid transfer operator. We design such an operator and prove mesh independent convergence of the variable V-cycle algorithm. We then extend this multigrid framework to the hybridized local discontinuous Galerkin method, and yield similar mesh independent convergence results. Numerical experiments are presented to indicate the asymptotically optimal performance of our algorithm, as well as the performance comparison among different hybridized finite element methods for targeted problems. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2009.
Local:
Adviser: Gopalakrishnan, Jayadeep.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28
Statement of Responsibility:
by Shuguang Tan.

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University of Florida
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University of Florida
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Copyright Tan, Shuguang. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
2/28/2010
Resource Identifier:
489236244 ( OCLC )
Classification:
LD1780 2009 ( lcc )

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IamgreatlythankfulforenormoushelpfromthefollowingpeoplewhoserveinmyPhDsupervisorycommittee,especiallymyadvisor: Prof.JaydeepGopalakrishnan,DepartmentofMathematics(advisor,committeechair) Prof.WilliamHager,DepartmentofMathematics Prof.SergeiPilyugin,DepartmentofMathematics Prof.TrevorPark,DepartmentofStatistics Prof.AlperUngor,DepartmentofComputerScience 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 2HYBRIDIZEDFINITEELEMENTMETHODS .................. 13 2.1AStandardFiniteElementMethod ...................... 14 2.2MixedMethod ................................. 16 2.3HybridizedMixedMethod ........................... 18 2.4HybridizedDiscontinuousGalerkinMethod .................. 23 2.4.1HDGMethod .............................. 24 2.4.2HIPMethod ............................... 26 3MULTIGRIDALGORITHMFORTHEHRTMETHOD ............. 29 3.1MultigridAlgorithm .............................. 29 3.2ProofoftheConvergenceResult ........................ 34 3.2.1VericationofCondition 3.4 ...................... 35 3.2.2VericationofCondition 3.5 ...................... 36 3.2.3VericationofCondition 3.6 ...................... 46 3.3NumericalExperiments ............................. 49 4ESTIMATESFORTHEHDGMETHOD ..................... 55 4.1EstimatesforHDG ............................... 55 4.1.1StabilityoftheHDGLocalSolvers .................. 57 4.1.2ErrorEstimatesfortheHDGMethod ................. 63 4.1.3ConditioningoftheHDGMethod ................... 65 4.2EstimatesforHIP ................................ 69 4.3NumericalExperiments ............................. 71 5MULTIGRIDFORHDG ............................... 78 5.1MultigridConvergenceAnalysis ........................ 80 5.1.1NormofProlongationOperator(Condition 3.4 ) ............ 81 5.1.2RegularityandApproximationProperty(Condition 3.5 ) ....... 84 5.2NumericalExperiments ............................. 93 5

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.................................... 95 REFERENCES ....................................... 97 BIOGRAPHICALSKETCH ................................ 100 6

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Table page 3-1Performancecomparisonbetweenconjugategradientandandmultigrid. ..... 50 3-2DiscretizationerrorsforthehybridizedmixedRaviart-Thomasmethod. .... 51 3-3Applicationofthemultigridalgorithmtoaproblemonthenon-convexdomain 52 3-4V-cyclewithconstantnumberofsmoothings. ................... 52 3-5Failureofcertainintergridoperators ........................ 54 4-1DiscretizationerrorkPuhkaofHDG(d=0)forvaryingandh. ....... 72 4-2DiscretizationerrorkuuhkL2ofHDG(d=0)forvaryingandh. ....... 72 4-3Discretizationerrork~q~qhkL2ofHDG(d=0)forvaryingandh. ....... 73 4-4DiscretizationerrorkPuhkaofHDG(d=1)forvaryingandh. ....... 73 4-5DiscretizationerrorkuuhkL2ofHDG(d=1)forvaryingandh. ....... 73 4-6Discretizationerrork~q~qhkL2ofHDG(d=1)forvaryingandh. ....... 73 5-1ThenumberofmultigriditerationsforHDG(d=0)withdierent 94 5-2ThenumberofmultigriditerationsforHDG(d=1)withdierent 94 7

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Figure page 2-1Illustrationofnotations ............................... 19 3-1Reningmesh:initialmeshontheleft,andrenedmeshontheright ...... 49 3-2Thenon-convexdomainusedinexperiments. ................... 51 3-3Arenedtriangleusedtodescribethefailedintergridtransferoperators .... 53 4-1ComparisonoflogkuuhkL2:CGvsHRTvsHDG ................ 75 4-2Comparisonoflogk~q~qhkL2:HRTvsHDG .................... 76 4-3ComparisonoflogkPuhka:HRTvsHDG ................... 77 8

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WeconsidertheapplicationofavariableV-cyclemultigridalgorithmforthehybridizedmixedmethodforsecondorderellipticboundaryvalueproblems.OuralgorithmdiersfrompreviousworksonmultigridforthemixedmethodinthatitistargetedatecientlysolvingthematrixsystemfortheLagrangemultiplierofthemethod.SincethemixedmethodisbestimplementedbyrstsolvingfortheLagrangemultiplierandrecoveringtheremainingunknownslocally,ouralgorithmismoreusefulinpractice.Thecriticalingredientinthealgorithmisasuitableintergridtransferoperator.WedesignsuchanoperatorandprovemeshindependentconvergenceofthevariableV-cyclealgorithm.WethenextendthismultigridframeworktothehybridizedlocaldiscontinuousGalerkinmethod,andyieldsimilarmeshindependentconvergenceresults.Numericalexperimentsarepresentedtoindicatetheasymptoticallyoptimalperformanceofouralgorithm,aswellastheperformancecomparisonamongdierenthybridizedniteelementmethodsfortargetedproblems. 9

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Theniteelementmethod(FEM)isoneofthemostoftenusedtoolsfornumericallysolvingpartialdierentialequations(PDE)oncomplicateddomains.Dependingontheimportanceofthepropertiesneededinanapproximatesolution,onechoosestherightkindofniteelementmethodforvariousapplications.HybridizedniteelementmethodshaverecentlyemergedasapowerfulsubclassofFEMwiththeabilitytoecientlyyieldapproximatesolutionswithinterestingproperties.Inthisstudy,weshallbeconcernedprimarilywithacceleratingthesolutionprocesswhenhybridizedniteelementsareused. Likeotherniteelementmethods,thehybridizedniteelementmethodsyieldmatrixsystemswithconditionnumberthatgrowsasmeshsizedecreases.Henceitisnecessarytousepreconditionediterativesolversorfastlinearsolverslikemultigrid(MG)algorithmstoobtainthesolutioneciently.However,theapplicabilityofsuchtechniquesforthehybridizedmethodshasnotbeeninvestigateduptonow.Inthisstudy,weshalldevelopmultigridalgorithmsforecientsolutionofthesystemsresultingfromdiscretizationthroughthehybridizedschemes. Whenlineariterativemethodsareappliedtosolvesuchsystems,wecanthinkthattheerrorconsistsoftwoparts:lowfrequencycomponentsandhighfrequencycomponents,correspondingtosmalleigenvaluesandlargeeigenvaluesrespectively.ClassicalmethodssuchastheGauss-Seideliteration,reducehighfrequencycomponentsquickly,butforthoselowfrequencycomponents,theconvergencebecomesreallyslowwhenthemeshsizedecreases.Amultigridmethod,ontheotherhand,usesadierentmechanismtoovercomethisdiculty.Generallyspeaking,itmaintainsasequenceofgridsstartingfromthecoarsesttothenest,withthelastonebeingthemeshonwhichtheproblemneedstobesolved.Amultigridmethodstillusesclassicaliterationstoeliminatehighfrequencycomponentsonnergrids(calledsmoothing),butforlowfrequencycomponents,ittransfersthemtocoarsergridstodothereduction(calledcorrection).Sincethenumber 10

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4ofthefollowingnermesh),therequiredamountofworkisgreatlyreduced.Thatiswhymultigridmethodhasoptimalperformanceoverallotherclassicaliterativemethods. Thedicultiesinadaptingmultigridtechniquestohybridizedniteelementschemesariseduetotwonon-standardfeaturesofhybridizedmethods: 1. Thespacesusedtondtheapproximatesolutionsconsistoffunctionsdenedontheedgesofa(twodimensional)meshoftriangles.Incontrast,otherniteelementmethodsusefunctionsdenedonelements(triangles)ofamesh. 2. Theapproximatesolutiongivenbyahybridizedmethodsatisesavariationalformulationinvolvingameshdependentbilinearform.IncontrastmanyofthestandardniteelementapproximationsarecharacterizedviaavariationalequationthatmakesenseinaSobolevspace(withoutreferringtoamesh). Becauseofthesefeatures,whenadaptingmultigridalgorithmstohybridizedniteelementmethods,wemustdesignthecomponentsofthealgorithmtoworkwithnon-nestedmultilevelniteelementspaces,andnon-inheritedbilinearforms.WeshalluseanabstracttheoryforthesocalledvariableV-cyclealgorithmwhichhasoftenprovedsuccessfulinanalyzingmultigridadaptationstootherproblemswithsimilardiculties. Thetwomainingredientsofanymultigridalgorithmaresmoothersandintergridtransferoperators.Fortheproblemsweshallconsider,smoothersdonotposenewchallenges.However,thedesignofproperintergridtransferoperatorstailoredtothehybridizedschemesarecriticalforthesuccessofthemultigridalgorithm.Theseareoperatorsusedwithinthealgorithmtomovedatafromacoarsergridtoanergrid,andviceversa.Sincehybridizedmethodsusefunctionsdenedonmeshedges,wemustdesignnon-trivialintergridtransferoperators.Asweshallsee,manyofthe\obvious"choicesdonotresultinecient,orevenconvergent,multigridalgorithms. Thenextchapterisdevotedtohybridizedniteelementmethods.Wegiveageneralintroductiontoniteelementtechniques,includingthe"standard"FEM,andwellknown 11

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Thepurposeofthischapteristointroduceaclassofmethodscalledhybridizedniteelementmethodsforboundaryvalueproblems,whichformsthemainsubjectofourresearch.Toputthismethodinproperperspective,webeginthischapterwithanintroductiontothemostbasicandstandardniteelementmethod,whichcanbetracedbacktoanearlypaperofCourant[ 21 ].Then,weshallpresenttheso-calledmixedniteelementmethod,whichwasintroducedasameanstoobtainbetteruxapproximations[ 12 31 ].Thesediscussionswillprovidethenecessaryleadintodiscussthehybridizedmethodsinthelatersections,beginningwiththehybridizedmixedmethod[ 2 17 ],andlaterthehybridizeddiscontinuousGalerkin(DG)methods[ 19 ]. Letusdenesomenotationshererst,whichwillbeusedthroughouttherestofthedissertation.LetbeapolygonaldomaininR2.A\subdivision"ofisanitecollectionofclosedsetscalled\elements"fKigsuchthattheirinteriorsaredisjointandtheirunionistheclosureof.Fortheniteelementmethodswehaveinmind,werequireasubdivisionwiththefurtherpropertythatallelementsKiaretrianglesandnovertexofanytriangleliesintheinteriorofanedgeofanothertriangle.Suchsubdivisionsarecalled\triangulations".WedenoteatriangulationofbyTh.Thesubscripthreferstoh=maxK2ThhK;wherehK=diam(K):

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whereisasabove,andf2L2().Thewellknownweakformulationofthisproblemistondu2H10()suchthata(u;v)=F(v);8v2H10() (2.2) wherea(u;v)=Z~ru~rvdx;F(v)=Zfvdx;H10()=fv2H1():vj@=0g;andH1()=fv2L2():@iv2L2();8ig: TheessentialideaofniteelementmethodsistoreplaceH10()withanite-dimensionalsubspaceVh,andthenseekanapproximationuhtotheexactsolutionufromthissubspace.Forexample,wecanchooseVh=fv2C():vj@=0;andvjKislinear;8K2Thg;

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Withthebasisabove,wecanwritetheapproximationasuh=nXi=1cii 2.2 ),wehaveZ~r(nXi=1cii)~rvdx=Zfvdx;8v2Vh;i.e.nXi=1ciZ~ri~rvdx=Zfvdx;8v2Vh: whereAi;j=R~ri~rjdx,Fi=Rfidx,andCisthevectorofcoecientsfcig.SincetheonlyunknownhereisC,bysolvingthematrixsystem,wecanndtheapproximationuhinVhtotheactualsolutionu.Moreover,thematrixAissymmetricandpositivedenite,allowingustosolveitconvenientlywithiterativetechniquesliketheConjugateGradientmethod[ 28 ]. Thefollowingwellknowntheorem[ 14 ]statesthetwopropertiesmostpertinenttothisstudy,namelytheconditioningofthesystem( 2.3 ),andanapriorierrorestimateforthediscreteapproximation.WeshallusekkHk(W)andjjHk(W)todenoteSobolevnormsandseminormsdenedonsomedomainWthroughoutthisstudy. 2.1 ,thenthespectralconditionnumberofthematrixAin( 2.3 )satises(A)Ch2;

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2.2 tellusthatifweusetheConjugateGradientmethodtosolve( 2.3 ),wecanexpectthenumberofiterationstogrowashdecreases.Specically,theconvergencerateoftheConjugateGradientmethodintheA-normis(p 2.3 )wemustresorteithertopreconditioning,ortotheconstructionofoptimallineariterativesolverslikemultigridmethods.Ouraimistostudytheapplicabilityofmultigridtechniquestosystemsanalogousto,butmorecomplexthan( 2.3 ),arisingfromhybridizedniteelementmethods(seeChapter 3 ). TheniteelementmethodintroducedaboveisanexampleofaGalerkinmethod.SinceitusesaniteelementspaceVhofcontinuousfunctions,itisaexampleofacontin-uousGalerkin(CG)method.MethodsthatusediscontinuousniteelementfunctionswillbecalleddiscontinuousGalerkin(DG)methods.Weshallseeexamplesofsuchmethodsinlatersections.Anengineeringapproachtotheconstructionofniteelementmethodscanfoundin[ 32 ]. 16

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Stillconsiderthemodelproblem( 2.1 ).Ifweintroduceanewvariable~q,denedas~q=~ru; 2.1 )canbereformulatedas Todistinguish,peopleoftenreferuastheprimalsolution,and~qastheuxsolution. TheRTmixedmethodseeksanapproximation(~qh;uh)totheexactsolution(~q;u)of( 2.4 ).LetPd(K)denotethespaceofpolynomialsonKofdegreeatmostd,andRd(K)bethecorrespondingRTspace,i.e.,Rd(K)=Pd(K)Pd(K)+~xPd(K): 17

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2.5 )givesrisetoamatrixsystemofblockform whereAi;j=R~vi~vjdx,Bi;j=Rwi~r~vjdx,Fi=Rfwidx,andQandUarethecoecientvectorsfor~qhanduhwithrespecttotheircorrespondingbasisf~vigandfwjg,respectively.Wecanthensolvethelinearsystemtogetanapproximationto(~q;u). Notethatincontrastto( 2.3 ),themixedmethodyieldsthesystem( 2.6 )thatisindenite.TheConjugateGradientmethodisnotappropriateforsuchsystems.WemustuseGMRES[ 33 ]orsimilarmethods.Whileitispossibletotailorpreconditioningstrategiesforthismethod,theyareconsiderablymoreinvolved,duetotheindenitenessofthesystem.Incontrast,thehybridizedversionofthemixedmethod,tobediscussedinthenextsection,resultsinasymmetricpositivedenitesystemthatgivestheexactlythesamesolutionasthemixedmethod.Apriorierrorestimateisavailableforthemixedmethod[ 18 23 ]. 2.6 ).Sincethesystemisnotpositivedenite,solvingforQandUisnotalwayseasy.AlthoughbyeliminationofQfromtheequations,wecanachieveapositivedenitesystem,thiswillrequiretheinversionofA,andtheinversematrixistypicallyafullmatrixofbigsize.Fortunately,anadvancedtechniquecalledhybridizationhelpsusovercomethisdiculty.Letusintroducethehybridizedmixedmethod,basedonthesamemodelproblem( 2.4 ). Themethodintroducesanadditionalunknownh,calledLagrangemultiplier,whichturnsouttobenothingbutanapproximationtothetraceofuoneachedgee2Eih.Aswewillshownext,itsintroductioncanactuallyeliminateboth~qhanduh,reducing 18

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Here[[]]iscalledthejumpoperator,denedasbelow. 2-1 ). Illustrationofnotations Itisawell-knownfact[ 2 17 ]thattheHRTmethodgivestheexactlysamesolutionsastheoriginalmixedmethod,asstatedinthefollowingtheorem. 19

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2.7 )isuniquelysolvable,and(~qh;uh)in( 2.7 )coincidewith(~qh;uh)in( 2.4 )fromthemixedmethod. 2.7 ),itmayseemthatweneedtosolveanevenmorecomplicatedequationsystemthantheoriginalmixedmethod.Butthatisreallynotthecase.Toexplainwhy,letusdenetwoso-calledliftingoperatorsasfollows. Giveneverym2L2(Eh),therstliftingpairoffunctions(~QRTm;URTm)2VhWhsatises andgiveneveryf2L2(),thesecondliftingpairoffunctions(~QRTf;URTf)2VhWhsatises forall(~v;w)2VhWh. OnethingtoknowisthedierencebetweentheFEMspacesfortheRTmethodandtheHRTmethod.IncaseoftheRTmethod,VhH(div;)impliesthatfunctionsinsideitneedtohavecontinuousnormalcomponentsacrossboundaries,whilefortheHRTmethod,thisrestrictionisliftedbysimplyrequiringVhL2()L2().Alsonotethatthesemappingsdenedaboveareuniquelydeterminedoneachelementbecauseofthesurjectivityofthemap(r):Vh7!Whrestrictedtoanelement.Moreimportantly,sincethefunctionsinVhWhhavenocontinuityconstraintsacrosselements,thecomputationoftheseliftingpairscanbedoneelementbyelementinadecoupledway.Suchcomputations,beinglocal,areinexpensive. 20

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17 ]. 2.7 ).Then

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2.11 )resultsinamatrixsystemARTC=BRT Byvirtueofthistheorem,insteadofsolving( 2.7 )directly,wecansolve( 2.11 )tondtheLagrangemultiplierh,andthenuse( 2.10 )torecoverthe~qhanduh.Thisrecoveryislocal,astheapplicationof~QRTandURTarelocaloperations,andconsequentlyofnegligiblecost,comparedtotheglobalinversionrequiredtondh. Thevariationalcharacterization( 2.11 )shouldbecomparedwith( 2.2 ).While( 2.2 )makessensenotonlyontheniteelementspace,butalsoonthewholeSobolevspaceH1(),theformulation( 2.11 )forthehybridizedRTmethodonlymakessenseonthespecicmeshunderconsideration.Thisisanimportantdierencethatwewillneedtokeepinmindduringmultigridanalysis. Notethatcomparedtothemixedmethod,thehybridizedversionhasthefollowingadvantages: 1. Thesizeofthematrixsystemcomingfrom( 2.11 )issmallerthantheonefromthemixedmethod( 2.4 )sincethespaceMhisdenedonEh,not. 2. Thematrixfrom( 2.11 )issymmetricpositivedenite,andhencewecansolveitwithfastiterativetechniquessuchastheConjugateGradientmethod. 22

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Oncehiscomputed,theothercomponentsofthesolutionpair,namely~QRTf,~QRTg,URTfandURTg,canbecomputedinexpensivelyinacompletelylocalfashion(elementbyelement),asseenfrom( 2.10 ). 4. 2 ]. 2.6 )canbediagonalized,whichmakesiteasiertocomputetheinverse.suchaniteelementmethodiscalleddiscontinuousGalerkin(DG)method.Similarasinthemixedmethod,byhybridizingDGmethods,wecangetevenbetterresults. Weconsideramoregeneralversionofthemodelproblem( 2.4 ): wheref2L2()andg2H1=2(@). TherearetwokindsofimportantDGmethods:localdiscontinuousGalerkinmethodandinteriorpenaltymethod.Basedonthegeneralframeworkofthehybridizedmixedmethod,wewillbrieyintroducethehybridizedversionsofthesetwomethods. Beforewecontinue,letusmakeafewconventions. 23

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)willalwaysbeexplicitlymentioned. 24

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Then,similarlyasintheHRTmethoddiscussedinx ,wedenetwoliftingoperatorslocallyoneachmeshelementK. Giveneverym2L2(Eh)andeveryK2Th,therstliftingpairoffunctions(~QDGm;UDGm)2VhWhrestrictedonKisinV(K)W(K),andsatises: (~QDGm;~v)K(UDGm;r~v)K=(m;~v~n)@K;forall~v2V(K);(~rw;~QDGm)K+(w;(~QDGm~n+K(UDGmm))@K=0;forallw2W(K):(2.13) Giveneveryf2L2()andeveryK2Th,thesecondliftingpairoffunctions(~QDGf;UDGf)2VhWhrestrictedonKisinV(K)W(K),andsatises: (~QDGf;~v)K(UDGf;r~v)K=0;forall~v2V(K);(~rw;~QDGf)K+(w;(~QDGf~n+K(UDGf))@K=(f;w)K;forallw2W(K):(2.14) TheHDGmethodcanbeviewedasseekingtheapproximation(~qh;uh;h)in(VhWhMh)satisfying (~qh;~v)(uh;r~v)+XK2Th(h;~v~n)@K=(g;~v~n)@forall~v2Vh;(~qh;~rw)+XK2Th(w;bqh~n)@K=(f;w)forallw2Wh;XK2Th(m;bqh~n)@K=0forallm2Mh;(2.15) wherebqh,( 2.4 forthejumpoperator[[]]).AsimilarcharacterizationtheoremasfortheHRTmethodisgivenasfollows,withadditionaltermsinvolvingg: 25

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2.15 ).Then 2.17 )resultsinamatrixsystemADGC=BDG

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,wedenetwoliftingoperatorslocallyoneachmeshelementK. Giveneverym2L2(Eh)andeveryK2Th,therstliftingpairoffunctions(~QIPm;UIPm)2VhWhrestrictedonKisinV(K)W(K),andsatises: (~QIPm;~v)K(UIPm;r~v)K=(m;~v~n)@K;forall~v2V(K);(~rw;~QIPm)K+(w;a~rUIPm~n+K(UIPmm))@K=0;forallw2W(K):(2.18) Giveneveryf2L2()andeveryK2Th,thesecondliftingpairoffunctions(~QIPf;UIPf)2VhWhrestrictedonKisinV(K)W(K),andsatises: (~QIPf;~v)K(UIPf;r~v)K=0;forall~v2V(K);(~rw;~QIPf)K+(w;a~rUIPf~n+KUIPf)@K=(f;w)K;forallw2W(K):(2.19) AsintheHDGmethod,isdouble-valuedoneachinteriormeshfaceofEh. TheHIPmethodcanbeviewedasseekingtheapproximation(~qh;uh;h)in(VhWhMh)satisfying (~qh;~v)(uh;r~v)+XK2Th(h;~v~n)@K=(g;~v~n)@forall~v2Vh;(~qh;~rw)+XK2Th(w;bqh~n)@K=(f;w)forallw2Wh;XK2Th(m;bqh~n)@K=0forallm2Mh;(2.20) wherebqh,( 2.6 )isgivenasfollows: 2.20 ).Then 27

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2.20 )resultsinamatrixsystemAIPC=BIP 28

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Inthischapter,wewilldevelopamultigridalgorithmforsolvingthelinearsystemarisingfromthehybridizedmixedmethod.AswehavealreadyseeninSection 2.3 ,allsolutioncomponentscanberecoveredoncewendhinMhsatisfyinga(h;)=b();forall2Mh: 24 ]. 2.1 holds.ThenthespectralconditionnumberofARTsatises(ART)Ch2: 1 ],theGauss-Seidelmethod[ 27 ],ortheun-preconditionedConjugateGradientmethod[ 28 ]willyieldincreasingiterationcountsashgoesto0.Therefore,weneediterativesolutionstrategiesthatdonotdeteriorateinperformancewhenthemeshsizedecreases.Inthissection,wegiveonesuchtechnique.Thealgorithmtsintotheabstractframeworkof[ 9 10 ]asoneoftheirabstractvariableV-cyclealgorithms.Theiralgorithm,withtheabstractcomponentsparticularizedtoourapplicationisgivenbelow,followingwhichwestateourmainresultontheconvergenceofthealgorithm. Theresultsofthischapterhaveappearedin[ 25 ]. 29

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EachMkisendowedwithtwobilinearforms,(;)k,andak(;).While(;)kisjustthestandardL2()-innerproductfork=0;:::;J1;atlevelJitisameshdependentL2-likeinnerproductdenedby (;)J=XK2TJjKj j@KjZ@K;(3.1) wherejjdenotesthemeasure.TheotherbilinearformonMkMkisdenedbyak(u;v)=8>><>>:Z~ru~rv;k=0;1;;J1;Z~QRTu~QRTv;k=J;

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Themaincomplicationinanon-nestedsettingisthenecessityofdesigningappropriateintergridtransferoperatorsformovingdatabackandforthbetweenthemultilevelgrids.WedenetheprolongationoperatorIk:Mk1!Mk,(k=2;:::;J)by whereMJ:L2(EJ)!MJistheL2(EJ)-orthogonalprojectionontoMJ.Itisimportanttonotethattherearemanynaivechoicesofintergridtransferoperatorsthatdoesnotworkinourapplication.InSection 3.3 ,weshallshownumericalexperimentswithcertain\obvious"transferoperatorsthatleadtoslowconvergenceofmultigrid.Thereversemovementofdata,fromnetocoarselevels,isachievedthroughtherestrictionoperatorQk1:Mk!Mk1,denedby(Qk1!;')k1=(!;Ik')k;forall';!2Mk1: 31

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Thealgorithmgivenbelowperformsmkpre-andpost-smoothingsatlevelk.Ourconvergenceresultisunderanassumptionthatthenumberofsmoothingsincreaseinaspecicway(detailedinTheorem 3.3 )asweproceedtothecoarserlevels. wherethemapMGk(;):MkMk7!Mkisdenedrecursivelyasfollows. 1. First,atthecoarsestlevel,setMG1(u;f)=A11f. 2. Next,fork2,deneMGk(u(i);f)bythefollowingsteps: (a) Setv(0)=u(i). (b) Setresidualrk=fAkv(mk): 32

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(f) Finally,denethenextiteratebysettingMGk(u(i);f)=w(2mk): 10 ]thatEJisalinearoperatoradmittingarecursiveexpression.Usingtheabstracttheoryof[ 9 10 ],weprovethatthisiterativeerrordecreasesgeometricallyatamesh-independentrate,asstatedinthenexttheorem. 3.2 satises0aJ(EJu;u)aJ(u;u);forallu2MJ: 33

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WealsobrieyexplainhowtouseAlgorithm 3.2 asapreconditioner.Sincethealgorithmdenesalineariteration,theoperatorBJ:MJ7!MJdenedbyBJg=MGJ(0;g);forallg2MJ 3.3 showsthat(BJAJ)isboundedaboveandbelowbymeshindependentconstants.HenceBJisanoptimalpreconditioner. 3.3 .Weshallusetheabstractmultigridtheoryof[ 9 10 ]whichallowstheuseofnon-inheritedformsandnon-nestedspaces.Accordingtothistheory,onceweverifythefollowingthreeconditions,theproofofTheorem 3.3 iscomplete.

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Theremainderofthissectionisdividedintothreesubsections,eachdevotedtothevericationofoneoftheaboveconditions. 3.4 foranyK2TJ.Then,bythedenitionofthelifting~QRT(),ZK~QRT(IJvJ1)~r=Z@K(IJvJ1)~r~n;8~r2R0d(K): 3.2 ),therighthandsideabovecanberewrittenasZ@K(IJvJ1)~r~n=Z@K(MJvJ1)~r~n=Z@KvJ1~r~n=ZK(~rvJ1)~r; 35

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3.2 ),wehaveZ@K(IJvJ1)~r~n=Z@KvJ1~r~n=ZK(~rvJ1)~r; ThevericationofCondition 3.4 isnowcompletedbyobservingthatbecauseofLemma 3.7 ,aJ(IJv;IJv)=(~QRT(IJv);~QRT(IJv))=(~rvJ1;~rvJ1)=aJ1(vJ1;vJ1): 3.1 ). 3.5 3.5 typicallyfollowasaconsequenceofsomeregularityresultsfortheunderlyingboundaryvalueproblem,combinedwiththeapproximationpropertiesoftheniteelementspaces.Itiswell-knownthatCondition 3.5 holdsfork=0;1:::;J1[ 6 ],whichispartofthestandardfullregularitybasedproofsofmultigridconvergenceforthecontinuousGalerkinmethod[ 3 5 10 ].SoweonlyneedtoverifyCondition 3.5 withk=J. Forthis,weneedanumberofintermediatelemmasthatestablishpropertiesofvariouslocaloperators.LetusbeginwiththelocalliftingoperatorURT()denedearlier. Proof. 2.8 ),wehaveZK~QRT(IJw)~rZKURT(IJw)r~r=Z@KIJw(~r~n);8~r2Rd(K):

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3.7 ,thisimpliesZK(URT(IJw)w)r~r=0;8~r2Rd(K): Next,weneedtodeneanewlocaloperatorthatmapsapairofinteriorandboundaryfunctionsintoonefunction.LetLd(K)=fp2Pd+3(K):pje2Pd+(e);8edgeeofKg,whered+=8><>:d+1;ifdiseven,d+2;ifdisodd. Supposewearegivenp2L2(K)and2L2(@K).Considerafunction(p;)2Ld(K)thatsatises forallthethreeedgeseofK.Thatsuchaisuniqueisprovednext.Asusual,whenperformingstandardscalingarguments,weobtainconstantsthatdependontheshaperegularityofthemesh,namelyonaxedconstantwhichisthemaximumofdiam(K)=KoverallelementsK,whereKdenotesthediameterofthelargestballinscribedinK. 3.5 ).Furthermore,thereareconstantsC1andC2dependingonlyontheshaperegularityconstantsuchthatC1k(p;)kL2(K)kpkL2(K)+j@Kj1=2kkL2(@K)C2k(p;)kL2(K)

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Proof. 3.5 )formsasquaresystemfor(p;).Indeed,thenumberofequationsinthesystem( 3.5 )equals dim(Pd(K))+3dim(Pd+1(e))=1 2(d+1)(d+2)+3d+:(3.6) Ontheotherhand,thenumberofdegreesoffreedomofLd(K)canbecountedbyaddingtogetherthedimensionofP1(K)(equaling3),thedimensionofthespaceofalledgebubblesofLd(K)(equaling3(d+1)),andthedimensionofinteriorbubblesofLd(K)(equaling(d+1)(d+2)=2).Thus,dim(Ld(K))=3+3(d+1)+1 2(d+1)(d+2); 3.6 ).Thus( 3.5 )isasquaresystem. Toprovethatthereisaunique(p;)satisfying( 3.5 ),itnowsucestoshowthatifpandvanish,theonlysolutionof( 3.5 )istrivial.Tothisend,considerainLd(K)satisfyingZKs=0;8s2Pd(K) (3.7)Ze=0;8e;82Pd+1(e): Thelastequation( 3.8 )impliesthatoneachedgee,jeisapolynomialonPd+(e)thatisorthogonaltoallPd+1(e).HencejemustbetheLegendrepolynomialofdegreed+.Nomatterwhatdis,d+isalwaysodd,hencejeisanoddfunctionontheedgee.Sincethisholdsforallthreeedges,andsincemustbecontinuouson@K,weconcludethatvanisheson@K. Since2Ldvanishesonallthethreeedges,itmusthavetheform=123pd;forsomepd2Pd(K) 38

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3.7 )impliesZK(123)pds=0;8s2Pd(K): 3.5 ). Thenormestimateofthelemmafollowsbecauseifpandareasinthestatementofthelemma,then(p;)=0ifandonlyifp=0and=0.Thus,onaxedreferenceelement^KwehaveC1k(p;)kL2(^K)kpk2L2(^K)+kk2L2(@^K))1=2C2k(p;)kL2(^K); UsingtheabovedenedelementspaceLd(K)oneachmeshelement,wecandeneanewliftingoffromtheelementboundariesintotheelementinteriorsby NotethatSisinH(div;)sinceitsnormalcomponentsarecontinuousbecauseof( 3.5 ).ThenextlemmaestablishesafewpropertiesofSthatweneed.Initsstatement,andintheremainder,kkadenotesthe\energy"-likenormonthenestlevel,i.e.,kk2a=aJ(;)forall2MJ; 3.1 ). 39

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Proof. 3.10 ),rstobservethatiftakesaconstantvalueontheboundaryofsomemeshelement@K,thenStakesthesameconstantvalueonK(thisfollowsfromLemma 3.8 ).Hence,forany,wehave~r(SK)=0;where=1 3.9 18 ,Lemma3.3]Cj@Kj1=2kkL2(@K): 24 ,Theorem2.2],wegetXK2TJk~r(S)k2L2(K)CXK2TJj@Kj1kk2L2(@K)Cah(;) whichproves( 3.10 ). Theproofof( 3.11 )isastraightforwardconsequenceofLemma 3.9 Theidentity( 3.12 )isobviousfrom( 3.5 )and( 3.9 ). WeneedonemoreintermediatemapbeforewecangiveourproofofCondition 3.5 .Todescribethismap,rstwedeneainMJforeveryinMJby (S;S)=(~QRT;~QRT);82MJ:(3.13) ThisequationisuniquelysolvableforinMJ,becauseiftheright-handsideiszero,thenS=0,so=0bytheestimate( 3.11 )ofLemma 3.10 .Next,letf=URT.ThemapweuseinthelaterproofisamapfromMJintoMJ,whichfornotationalsimplicity,we 40

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Thefollowinglemmarevealstherelationshipbetween;~and. 3.15 ),kSkL2()=sup2MJ(S;S) 3.13 )=sup2MJ(AJ;)J 3.11 )CkAJkJ: 3.16 ),notethataJ(;)=(S;S)aJ(~;)=(WhS;S); 3.14 ),therighthandsidefunctionisf=URT=Wh(S) (3.17) by( 3.12 ).Subtracting,andsetting=~,wegetk~k2a=((IWh)S;S(~))=((IWh)S;(IWh)S(~)):

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11 ]kuWhukL2(K)ChJjujH1(K),wegetk~k2aXK2TJCh2JjSj2H1(K)1=2XK2TJCh2JjS(~)j2H1(K)1=2XK2TJCkSk2L2(K)1=2XK2TJCh2JjS(~)j2H1(K)1=2CkAJkJXK2TJCh2JjS(~)j2H1(K)1=2 3.15 ).Finally,applyingtheestimate( 3.10 )ofLemma 3.10 onthelasttermabove,andcancelingthecommonfactor,weobtain( 3.16 ). Withthesepreparations,wecannownishthevericationofCondition 3.5 3.5 with=1. First,byLemma 3.7 Second,theexpressionontherighthandsideabovesatises (~QRT+~r(PJ1);~rvJ1)=0;8vJ12MJ1;(3.19) because(~r(PJ1);~rvJ1)=aJ1(PJ1;vJ1)=aJ(;IJvJ1)=(~QRT;~QRT(IJvJ1))=(~QRT;~rvJ1):

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anditsucestoestimate~QRT+~r(PJ1):Here,andintheremainder,weusekk(withoutanysubscripts)aswellaskkL2()todenotetheL2()-norm. Tobegintheestimation,wesplit~QRT+~r(PJ1)intomanyterms,labelingeachoneasfollows:~QRT+~r(PJ1)=~QRT(~)(termA) (3.21)+~QRT~~QRT(MJ~u)(termB)+~QRT(MJ~u)+~r~u(termC)+~rPJ1~~r~u(termD)+~rPJ1(~);(termE) where~isasdenedin( 3.14 )and~uistheuniquefunctioninH10()thatsolves(~r~u;~rv)=(f;v);8v2H10(): byawellknownregularitytheorem[ 26 ]. Thersttermcanbeestimatedbyk(termA)k=k~QRT(~)k=k~kaChJkAJkJ 3.16 )ofLemma 3.11 .Forthenextterm,rstobservethatduetothecharacterizationofLagrangemultipliersgivenbyTheorem 2.6 ,~isthehybridizedmixedmethodapproximationto~u.HencebyapreviouslyestablishedLagrangemultipliererror 43

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18 ,Theorem3.1], where~q=~r~uandR~qistheRaviart-Thomasinterpolantof~q.Bythestandarderrorestimatesforthisinterpolant,weimmediatelyndthatk(termB)k=k~QRT~~QRT(MJ~u)kChJj~ujH2()ChJkfkL2()by( 3.22 )=ChJkWh(S)kL2()ChJkSkL2()ChJkAJkJ; 3.15 )ofLemma 3.11 Weproceedtoanalyzethenextterm.Forthis,recallthedivergencefreesubspaceR0d(K)denedin( 3.4 ).Bythedenitionof~QRT(),ZK~QRT(MJ~u)~r=Z@K(MJ~u)~r~n=Z@K~u~r~n=ZK~r~u~r Now,suppose~uJ12MJ1istheexactsolutionof(~r~uJ1;~rv)=(f;v);8v2MJ1: 3.24 ),wehave~QRT(MJ~u)+~r~u;~QRT(MJ~u)+~ruJ1K=0;

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3.22 ),and( 3.15 )ofLemma 3.11 For(termD),wewillrstshowthatPJ1~coincideswiththe~uJ1denedabove.Indeed,forallwJ12MJ1,wehave(~rPJ1~;~rwJ1)=(~QRT~;~rwJ1)by( 3.19 )=(~QRT~;~QRT(IJwJ1))byLemma 3.7 =(f;URT(IJwJ1))by( 3.14 )andTheorem 2.6 =(f;wJ1)byLemma 3.8 and( 3.17 ): 3.25 ). Forthenalterm,werstnotethatifwechoose=andvJ1=PJ1in( 3.19 ),thenwehavek~rPJ1k2=(~QRT;~rPJ1);

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3.16 )ofLemma 3.11 Returningto( 3.20 )andcombiningtheestimatesforeachofthetermsabove,weobtainaJ((IIJPJ1);)=k~QRT((IIJPJ1))k2Ch2JkAJk2J 24 ,Theorem2.3],weknowthatJCh2J.HencetheaboveinequalityprovesCondition 3.5 with=1. 3.6 10 ].Forthehighestlevelk=J,theargumentsarealsofairlystandard.Nonetheless,wewillsketchtheproofforthiscasenow. BoththeJacobiandGauss-Seideliterations,whicharewellknownclassicaliterations,canberewrittenusingthemodern\subspacedecomposition"framework.TodisplaythisdecompositionforthenestlevelspaceMJ,letiJ,i=1;2;:::;NJdenotealocalbasisforMJwiththepropertythateachiJissupportedonlyononemeshedge.Further,letMJ;i=spanfiJg.ThenthesubspacedecompositionofMJisMJ=NJXi=1MJ;i

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TheGauss-SeideloperatoroneachMJisdenedas Ontheremaininglevels,smoothersJkandGkcanbewrittenoutusingthestandardsubspacedecompositionsoftheconformingniteelementspaces. WebeginwithasimplelemmaonthestabilityofthedecompositioninthemeshdependentL2-likenormonEJ. 47

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3.26 ),byawellknownlemmaonadditiveoperators[ 10 ],splittingv=PNJi=1ciiJ,(J1Jv;v)=NJXi=1aJ(ciiJ;ciiJ)=NJXi=1c2i(AJiJ;iJ)JNJXi=1c2iJ(iJ;iJ)J=JNJXi=1c2ikiJk2J!CJkNJXi=1ciiJk2J=CJkvk2J; 3.12 Fromtheabovelemma,thesmoothingconditionscanbeveriedbystandardarguments.Indeed,theonlyotheringredientneededisaninequalityoftheformNJXj=1NJXl=1jaJ(vj;wl)jNJXj=1aJ(vj;vj)!1=2NJXl=1aJ(wl;wl)!1=2 7 8 10 ]thefollowinglemma: 1. ifRJ=JJ,then(~R1Jv;v)J1(J1Jv;v)J;8v2MJ: ifRJ=GJ,then(~R1Jv;v)J(J1Jv;v)J;8v2MJ: 3.6 notethat!kvkk

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3.14 withLemma 3.13 .Fortheremainingk,theestimateisstandardforpointJacobiandGauss-Seideloperators.ThusCondition 3.6 isveriedforallk. Fortherstexperimentthatweshallnowdescribe,westartedwithacoarsemeshT1generatedbythepublicdomainmeshingsoftwareTriangle[ 35 ],andthenproducedasequenceofrenementsT2;T3;:::;TJbyconnectingthemidpointsofedges,asexplainedbefore.ThedomainandthersttwomeshesareshowninFigure 3-1 .Supposeweneedtosolvethemodelproblem( 2.4 )onthenestmeshlevelTJforvariouschoicesofJ.Theexactsolutionisu(x;y)=sin(x)ey=2.ThisproblemrequiresanonzeroDirichletboundaryconditionu=gon@,whichentailstheadditionofthetermR@g~v~nontherighthandsideoftherstequationof( 2.7 ).Butthemultigridalgorithmisunaected. Reningmesh:initialmeshontheleft,andrenedmeshontheright.Cornercoordinatorsintheinitialmeshare(0;0);(1;0);(0:8;0:7)and(0;0:5). 49

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3.2 withmk=2Jk.Allexperimentsaredonewiththelowestordermethod,i.e.,d=0.Westartwiththezerofunctionastheinitialiterateandstoptheiterationswhentheinitialerrorisreducedbyafactorof108.WelisttheresultsinTable 3-1 .AllexperimentsarerunonIntelCoreDuoprocessor(CPU@1:73GHz,512MbRAM).Aftersolvingthesystemfor,werecoverbothuand~qasdescribedinx .ThreedierentkindsofdiscretizationerrorsarereportedinTable 3-2 .Theyshowtheconvergenceofniteelementerrorsinaccordancewiththeknowntheoreticalresults[ 2 18 ],andassuchmaybeconsideredtobeavalidationofourcode. Table3-1. Performancecomparisonbetweenunpreconditionedconjugategradientmethodandthemultigridmethod.(Entriesmarked*indicatesunavailabledataduetoexcessivecomputationaltime.) Sizeconjugatemultigrid IterationscpusecsIterationscpusecs 74480.00200.013161010.02260.0213042070.19310.1052964181.64330.392134483317.66342.18856961658148.03349.6034342432831209.653440.681374976650310190.0034163.085502464**34668.6322014976**342607.65 Ascanbeseenfromthelastcolumnofthetable,whenthesizeofthematrixincreasesbyafactorofabout4(whichhappenswhenhishalved),thenumberofmultigriditerationsaswellasthecputimeinsecondsalsoincreasesbyafactorof4.ThisindicatesthatourmultigridalgorithmindeedgivesaniterativeprocesswiththeasymptoticallyoptimalO(N)cost,whereNisthenumberofunknowns.Atthesametime,thecostincreasesbyafactorofaround8fortheconjugategradientmethod,eachtimethemeshsizeishalved.Thisclearlydemonstratesthebenetsofthemultigrid 50

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DiscretizationerrorsforthehybridizedmixedRaviart-Thomasmethod. hkPuhkakuuhkL2k~q~qhkL2 algorithm.Alsonoticethatthenumberofmultigriditerationsseemsboundedevenasthematrixsizegetsverylarge.ThisisinaccordancewiththeconclusionofTheorem 3.3 .Inotherwords,theerrorreductionfactorseemstobeindependentofmeshsize,whichisinaccordancewiththeconclusionofTheorem 3.3 Next,wepresentanexampledesignedtocheckifthesucientconditionthatisconvex(inTheorem 3.3 )isnecessary.WerepeatedtheexperimentswiththedomainasshowninFigure 3-2 .Table 3-3 givestheexperimentaldata.Thenumbersofmultigriditerationstillseemstoremainbounded.Weconcludethatourmultigridalgorithmcanbeeectiveevenwhenisnotconvex. Thenon-convexdomainusedinexperiments. 51

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Applicationofthemultigridalgorithmtoaproblemonthenon-convexdomainofFigure 3-2 Size116496204883203353613465653964821606408646656 Iterations232730323233333333 Next,weinvestigateexibilityinregardtothenumberofsmoothingsmk.Theorem 3.3 assumesthatthenumberofsmoothingsincreasesgeometricallyaswedecreasetherenementlevelk.Wenowrepeattherstexperiment,butinsteadofsettingmk=2Jk,wenowxmktobeoneforallk.Table 3-4 indicatesthattheV-cyclealgorithmcontinuestoexhibitmeshindependentconvergence. Table3-4. V-cyclewithconstantnumberofsmoothings. Size7431613045296213448569634342413749765502464 Iterations212631343434353535 Finally,wegivenumericalresultswhenwereplaceourintergridtransferoperatorwithtwoseeminglyplausibleintergridtransferoperatorsinthevariableV-cycle.Theseoperatorsfail,asweshallsee,buttheyprovideinsightintowhatoneshouldavoidwhenconstructingagoodprolongation.ConsiderAlgorithm 3.2 withthenonnestedmultilevelspacesMk=f2L2(Eik):je2P0(e);foralle2Eikgandnon-inheritedformsateverylevelgivenbyak(u;v)=R~QRTu~QRTv,wherenowtheliftings~QRT()aredenedwithrespecttoTk.Theformsak(;)andabaseinnerproductasin( 3.1 )generalizedtoalllevelsk,denethemultileveloperatorsAkandQkinthealgorithm.Inotherwordsthelowestorderhybridizedmixedmethodisusedtodenethespacesandformsateveryrenementlevelinthealgorithm.ThenweconsidertwodierentintergridtransferoperatorsIk:Mk1!Mk,givenasfollows.ConsideratriangleofmeshTk1,forinstance,thetriangleT=ABCshowninFigure 3-3 .Lete= 52

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Arenedtrianglewiththenotationsusedtodescribethefailedintergridtransferoperators. alltheneredgesasfollows:I(1)kAB=8>>>><>>>>:URT(AB)onthe3newedges( DF; EF)1onthe2newedges( BD)0ontheother4newedges( CF; BE; CE) whereURT()istheliftingoperatordenedin( 2.8 ),butnowwithrespecttothemeshTk1.ThesecondprolongationcandidateweshallconsiderisI(2)k:Mk1!MkisdenedbyI(2)kAB=8>>>><>>>>:1=2onthe2newedges( DF)1onthe2newedges( BD)0ontheother5newedges( CF; BE; CE; EF) Eachoftheseoperatorsgivesadierentmultigridalgorithm.WereportontheperformanceoftheV-cyclealgorithmwiththesetwoprolongationcandidatesandaxednumberofsmoothingsmk=1inTable 3-5 .Clearly,theresultsaredismal. WebelievethatthefailureisduetothefactthatprolongationoperatorslikeI(1)kandI(2)kincreaseenergyuponcontinualtransferofacoarsegridfunctiontoincreasingly 53

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Failureofcertainintergridoperators.(Anentry*indicatesthattheiterationdiverged.) SizeMGiterationcounts withI(1)kwithI(2)k nerlevels.Incontrast,thesuccessfulprolongationIkthatweanalyzed,doesnotincreaseenergy,ascanbeseenfromCondition 3.4 ,whichweveriedforourIk. 54

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ThischapterisdevotedtoastudyoftheHDGmethodintroducedinSection 2.4.1 .ThiswillformpreliminarymaterialforthestudyofthemultigridmethodsforHDGmethodspresentedinthenextchapter. HDGmethodswerediscoveredin[ 19 ].Thersterroranalysisofsuchmethodswasprovidedin[ 15 ].Thisanalysiswasimprovedusingaspecialprojectiontailoredtothemethodin[ 20 ].Theerrorestimatesweshallpresentinthischapterareprovedusingthesameprojection.Theseestimatesareslightlydierentfromthosein[ 20 ](and[ 15 ])inthatoursholdunderlessregularityassumptions.Suchlowregularityestimatesarenecessaryforthemultigridanalysislater,wherewedonotplaceconvexityorothersuchassumptionsonthedomainthatguaranteesolutionsofhigherregularity. Apartfromerrorestimates,wealsoproveboundsonthespectrumoftheoperatorsarisingfromtheHDGmethod.ThisyieldsaconditionnumberestimateforthematrixsystemresultingfromtheHDGmethod,indicatinghowtheperformanceofclassicaliterativetechniques(likeGauss-Seidelorconjugategradientmethods)deterioratewithsystemsize.TheyserveasapreludetothenecessityoftechniqueslikemultigridtoacceleratesolutionofHDGmethods.OurtechniquesalsoeasilyyieldaconditionnumberboundontheHIPmethod,asweshow.WhileconditionnumberboundsfortheHRTmethodswereknownpreviously,nosuchboundswereestablishedfortheHDGorHIPmethodspreviously. .Itsdeningequationsystemis( 2.17 ).Infact,itwillbenotationallyecienttorewritethisformintermsofthefollowingoperators.DeneA:Vh7!Vh,B:Vh7!Wh,C:Vh7!Mh,R:Wh7!Wh, 55

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forsome~gh2Vhandfh2Wh,wherethesuperscript\t"denotestheadjointwithrespectto(;)horh;i@hasappropriate.Itiseasytoseethat( 2.15 )canberewrittenastheabovesystemwithfh=Whf,whereWhdenotestheL2()-orthogonalprojectionintoWh,and~ghsettotheuniquefunctioninVhsatisfying (~gh;~v)h=hg;~v~ni@forall~v2Vh:(4.2) Notethatinthelowestordercasek=0,theoperatorBiszero,butthesystemcontinuestobeuniquelysolvable. Theresultontheabovementionedeliminationcanbedescribedusingadditional\local"operators~QDGV:Vh7!Vh,~QDGW:Wh7!Vh,UDGV:Vh7!Wh,UDGW:Wh7!Wh,whoseactionisdenedbysolvingthefollowingsystems0B@ABtBR1CA0B@~QDGV~ghUDGV~gh1CA=0B@~gh01CA0B@ABtBR1CA0B@~QDGWfhUDGWfh1CA=0B@0fh1CA forall~gh2Vhandfh2Wh. Onecanthenreadilyverifythat~QDGm=~QDGV(Ctm)~QDGW(Stm)andUDGm=UDGV(Ctm)UDGW(Stm),where(~QDGm;UDGm)isdenedin( 2.13 ).Forexample,( 4.3 ) 56

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(~QDGm;~v)K(UDGm;r~v)K=hm;~v~ni@Kforall~v2Vh; whichisequivalentto( 2.13 ). Basedontheseoperatorsabove,wewillobtainestimatesonthestability,conditioning,anddiscretizationerrorsoftheHDGmethod.OurtechniqueconsistsofrstobtainingboundsforvariouslocalsolutionoperatorsoftheHDGmethod.Thelocalboundsthenimplyglobalbounds,suchasboundsforthediscretizationerrorsandthelengthofthespectrum. 4.3 ),withtheobviousmodicationoftherighthandside.ItisinstructivetocompareTheorem 4.1 withasimilarresultfortheseHRToperators,asprovedin[ 18 ,Lemma3.3].Forinstance,onepairofinequalitiesof[ 18 ,Lemma3.3]isk~QRTkKCh1Kkkh;K;kURTkKCkkh;K; 57

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4.5 ),whenisofunitsize.Moreinterestingly,cf.( 4.6 )withk~QRTWfkKChKkfkK;kURTWfkKCh2KkfkK; 18 ,Lemma3.3].Observethatifisofunitsize,theselocalHRToperatorsaremorestablethanthecorrespondingHDGones.Indeed,whileURTWfdampsperturbationsinfbyO(h2),thecorrespondingHDGoperator,namelyUDGWf,dampsitbyonlyO(h)because(dKh)2=O(1=h). WewillnowdevelopaseriesofintermediateresultstoproveTheorem 4.1 intheremainderofthissubsection. 2.8 )bypartsonameshelementKwithm=,wehave (~QRT+~rURT;~v)K=hURT;~v~ni@K:(4.9) Thereisan~vinRd(K)suchthat~v~n=URTon@Kand(~v;~pd1)K=0forall~pd1inPd1(K)n(thisisobviousfromthewell-knowndegreesoffreedomofthespaceRd(K)).Additionally,byascalingargumentitisimmediatethat Withthis~vin( 4.9 ),weobtainkURTk2@K=(~QRT;~v)K; 4.10 ).

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^Ck^wk^Ksup~v2Pd(^K)j(^w;r~v)^Kj k~vk^K+k^wk^F;8^w2Pd(K);(4.11) foranyface^Fof^K.Thisfollowsbyequivalenceofnorms.Thattherighthandsideindeeddenesanormcanbeseenasfollows:divergenceisasurjectivemapfromPd(K)ntoPd1(K).Henceifthesupremumiszero,then^wisorthogonaltoPd1(K),inwhichcase^wiszeroonceitvanishesonanyface^F(see[ 20 ,Lemma2.1]).Thelemmafollowsbymapping( 4.11 )toanysimplexKandusingstandardscalingarguments. (iv) 59

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2.8 )from( 2.13 )wehave ((~QDG~QRT);~v)K(UDGURT;r~v)K=0 (4.17a)(r(~QDG~QRT);w)K+h(UDGURT);wi@K=h(URT);wi@K forall~v2Pd(K)nandforallwinPd(K).Notethatsincer~QRT=0,thelifting~QRTisinfactinPd(K)n.Hencef~QDG~QRT;UDGURTgformstheuniquesolutionof( 4.17 ). First,letusprovetherstassertion (i) ofthelemma.Indeed,iftakesaconstantvalueon@K,thenitiswellknownthatURTequalsthesameconstant[ 17 ]and~QRT=0,sotherighthandsideof( 4.17 )vanishes.HenceURTUDGand~QRT~QDGalsovanish,thusproving (i) Thestatement (ii) isprovedbythesametechniqueas (i) .TheonlydierenceisthattheanalogousresultfortheRTcaseislesswellknown,soletusrstshowit,namelyURTj@K=j@Kwhend>0andj@Kequalsthetraceofsomev2P1(K).Inlightof( 3.7 ),equation( 2.8 )becomes(~rv;~v)(URT;r~v)=hv;~v~ni@K=(~rv;~v)(v;r~v): (i) ,thesolutionof( 4.17 )vanishesinthiscasealso,andwehaveprovenitem (ii) Next,letusprovetheestimates.Setting~v=~QDG~QRTandw=UDGURTin( 4.17 ),wehavek~QDG~QRTk2K+kUDGURTk2;@K=h(URT);UDGURTi@K: 60

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4.18 )implieskUDGURTk;@KkURTk;@K; 4.12 )followsbytriangleinequalityandLemma 4.2 ApplyingCauchy-Schwarzinequalitytotherighthandsideof( 4.18 ),k~QDG~QRTk2K+kUDGURTk2;@KCh1=2Kk~QRTkKmaxK1=2kUDGURTk;@K 4.2 .Clearly( 4.13 )immediatelyfollows. Itremainsonlytoprove( 4.14 ).LetFmaxdenoteafaceofKwhere=maxK:ThenmaxKkUDGURTk2Fmax=kUDGURTk2;FmaxkUDGURTk2;@KCmaxKhKk~QRTk2K; 4.3 ,weobtainkUDGURTkKChKkBt(UDGURT)kK+h1=2KkUDGURTkFmaxChKk~QDG~QRTkK+hKk~QRTkK; 4.14 )follows.(Thisappliesevenifd=0,inwhichcasetheterminvolvingBtisabsent.)Thuswehaveproveditem (iii) Forthenalitem (iv) ,(JK~QDG;JK~v)K=h;JK~v~ni@Kby( 2.13 );=(~QRT;JK~v)K;by( 2.8 ); 4.15 ),as~QRTisintherangeofJKby( 2.8 ).Theestimate( 4.16 )isthenobviousasorthogonalprojectorshaveunitnorm. 61

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4.1 4.13 )ofLemma 4.4 4.8 ): 4.13 )inplaceof( 4.13 ). Next,consider~QDGWf;UDGWf.Fromtheirdenitions,itiseasytoseethat (~QDGWf;~QDGWf)K+hUDGWf;UDGWfi@K=(UDGWf;f)K:(4.19) LetFmaxdenoteafaceofKwhere=maxK:Then,kUDGWfkKC(hKkBtUDGWfkK+h1=2KkUDGWfkFmax)byLemma 4.3 ,C(hKk~QDGWfkK+h1=2KkUDGWfkFmax)asA~QDGWf+BtUDGWf=0,ChK(k~QDGWfkK+(maxKhK)1=2kUDGWfk;@K)ChK((UDGWf;f)1=2K+(maxKhK)1=2(UDGWf;f)1=2K)by( 4.19 );ChK(1+(maxKhK)1=2)kUDGWfk1=2Kkfk1=2K; 4.19 ),weimmediatelygetthestatedboundfor~QDGWfaswell. Finally,toprove( 4.7 ),westartfromthefollowingeasyconsequenceofthedenitionsof~QDGV~g;UDGV~g: Sinceitisimmediatefromtheabovethatk~QDGV~gkKCk~gkK,itonlyremainstoprovetheboundforUDGV~g.ByLemma 4.3 ,kUDGV~gkKChKkBtUDGV~gkK+Ch1=2KkUDGV~gkFmaxChK(kA~QDGV~gkK+k~gkK)+ChK(maxKhK)1=2k~QDGV~gk;@K;

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4.20 )intheabove. 20 ].Here,asanapplicationoftheestimatesweprovedinx ,weprovetwonewerrorestimatesnotin[ 20 ]. Theproofisquickandeasyonceweusethespecialprojectionof[ 20 ].Theprojection,denotedbyh(~q;u),isintotheproductspaceVhWh,anditsdomainisasubspaceofH(div;)L2()consistingofsucientlyregularfunctions,e.g.,H(div;)\Hs()nHs()fors>1=2: 20 ]: forall1=2
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2.12 )be(~q;u),andthediscretesolutionsatisfying( 2.17 )be(~qh;uh;h).Then,thefollowingerrorestimateshold:k~q~qhk2k~qVh~qk; 20 ]: 20 ,Lemma3.1]. 4.5 20 ](andtheproofiseasy),soweonlyprovetheremainingtwo. Toprove( 4.23 ),weapplyTheorem 2.10 to( 4.25 )ofLemma 4.6 .Then,wendthat"hsatisesah("h;)=(~eh;~QDG)=((Vh~q~q);~QDG) forallinMh.Hence( 4.23 )followsbychoosing="handapplyingtheCauchy-Schwarzinequality. 64

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4.24 ),weapplythelocalrecoveryequation( 2.16 )ofTheorem 2.10 to( 4.25 ),whichgives"uh=UDG"h+UDGV~eh:Therefore,k"uhkKkUDG"hkK+kUDGV~ehkKcKhCk"hkh;K+CdKhhKk~ehkK; 4.1 .SincedKhhKCh1=2K(maxK)1=2,kuuhk2KC(cKh)2k"hk2h;K+C(maxK)1hKkVh~q~qk2K+CkuWhuk2K: 4.9 ,weobtainkuuhk2hb(1)Ck"hk2a+Cb(2)kVh~q~qk2h+CkuWhuk2h: 4.24 )usingthepreviousestimate( 4.23 )for"h. 20 ]underadditionalregularityassumptions.Theonlyregularityrequirementfortheestimates( 4.24 )and( 4.23 )toholdisthat(~q;u)isinthedomainofh,whereastheanalysisin[ 20 ]assumesinadditionthefullregularityconditionsuitableforanAubin-Nitschetypeargument. 3.1 ).Themainresultofthissubsectionisthefollowing.

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4.8 imply(2)hCh2: TheproofofTheorem 4.8 reliesonthetwolemmasbelow.Tostatethem,weneedanadditionalnormpreviouslydenedinNotation 2.9 .NotethatsincekkhisanL2-likenorm,jjjjjjhisanH1-likenorm,andsincefunctionsinMhcanbethoughtofashavingzeroboundaryconditionson@,itisnotsurprisingthatthefollowingPoincare-typeinequalityholds: 24 ,ProofofTheorem2.3]. 66

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24 ],theproofoflemmacanbecompletedinstantlybyjjjjjjh;KCk~QRTkKCk~QDGkK 4.16 )ofLemma 4.4 LetTKbetheaneisomorphismmappingthereferenceunitsimplex^Kone-oneontoK.IthastheformTK(^x)=MK^x+bforsomennmatrixMK.WewillalsoneedthePiolamapKmappingfunctionsonKto^K,denedbyK(~v)=(detMK)1M1K~vTK.Westartbyletting^=TKandrecallingthatthereisafunction~v^inPk(K)nsuchthatr~v^=0;in^K;~v^~n=^m^K(^);on@^K;andk~v^k^KCk^m^K(^)k^K: 13 ]appliedto^m^K(^);orevenbymoreelementaryobservations.Nextlet~v=1K(~v^).BythewellknownpropertiesofthePiolamap[ 13 ],weknowthatr~v=0andh;~v~ni@K=h^;~v^~ni@^K: 4.4a ),weget(~QDG;~v)K=h;~v~ni@K=h^;~v^~ni@^K=(^m^K(^);~v^~n)@^K=k^m^K(^)k2@^K:

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4.8 4.12 )and( 4.16 )ofLemma 4.4 toconcludethatkUDGk2;@KCmaxKhKk~QRTk2KCmaxKhKk~QDGk2K: 4.1 .Thus,theupperboundfollows. Forthelowerbound,wecombinetheestimatesofLemmas 4.9 and 4.10 toobtainkk2hC0jjjjjj2hC0Ck~QDGk2hCC0ah(;); 68

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Settingm=;~v=~QDG;w=UDGin( 2.13 ),wehaveaDGh(;)jK=k~QDGkK+kUDGk;@K=(;~QDG~n)@K(;(UDG))@K: wherethelastinequalitycomesfromascalingargument,andj(;(UDG))@Kj(maxK)1=2h1=2(kk@Kh1=2)k1=2(UDG)k@K:

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19 ]:aIPh(;)=XK(k~QDGk2Kk~QDG+~rUDGk2K+kUDGk2;@K) (4.30)=XK(k~rUDGk2K2(UDG;~rUDG~n)@K+kUDGk2;@K) (4.31)=XK((;~rUDG~n)@K+(;K(UDG))@K): Firstletusprovethelowerbound.ByLemma 4.9 and 4.10 ,wejustneedtoshowk~QDGk2KaIPh(;).From( 4.30 ),itisclearthatitsucestoprove Tothisend,startingwiththefollowingidentity[ 19 ,Lemma2.2] (~QDG+~rUDG;~v)K=(UDG;~v~n)@K;8~v2V(K);(4.34) andalocalscalingargument,wehavek~QDG+~rUDGk2KC(minKhK)1=2kUDGk;@Kk~QDG+~rUDGkK; 4.33 )followsprovidedminKhK1=C2. 70

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4.32 ),aIPh(;)=XK(;~rUDG~n)@K+(;K(UDG))@K)Ch1XK(kk@Kh1=2K)(k~rUDGk2K+maxKhKkUDGk2;@K)1=2Ch1XKkk2@KhK!1=2XK(k~rUDGk2K+kUDGk2;@K)!1=2; Forthisweuse( 4.31 )andj(UDG;~rUDG~n)@KjC(minKhK)1=2kUDGk;@Kk~rUDGkK^C(kUDGk2;@K+k~rUDGk2K); 4.31 ),weseethat( 4.35 )follows.Thustheproofiscomplete. 4.13 ,weputstrongrestrictionsonthechoicesoftofavorthearguments.Butwesuspectthatthisestimate( 4.29 )maystillholdforlessrestrictive.Thatconjecturewillbeleftforfutureinvestigation. .Wewillalsoinvestigatethedependenceoftheerrorsonthepenaltyparameter.WeimplementedtheHDGmethodandperformedexperimentswiththesamemeshesanddomainasinx .Theexacttestsolutionisagainu(x;y)=sin(x)ey=2. 71

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4-1 to 4-6 ,namely,kuuhkL2,k~q~qhkL2andkPuhka,wherePuistheL2orthogonalprojectionontoMh.Asseenfromthosetables,forxed,alldiscretizationerrorsconvergeinO(h)ford=0,andO(h2)ford=1.TheseagreewiththeknowntheoreticalestimatesonHDGerrors[ 20 ,Theorem2.1]aswellasthetheoremweprovedearlier,namelyTheorem 4.5 .Noticehowtherateofconvergenceslowsdownifissetto1=h.ThishasbeenatraditionalchoiceinmanyDGmethods.However,forHDGmethods,itisimportantthatthepenaltynotbesetto1=hasseenfromthelastrowofTables 4-1 through 4-6 Table4-1. DiscretizationerrorkPuhkaofHDG(d=0)forvaryingandh. 1=20.903880.005990.002840.001400.000690.0003410.870650.011860.005660.002790.001390.0006920.814770.023190.011190.005550.002770.0013940.732730.044490.021910.010980.005520.0027780.636610.070420.038530.020530.010670.00544 Table4-2. DiscretizationerrorkuuhkL2ofHDG(d=0)forvaryingandh. 1=20.128230.034820.017410.008710.004350.0021810.121150.034820.017410.008710.004360.0021820.110200.034840.017430.008720.004360.0021840.096260.034890.017470.008750.004370.0021980.083050.035060.017620.008850.004430.00222 NextwetestandcomparetheperformanceamongthecontinuousGalerkin(CG)method,theHRTmethodandtheHDGmethodwithdierent.Thecomparisonsarebasedonthreekindsofdierentdiscretizationerrors,namely,theprimalsolutionerrorkuuhkL2,theuxsolutionerrork~q~qhkL2,andthetracesolutionerrorkhka. 72

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Discretizationerrork~q~qhkL2ofHDG(d=0)forvaryingandh. 1=21.215710.005930.002830.001390.000690.0003511.127910.011590.005590.002780.001390.0006920.986720.022160.010940.005490.002760.0013940.792810.040780.020930.010730.005460.0027680.580040.070410.038540.020540.010670.00545 Table4-4. DiscretizationerrorkPuhkaofHDG(d=1)forvaryingandh. 1=20.001700.000470.000123.2e-058.0e-062.0e-0610.002140.000580.000153.8e-059.5e-062.4e-0620.003320.000870.000225.6e-051.4e-053.5e-0640.005880.001530.000399.8e-052.5e-056.2e-0680.010840.002890.000740.000194.7e-051.2e-05 Table4-5. DiscretizationerrorkuuhkL2ofHDG(d=1)forvaryingandh. 1=20.003200.000790.000194.9e-051.2e-053.1e-0610.002640.000660.000164.1e-051.0e-052.6e-0620.002470.000620.000163.9e-059.7e-062.4e-0640.002430.000610.000153.8e-059.5e-062.4e-0680.002460.000610.000153.8e-059.5e-062.4e-06 Table4-6. Discretizationerrork~q~qhkL2ofHDG(d=1)forvaryingandh. 1=20.004540.001150.000297.3e-051.8e-054.6e-0610.004720.001200.000307.7e-051.9e-054.8e-0620.005380.001390.000358.9e-052.2e-055.6e-0640.007250.001920.000490.000133.2e-057.9e-0680.011270.003150.000840.000225.5e-051.4e-05 73

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4-1 .xaxisrepresentsthelogarithmofthedegreesoffreedom,andyaxisrepresentslog(kuuhkL2).TheCGmethod(d=1)usestheniteelementspaceconsistingofcontinuouspiecewiselinearfunctionstoapproximatetheexactsolution.Incontrast,TheHDGmethod(d=1)usestheniteelementspaceofdiscontinuouspiecewiselinearfunctions.Welistthreecasescorrespondingdierent(1/10000,1and10000)forthelatter.TheresultsshowthattheCGmethodyieldsthebestapproximationsfortheprimalvariableu.However,theCGmethoddoesnotyieldgooduxes(whichisonetheprimarymotivationsforconstructingmixedandhybridDGmethodsbasedonthedualform). Next,wecomparek~q~qhkL2amongtheHRTmethod(d=0)andtheHDGmethod(d=1)inFigure 4-2 .Bothmethodsuselinearfunctionstoapproximatetheuxsolution~q,andthedierenceisthatthelatterusesthefullspaceoflinearpiecewisefunctions,whiletheformerusesonlyasubspaceofit(thenormalcomponentsneedtobecontinuousacrosstheinteriorfaces).WeseethatwhiletheperformanceoftheHDGmethodremainsrelativelyunalteredasgoesfromunitsizetozero,itdeterioratesifapproachesinnity. WeproceedtocomparekhkaamongtheHRTmethod(d=0)andtheHDGmethod(d=0)inFigure 4-3 .Bothmethodsusethespaceofpiecewiseconstantfunctionstoapproximatethetracesolution.TheHDGmethodsseemtohavelargererrorsoncoarsermeshes,buttheirperformancequicklybecomecomparabletotheHRTmethodwhengoingtonermeshes. Inallthesegraphs,itmayseemthatthesmalleris,thebettertheapproximationgets,whichconrmstheresultsasindicatedinTable 4-4 .Moreover,thebiggeris,theclosertheHDGcurvegetstotheCGcurve.Asamatteroffact,theCGmethodcanbeconsideredasalimitingcaseoftheHDGmethodbylettinggotoinnity[ 19 ]. 74

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ComparisonoflogkuuhkL2amongtheCGmethod(d=1),theHRTmethod(d=0)andtheHDGmethod(d=1)withdierent. 75

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Comparisonoflogk~q~qhkL2amongthetheHRTmethod(d=0)andtheHDGmethod(d=1)withdierent. 76

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ComparisonoflogkPuhkaamongthetheHRTmethod(d=0)andtheHDGmethod(d=0)withdierent. 77

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ThemaincontributionofthischapterisaprovablyecientmultigridalgorithmfortheHDGmethod.WhilethealgorithmitselfisastraightforwardgeneralizationofwhatwepresentedinChapter 3 fortheHRTmethod,itsanalysisismoreinvolvedduetocomplexities(suchas-dependence)intheestimatesfortheHDGmethod.Furthermore,werelaxtheregularityassumptionsontheoriginalboundaryvalueprobleminthischapter,resultinginamoregenerallyapplicableconvergencetheorem,butnecessitatingmoretechnicalproofs.Theresultsofthischapterwillbepublishedin[ 16 ]. WeusethesamealgorithmsettingasintheHRTcase,specicallywecontinuetouseAlgorithm 3.2 .OurmaingoalistoproveaconvergenceresultforthisalgorithmappliedtotheHDGcase.Again,weusetheabstractmultigridtheoryin[ 10 ],sothattheconvergenceproofreducestothevericationofCondition 3.4 throughCondition 3.6 .NotethattheproofofCondition 3.6 remainsthesameforHRTandHDG(smoothingsubspacesareidentical),soitleavesusonlytherst2conditionstoverify. RecallingthesettingofAlgorithm 3.2 ,notethattherearemanysimilaritieswhenitisadaptedtotheHDGsetting.Asinx ,werequireamultilevelhierarchyofmeshesandspacesandthatthemeshThinwhichthesolutionissought,isobtainedbysuccessiverenementsofacoarsemeshT1asinx .TheHDGmultilevelspacesMkareidenticaltothoseintheHRTsetting.Themaindierenceisthatthebilinearformak(;)andtheresultingmatrices(andoperatorsAk)arenowobtainedfromtheHDGmethod.ThesmoothersdependonAk,sotheymustalsobemodied.TheintergridtransferoperatorIkremainsthesame. Thefollowingisthemainresultofthischapter.Itisprovedunderthefollowingassumptions. 78

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Theboundaryvalueproblem( 2.12 )admitsthefollowingregularityestimateforitssolution: forsomenumber1=21=2.Notealsothatonce( 5.1 )holdswiths>1=2,wecanapplytheprojectionhto(~q;u). 3.2 modiedtoapplytotheHDGmethodasdescribedabove.SupposeAssumption 5.1 holds.Assumealsothatd1.Thenthereexistsapositive<1,independentofthemeshsizehJ,suchthattheerrorreducingoperatorEDGJofthealgorithmsatises0aJ(EDGJu;u)aJ(u;u);forallu2MJ: 5.2 maystillhold.Butthetheoreticalproofrequiressometediousmodications,whicharepostponedforfutureresearch.Throughouttherestofthischapter,wealwaysassumed>0. 79

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. Proof. 4.10 ,soitonlyremainstoprovetheupperbound.Forthis,notethatah(;)=k~QDGk2+kUDGk22k~QDG~QRTk2+2k~QRTk2+C(1+maxKh)k~QRTk2by( 4.12 )C(1)hk~QRTk2by( 3.24 ): 24 ]. 80

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4.4 (ii) .Ifd=0,itfollowsfromthedenitionof~QDG(IJw),namely( 4.4a ),whichreducesto(~QDG(IJw);~r)K0=hIJw;~r~ni@K=hw;~r~ni@K=(~rw;~r)K 5.2 )follows. Toprove( 5.3 ),weuse( 5.2 )inthedenitionofaJ(;)toget andobservethatthelasttermvanishesbecauseofLemma 4.4 (ii) Toprove( 5.4 ),weagainuse( 5.2 ):(~QDG+~rPJ1;~rw)=(~QDG;~rw)+(~rPJ1;~rw)=(~QDG;~QDG(IJw))+aJ1(PJ1;w)=(~QDG;~QDG(IJw))+aJ(;IJw)=h(UDG);UDG(IJw)wi: 4.4 (ii) andweget( 5.4 ). Finally,toprove( 5.5 ),aJ(IJPJ1;)=aJ(;)aJ1(PJ1;PJ1)=(~QDG;~QDG)(~rPJ1;~rPJ1)+kUDGk2=((~QDG+~rPJ1);(~QDG~rPJ1))+kUDGk2: 5.4 )wheneverd>0. 3.4 ) 81

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2.13 ),h(UDG(IJw)w);UDG(IJw)w0i@K=0: Applyingalocaltraceinequality, wherewehaveusedFriedrichsinequalityafterchoosingw0tobethemeanofwonK.Thisproves( 5.7 ). Inequality( 5.8 )followsfrom( 5.7 )bytriangleinequalityonceweprovethat SincethemeanofIJwwvanishesoneachfaceFof@K,applyingFriedrichsinequality,wehave 82

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5.13 )implies( 5.12 ). For( 5.9 ),weuseastandardlocalestimateforlinearfunctions,CkUDG(IJw)wkKhKk~r(UDG(IJw)w)kK+h1=2KkUDG(IJw)w)kF 5.10 ).Thenwenishtheproofasin( 5.11 ). 5.6 5.5 :aJ(IJv;IJv)=(~rv;~QDG(IJv))by( 5.3 )=(a~rv;~rv)by( 5.2 ): 5.2 )ofLemma 5.5 ,hencewecanput~QDG(IJv)=~rvinthedenitionofaj(;)togetaJ(IJv;IJv)=(~rv;~rv)+kUDG(IJv)IJvk2aJ1(v;v)+CmaxKhJk~rvk2; 5.8 )ofLemma 5.7 .Thisprovesthetheorem. 83

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3.5 ) 5.6 5.6 tothelastterm. Nextwedeneanewoperator,S,denedoneachedgeofthemesh.LetbetherestrictionofafunctioninMhon@KforsomemeshelementK.LetFidenotethefaceofKoppositetotheithvertexofK.ThendeneSKiinPk+1(K)by and,consideringallthen+1facesofK,dene(;)S=XK2Th1 5.14a )and( 5.14b )uniquelydeneaSKiinPk+1(K).Further-more,forallinMh,UDGjK=Wh(SKi);

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5.14 )formsasquaresystemforSi,toshowthatithasauniquesolution,itsucestoshowthattheonlysolutionwhentherighthandsidesarezeroisthetrivialsolution.Thatthisisindeedthecaseisanimmediateconsequenceof[ 20 ,Lemma2.1].Theidentity( 5.15 )isobviousfrom( 5.14b ).Letusprovetheremainingassertions. Weprove( 5.16 )byascalingargument.Tothisend,consideraxedreferencesimplex^K,withanarbitrarilychosenface^F,anddene^;^qbyh^;^q;i^F=h^;i^Fforall2Pk+1(^F);(^;^q;v)^K=(^q;v)^Kforallv2Pk(^K): SummingoverallfacesFi@K, wherewehaveusedTheorem 4.1 .Now,toobtain( 5.16 ),weneedonlysumoverallK. Toprove( 5.17 ),rstobservethatiftakesaconstantvalueontheboundaryofsomemeshelement@K,thenSKi.ThisisbecauseUDGbyLemma 4.4 (i) ,sothefunctionsatisesboththeequationsof( 5.14 ).Therefore,bytheuniquesolvabilityof( 5.14 ),SKi.Aconsequenceofthisfactisthatforany,wehave~r(SKi(mK())=0 85

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2.9 ).Therefore,k~r(SKi)kL2(K)=k~rSKi(mK())kL2(K);Ch1KkSKi(mK())kL2(K)(byaninverseinequality)Ch1K(1+(maxKhK)1=2)h1=2KkmK()kL2(@K)(by( 5.18 ))C(1=2)hjjjmK()jjjh;K; 5.17 )followsfromLemma 4.10 Next,wedeneamap7!~fromMJintoMJasfollows.First,giveninMJ,letbetheuniquefunctioninMJsatisfying (;)S=ah(;);82MJ:(5.19) ThisequationisuniquelysolvableforinMJ,becauseiftheright-handsideiszero,thenby( 5.16 )ofLemma 5.9 ,wehavethat=0.Next,letf=UDGanddene~2MJtobetheuniquesolutionoftheequation 5.21 )and( 5.22 )aresimilartotheproofof 5.10 .TheonlydierenceisthatwenowusetheestimatesofLemma 5.9 .Toprove( 5.21 ),rstobservethatf=WhSKi

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5.15 )ofLemma 5.9 .Therefore,kfk2=XK2Th1 5.21 ).Moreover,kk2S=sup2MJ(;)S 5.19 ),=sup2MJ(AJ;)J 5.16 )ofLemma 5.9 ,CkAJkJ; 5.21 ). Toprove( 5.22 ),letusrstnotethatwecanrewrite( 5.19 )and( 5.20 )asfollows:a(;)=XK2Th1 5.9 ,wherebyf=UDG=Wh(SKi)onanyelementK.Subtracting,andsetting=~,wegetk~k2a=XK2Th1 5.21 )and( 5.17 )ofLemma 5.9 .Cancelingthecommonfactorabove,weobtain( 5.22 ). 87

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5.23 ).Tothisend,givenanyinH10(),letJ1inMJ1denoteafunctionsatisfying Suchapproximationsarewellknowntoexist[ 34 ].Then,kfkH1()=sup2H10()(f;) 5.24 );(f;J1UDG(IJJ1))=0;byLemma 4.4 (ii) ,ifd>0,(f;UDG(IJJ1))=aJ(~;IJJ1)by( 5.20 );k~kaCdk~rJ1k;byTheorem 5.6 thetermsinthesupremumcanbeboundedaccordinglytogetthatkfkH1()Cdk~ka+ChJkfk: 5.21 )impliesthatkfkCkAJkJ,andsince( 5.22 )impliesk~kakka+ChJkAJkJ;

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4.8 .Hencewenishtheproof. Now,let~ubetheuniquefunctioninH10()thatsolves (~r~u;~rv)=(f;v);8v2H10();(5.25) andlet~uJ1betheuniquefunctioninMJ1satisfying (~r~uJ1;~rv)=(f;v);8v2MJ1:(5.26) Proof. by( 5.20 ).Now,ifd>0,byLemma 4.4 (ii) ,weknowthatUDG(IJw)w=0.Hencewehave(~rPJ1~;~rw)=(f;w)8w2MJ1; 89

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5.1 ,foranyinMJ,k~QDG+~rPJ1k2Ch2JkAJk2J+Ch2saJ(;)1skAJk2sJ: 5.10 ,kt2kk~PMh~ukak~qVh~qkcby( 4.23 )ofTheorem 4.5 ,Chs(j~qjHs()+maxKj~ujHs())by( 4.21 ), where~q=~r~u.Fort3,weuse( 4.5 )ofTheorem 4.1 togetthat 90

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Since~uJ1isastandardGalerkinapproximation[ 14 ]of~u,wehave Furthermore,astandarddualityargument[ 14 30 ]yields Summing( 5.29 )overallelementsandusing( 5.30 )and( 5.31 ),wecanestimatek~u~uJ1kh.Returningto( 5.28 )andusingthisbound,wehavekt3kC((1)h)1=2hsjujH1+s(): 5.2 )ofLemma 5.5 ,kt5kChJkAJkJ;ifd>0;kt6kCk~kaChkAJkJ;byLemmas 5.8 and 5.10 Combiningtheseestimatesforallti,weobtaink~QDG+~rPJ1k2Ch2JkAJk2J+C(1)hh2sJj~uj2H1+s()+Ch2s(j~qjHs()+maxKj~ujHs())2Ch2JkAJk2J+C(1+maxK2)h2skfk2H1+s() bytheregularityassumption( 5.1 ).SinceH1+s()isaninterpolationspace[ 4 ]inthescaleofintermediatespacesbetweenH1()andL2(),weknowthatkfk2H1+s()kfk2(1s)H1()kfk2s:

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5.23 )and( 5.21 )ofLemma 5.10 .Notethatd2(1s)C(1+(maxKh)1s).Returningto( 5.32 )andusingtheresultingbound,wethenobtaintheestimateofthelemma. 6 10 ]toholdforallk0: 5.12 ,letusrstinvestigatetheremainingterminvolvingUDG.Tothisend,thefollowinginequalitywillbehelpful:kUDG(IJPJ1)(IJPJ1)k;@KCp 4.12 )ofLemma 4.4 .ByLemma 3.7 ,wealsoknowthat~QDG(IJPJ1)=~rPJ1: wherethelastinequalitywasbecause~QDG+~r(PJ1)=JK(~QDG+~r(PJ1)):

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5.33 )togetak((IIkPk1)v;v)k~QDG+~rPJ1k2c+CmaxKhKk~QDG+~r(PJ1)k2KCk~QDG+~rPJ1k2c 5.12 withs=1. 3 .Weusethesamemeshandproblemsettingsasinx .Discretizationerrorsintheniteelementanalysisarepreviouslyreportedinx ,sointhissectionwewillonlyreportobservationsonthemultigridconvergence. Inordertocheckthattheiterationerrorinthemultigridcycleconvergesatarateindependentofthemeshsizeh,wedesigntherstexperimentinasuchwaythatweknowtheexactsolution,asshownbelow: 1. Wesetb=0,i.e.,trytosolvetheequationsystemAx=0withtheexactsolutionx=0. 2. Theinitialguessx0intheMGiterationoneachmulti-levelspaceMk;k=0;1;:::;J,issettobeIJIJ1I1v,wherevisthelinearcombinationoftheglobalbasisfunctionsinM0withthecoecients(1;:::;1)(i.e.,vtakesthevalue1forallinteriornodes,islinearonallmeshelements,iscontinuousacrosselements,anddecreasesto0ontheboundary). 3. WeuseGauss-Seideliterationasthesmoother.Thenumberofsmoothings,mk,equals2JkforHDG(d=0)and42JkforHDG(d=1),onmk. 4. Westopiterationseitherwhenkxixi1ka108kx0xka,orwhentheiterationcountreachesthelimit99. 93

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5-1 and 5-2 forvariousvaluesofillustratestheuniformconvergenceofthemultigriditerationweprovedinTheorem 5.2 .AswementionedinRemark 5.3 ,theexperimentresultsalsoindicatethatthemultigridV-cycleconvergesuniformlyeveninthecaseofd=0.Perhapssurprisinglythecase=1=halsoyieldedgoodmultigridconvergence(unlikethecaseofthediscretizationerrorconvergence{seex ). Table5-1. Thenumberofmultigriditerations(mk=2Jk)forHDG(d=0)withdierent.*indicatesdivergenceoraniterationcountexceedingthelimit. mesh012345 Table5-2. Thenumberofmultigriditerations(mk=42Jk)forHDG(d=1)withdierent. mesh012345 94

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Thismainproductofthisstudyisafastandecientiterativesolutiontechniqueforhybridizedniteelementmethods.Weadoptedthemultigridframeworkfordesigningsuchsolvers.Butattheinitialstages,wewerefacedwiththedicultiesarisingfromthefactthatthemultilevelspacesinallhybridizedmethodswerenon-nested.Infact,themultilevelfunctionsdidnotevensharethesamedomain.Wedevelopedamultilevelframeworkinwhichthisdicultycanbeovercome.Withinthisframework,weanalyzedtwospecicmethodsforwhichweareabletoprovepreciseresults.However,beyondtheregimeofproofs,theframeworkgivesonethekeyingredientstoformulateamultigridalgorithmforanyhybridizedmethod.Withtheinsightsfromournumericalexperiments,weputforththesetechniqueswithgoodcondence,evenfortheotherhybridizedmethods. Letussummarizethemaintheoreticalresults: 1. ConsideringthewellknownHRTmixedmethodrst,weprovedtheuniformconvergenceofamultigridV-cyclealgorithm.Althoughmultigridforthemixedmethodhasbeenstudiedpreviouslybymanyauthors,ouralgorithmisappliedtothecondensedhybridbilinearformandisthusthemostusefulpractically.Aproperlydesignedintergridtransferoperatorwasthekeyingredientthatopenedtheavenuetowardsasuccessfulalgorithm.Wealsoprovidednegativeresultsthatcertainseeminglyplausibleintergridoperatorsdonotwork. 2. Next,westudiedtherelativelyrecentclassofDGmethodscalledHDGmethods.Weprovednewerrorestimates.WeshowedthattheconditionnumberoftheHDGmatrixgrowslikeO(h2)asmeshsizehtendstozero.ThetechniquesdevelopedhereareanticipatedtohaveutilityinotherDGmethodsofthehybridtype. 3. Finally,weshowedthatamultigridalgorithmgaveanecientsolverforthematrixsystemsarisingfromtheHDGmethod.Werigorouslyprovedaconvergencerate 95

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Wehavesetapartforfutureresearchanumberofquestionsthataroseduringthisstudy.Forexample,althoughwehavefullproofsonthemultigridconvergencefortheHDGmethodwithd>0,thed=0partneedstobeinvestigatedfurther.Numericalexperimentshavestronglysuggestedthatouralgorithmalsoworkforthatcase.TheHDGd=0caseismorethanofacademicinterestasitusesspacessmallerthaneventhelowestordermixedmethod. ExtensionstoothermethodslikeHIParealsoofinterest.WehavealreadyprovedaconditionnumberestimatefortheHIPmethod(Theorem 4.13 ).However,errorestimatesandmultigridconvergenceboundsremainunknown. Finally,themodicationofadaptiveniteelementtechniquestohybridizedmethodsisaninterestingquestionofsomesignicance.AsHDGmethodsaregainingpopularity,severalgroupsaretryingthemoutforpracticalproblems.Treatmentofpracticalproblemsrequirenotonlyecientsolvers,butalsoadaptiverenementsandaposteriorierrorestimatorstoautomaticallyindicatewhichregionstorene.TheseareissuesthatcurrentlyremainopeninthearenaofHDGmethods. 96

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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] 97

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[17] [18] ,Erroranalysisofvariabledegreemixedmethodsforellipticproblemsviahybridization,Math.Comp.,74(2005),pp.1653{1677(electronic). [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] 98

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[32] [33] [34] [35] 99

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ShuguangTanwasborninChina.HereceivedhisB.S.inInformationSciencefromthePekingUniversityin2001.HecametoU.S.A.in2003forcontinuinggraduatestudy,andreceivedhisM.SinStatisticsandPh.D.inMathematicsconcurrentlyfromtheUniversityofFloridainthesummerof2009. 100