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Stabilized Crouzeix-Raviart element for the coupled Stokes and Darcy problem

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Stabilized Crouzeix-Raviart element for the coupled Stokes and Darcy problemfor,the,and,Darcy

    Stabilized Crouzeix-Raviart element for the

    coupled Stokes and Darcy problem App1.Math.Mech.Eng1.Ed.31(3),393-404(2010)

    DOI10.1007/s10483010-0312-z

    ?ShanghmUniversityandSpringer-Verlag

    BerlinHeidelberg2010

    AppliedMathematics

    andMechanics

    (EnglishEdition)

    StabilizedCrouzeix-RaviartelementforthecoupledStokesand Darcyproblem

    MinfuFENG(冯民富),RuishengQI(祁瑞生)

    RuiZHU(朱瑞),Bing-taoJU(鞠炳焘).

    (1.DepartmentofMathematics,SichuanUniversity,Chengdu610064,P.R.China; 2.DepartmentofInformationTechnology,CNOOCEnergyTechnologyand ServicesLimitedBeijingBranchCompa~ay,Beijing100027,P.R.China) (CommunicatedbyZhe-weiZHOU)

    AbstractThispaperintroducesanewstabilizedfiniteelementmethodforthecoupled StokesandDarcyproblembasedonthenonconformingCrouzeix-Raviartelement.Opti

    malerrorestimatesforthefluidvelocityandpressurearederived.Anumericalexample ispresentedtoverifythetheoreticalpredictions.

    KeywordsBeavers-JosephSaffmancondition,massconservation,balanceofforce coupledStokesandDarcyproblem

    ChineseLibraryClassification0242.21

    2000MathematicsSubjectClassification65N30,65N15

    1Introduction

    OurresearchinthispaperbeginswiththemodelofthecoupledStokesandDarcyproblem.

    ThemodelisbasedontheStokesequationinthefluidcoupledacrossaninterfacewiththe Darcyequationforthefiltrationvelocityintheporousmedia.Itiswidelyappliedinindustries. Forexample,wecanuseittosimulatetheprocedureofthepollutantsdischargedintostreams, lakes.andriversandmakingtheirwayintothewatersupply,andtodeterminewhetheradam builtoverariverissolidfromtheamountofwaterfiltrationthroughthedam. Recently,thecoupledproblemhasbecomearesearchtopicfrommathematicsandnumerical analysisviewpoint[1J.However.therestillexistseveralmaindifficultiesinthesolutiontothe coupledproblembyusingthefiniteelementapproximation.First,thecoupledproblemownsa commonrestraintwiththeStokesequationandtheDarcyequationthatvelocityandpressure spacesmustsatisfytheinf-supconditionofBabugka

    7JandBrezziIsJ.Second.finiteelement

    discretizationswithchoicesofspaceintworegionsaredifferentformanymethods~5,9-1o].Thus,

    differentspacesintworegionsleadtomanydifficultiesinmathematicaltheoryandnumerical analysis.Ontheotherhand,inI1,41,itwasarguedthatfiniteelementdiscretizationsbased onthesamefiniteelementspaceforbothregionswillhavesomeadvantageswithrespect toimplementation.However.theconstructionoftheelements[1,4Jisrathercomplicatedand ReceivedJun.12,2009/RevisedJan.10,2010

    ProjectsupportedbytheScienceandTechnologyFoundationofSichuanProvince (No.05GG006-0062)

    CorrespondingauthorMin-fuFENG,Professor,Ph.D.,E-maihfmf@wtjs.cn

    Min-fuFENG,RuishengQI,RuiZHU,andBing-taoJU

    thereforelessattractiveforengineeringpurposes.Third.thetechnicaldimcultyishowtotreat theinterfacecondition,especially,themassconservationcondition.Hence,itisinterestingto useonesimpleelementthatcouldbeimplementedeasilytoapproximatethecoupledproblem. AseeminglypromisingcandidateforsuchanelementistheCrouzeix-RaviartfCR1element, whichhasthefollowingniceproperties:incombinationwithpiecewiseconstantpressure.it satisfiestheinf-supconditionandelement

    wiseconservationofmass.Hence,theCRelement

iswidelyusedinmanyproblems,suchastheDarcy

    Stokesproblem[11J,theStokesproblem[12J,

    andtheelasticityproblem[13141.

    In111,inordertoensureconvergenceintheDarcy1imit

    andfuifilIKorn'sinequalitiesfortheDarcyStokesproblemthatisnotaninteffaceproblem.

    asimilarstabilizationlike131isintroduced.ForthecomplexityofthecoupledStokesand Darcyproblem,whentheCRelementisused,besidesthosedifficultiesintheDarcyStokes

    problem.thereareotherdimcultiesintreatingtheinterfacecondition.especially,inthemass conservationcondition.Inthispaper,weapplytheCRelementtothecoupledStokesand Darcyproblemwithanaddedstabilization,whichisalsosimilartotheoneinf131.Inorder totreattheinterfaceconditionwell,anewinterpolationthatisnotaCRinterpolationonthe entiredomainisintroduced.Wegivetheproofofstabilityandconvergenceofourmethod. Finally,wepresentanumericalexampletoshowtheperformanceofthemethodonthecoupled problem.

    2Modelofproblemandnotation

    LetQbeapolygonaldomaininR2,whichissubdividedintosubdomainsQ,andQ.,J (i=1,2,,ml;J=l,2,,m2)(seeFig.1).ThesubdomainsQ,.andQ,.areassumedto

    beboundedconnectedpolygonaldomainssuchthat

    nQ,=0,122,inQ,=0fori?J

    m1

    Q,'nQ.,.:,Q1=UQ,Q2==l

    DenotebyFijtheinterfacebetweentwosubdomains,andQJ.andletF:UFijand Fi=0fh/r.Denotebyu=(Ul,2)thefluidvelocityandbyP:(Pl,p2)thefluidpressure, where=InandPi=pIn.

    Fig.1Thediagramofthedomain

    WeassumethattheflowindomainQ1istheStokesequation.Therefore,thefollowing equationsaresatisfied:

    =

    flinQ1,

    (1)

U

    n1.r

    lI

    

    2.

    ,_IIJ'l?【

    StabilizedCrouzeix-RaviartelementforthecoupledStokesandDarcyproblem395

    wheree(u1)isthestraintensordefinedbye(u1)=l(Vul+w1). IntheregionQ2,theflowpressureandvelocitysatisfythethesinglephaseDarcyflow

    equations:

    wherekisthesymmetric,positivedefinite

    representthebodyforces,>0denotesthe

    normaltoF2.

    Q2,

    tensorboundedbelowandaboveuniformly,lt

    viscosityofthefluid,andnistheoutwardunit

    Asthepressureisuniqueuptoanadditiveconstant,weassume l/

    pdx=0

    l,

    LetnandbetheunitnormalandtangentialvectorstoFoutwardofQ1,respectively.The

    conditionsattheinterfaceFare

    1-n=u2?nonF,

    2pn?E(U1)?n=PlP2onF,

    2#n.E(1).7-=一百11.7-onr

    (4)

    (5)

    (6)

    Here,(4)representsthemassconservation,(5)representsthebalanceforce,andthecondition

    BeaverJosephSaffmanlaw(6)isthemostacceptedcondition[71andincludesafriction

    constantk>0thatcanbedeterminedexperimentally.

    WeintroducetheHilbertspaces

    and

    :={?H0(div,Q):ln?[H(Q1)].,vlr=0),

    :={?H0(div,Q):ln?【H(Qt)],Ir=0},

    Q:=(Q)={qEL2(Q):qdx=

    Fori=1,2,letbeanondegeneratequasiuniformtriangulationofQt,rbetheunionof

    theboundaries(exceptsomeboundariesbelongingtoF)ofallelementsin,andhidenote themaximumdiameterofelementsin.Weassumethemeshesatinterfacesinthesense:

    anyedgee?OKnF,whereK(?)belongstooneandonlyoneelementK(?).We introducethenonconformingCrouzeix-Raviartfiniteelementspace:

    :=:I?[P1()V?靠n;

    ?

    ds=.,e?Q.,

    lt..lv]ds

    e

    wherethejumps[V]andV?nonafaceearedefinedby 0,e?(ruF)/aQ2;

    M?:=!,

    e?,

    C:

    m

    +n

    ,????,,???

    396MinfuFENG,RuishengQI,RuiZHU,andBingtaoJU

    and

    :

s~m+(v(x+

    

    sn)-v(

    ,

    nifr

    wherenisanormalunitvectoroneandX?e.Ife?aQ.wechoosetheorientationofnto beoutwardwithrespecttoQ;otherwisenhasanarbitrarybutfixedorientation.Further,we

    introducethefollowingspaces:

    IKePo(K),dx=0,

    :?PI(K).,V,]ds=.,

    X=,i=1,2,

    e?哪弧),_1'2,

    whereXmaybeVhorV.

    Lemma2.1AssumeFij(i=1,2,,ml;J=1,2,,Irt2)arepolygonalinterface8of 21andQ2.Then,thereexistsaninterpolationoperatorRhfromVto. ProofLetrdenotetheCrouzeix-Raviartinterpolation rhi:[H(Qt)],i=1,2,

    andlet

    Rhly~=It,i=1,2

    Obviously,RhisaninterpolationfromVtoVhonlyif Rhv.n]ds=.,?VeEF

    Infact,

    Rhv,n]ds=s?n=n=ds_0jVeEF

    wherenisanormalvectorone.

    Remark2.1ThoughRistheCrouzeix-RaviartinterpolationinthedomainQt,Rh

    isnottheCRoneintheentiredomainfromthedefinitionofVh. LettingthediscretelydivergencefreespaceZhas

    Qh) Zh:={vh?Wh:(?Vh,qh):0,V帆?

    weshallshowthatthespacesYhandQhsatisfytheinf-supconditionforthecoupledproblem,

    andhenceZhisnonempty.

3Finiteelementformulation

    Inordertoformulateourfiniteelementmethod,wefirstintroducetheweakformulations

    of(1)(6).

    Let

    2e():e()d+k-lu2"v2dx+#a(1.7.)(1.T)ds, =

    StabilizedCrouzeix.RaviartelementforthecoupledStokesandDarcyproblem397

    whereE:E=?.

    Considerthebilinearform

    [,p),(V,q)=0(,V),?)n+(q,?)

    Then,theweakformulations(1)(6)taketheform:find(,P)?WQsuchthat B[(,p),(V,g)]=(f,V)+(g,g),V(",q)?W0Q,(8)

    wheretIQ.=and9I2t=gi.Notethatallthefreeinterfaceconditions(4)(6)areexpressed

    weakly.

    Weintroducethefollowingbilinearformonwhichwewillbaseourfiniteelementmethod:

    where

    and

    B^[(,p),(V,)]=ah(U,)+b(v,P)b(,q)+J(,),

    ).E(#k-lu2"v~dx+)ds,

    

    (h=--p

    (,)sr

    where>0and>0are

    Weproposethefollowing

    stabilizationparametersandh. finiteelementformulation:find =

    IeJ.

    n]ds

(,Ph)?VhQsuchthat

    B^[(u,ph),(Vh,qh)]=(f,Vh)+(g,qh),V(vh,qh)?Q^

    Remark3.1Unlikesomestabilizedterms[18-20]fortheStokesproblems,thetermj(u,V)

    isusedtocontroltherigidrotationwhichcausesalackofcoercivityfortheCrouzeixRarvriant

    approximationratherthanovercometheinf-supconditionrestraint.

    Lemma3.1ForVu?Zh.ehave

    ProofForV?,weget

    .

    vlK=0,VK?靠U

    o

    /KV.vdx=

    als_0

    Thus,?V?Q^.Takingq=?Vyields

    Thelemrnafollows

    0:|

    

    I.vl.dx=meas(K)lV:..?,

    ??u?

    MinfuFENG,RuishengQI,RuiZHU,andBing-taoJU 4Stability

    Inthissection,weshowtheinf-supstabilitywiththesuggestedjumpterm

    Wledefinethenorm

    (,p)lI:=.h(,)+J(,u)+IlV?z1I:+lIplI3

    Theorem4.1Forll(Vh,qh)llB?0,thereholds

    sup

    (Vh,qh)cvh×Qh

    ll(Vh,qh)lls"".(11)

    where8isaconstantindependento{themeshsize. ProofWewillprovethisinf-supconditionintwosteps.First,weprovethatthereexists

    (Wh,qh)?VhQhsuchthat Then,weprove

    cLL(~h,Ph)Ll~[(,),(Wh,).

    (^,%)IlBcll(Uh,ph)ll- Step1Wehave

    [(uh,),(,ph)]=ah(~h,Uh) ForanyPh?Qhc3(Q),thereexists?[(Q)]suchthat

    Since

    wefind

    

    V?Vp=Ph.

    IIVpll1cllPhllo

    (?Vp,1)h=(?RhVp,1)h, Bh

    [(Zth,Ph),(,.)lab(Uh,Rhlj(uh,Rh

    Further,wehave

    ?

    h)J(,"'

    andbyuseofthetraceinequality

    u)+:J(Rh,_R),u)+J(,),

    ?c()+hKIIK)),V?日()

    (see[21),wefind

    h,Rhvp)=(Rh,Rh

    (1..Rhll(K)+IlRhll(K)) Ilvpll~-<3

    (12)

    (13)

    (14)

    (15)

    (16)

(17)

    ?

    C

    <

    StabilizedCrouzeix-RaviartelementforthecoupledStokesandDarcyproblem399

    W_ealsohave

    

    1

    .(h,)

    Consequently,wehave nh(Uh~"lth)+1n^(Rh,R^) "lth~Uh)+

    7B^[(^,ph),(Uh,m)]+Bh[(Uh~Ph),(1,.)] (一号)h(Uh~Uh)+(一号)h)+(一鲁一笔). IfwechooseE1

    qh="t'Ph+?

    =2c,

    乱?.

    (18)

    C22,and7=?,(12)followswithWh=7h1RhVpand Step2Usingthesameargumentsoncemore,weimmediatelyfind

    (Wh,qh)JJB

    Theproofiscomplete 5Convergence

    f'~Uh--1Rh~ItV~Uhl +^,ItV~Uh

    I

    crl(Uh,Ph)lIB

    Infact,theproblem(10)isequivalenttotheproblem:find(Uh,Ph)?YhQsuchthat

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