A POSTERIORI ERROR ESTIMATE FOR THE H(div) CONFORMING MIXED FINITE ELEMENT FOR THE COUPLED DARCY-STOKES SYSTEM

Transcription

1 A POSTERIORI ERROR ESTIMATE FOR THE Hdiv CONFORMING MIXED FINITE ELEMENT FOR THE COUPLED DARCY-STOKES SYSTEM WENBIN CHEN AND YANQIU WANG Abstract An Hdiv conforming mixd finit lmnt mtod as bn proposd for t coupld Darcy-Stoks flow in [30], wic imposs normal continuity on t vlocity fild strongly across t Darcy-Stoks intrfac Hr, w dvlop an a postriori rror stimator for tis Hdiv conforming mixd mtod, and prov its global rliability and fficincy Du to t strong coupling on t intrfac, spcial tcniqus nd to b mployd in t proof Tis is t main diffrnc btwn tis papr and Babuška and Gatica s work [5], in wic ty analyzd an a postriori rror stimator for t mixd formulation using wakly coupld intrfac conditions 1 Introduction T coupld Darcy-Stoks problm is a wll-known and wll-studid problm, wic as many important applications W rfr to t nic ovrviw [19] and rfrncs trin for its pysical background, modling, and common numrical mtods On important issu in t modling of t coupld Darcy-Stoks flow is t tratmnt of t intrfac condition, wr t Stoks fluid mts t porous mdium In tis papr, w only considr t so calld Bavrs-Josp-Saffman condition, wic was xprimntally drivd by Bavrs and Josp in [7], modifid by Saffman in [40], and latr matmatically vrifid in [27, 28, 29, 36] Dpnding on wtr to us t primal formulation or t mixd formulation in t Darcy rgion, tr ar two popular ways to formulat t wak problm of t coupld Darcy-Stoks flow Hr w concntrat on t mixd formulation, wic as bn studid in [3, 21, 22, 23, 30, 31, 32, 38, 39] In [32], rigorous analysis of t mixd formulation and its wak xistnc av bn prsntd T autors studid two diffrnt mixd formulations T first on imposs t normal continuity of t vlocity fild on t intrfac wakly, by using a Lagrang multiplir; wil t scond on imposs t normal continuity strongly in t functional spac Latr w sall call ts two mixd formulations, rspctivly, t wakly coupld formulation and t strongly coupld formulation 1991 Matmatics Subjct Classification 65N15, 65N30, 76D07 Ky words and prass mixd finit lmnt mtods, coupld Darcy-Stoks problm, a postriori stimats Cn was supportd by t Natural Scinc Foundation of Cina , t Ministry of Education of Cina and t Stat Administration of Forign Exprts Affairs of Cina undr t 111 projct grant B08018 Wang tanks t Ky Laboratory of Matmatics for Nonlinar ScincsEZH , Fudan Univrsity, for t support during r visit 1

2 2 WENBIN CHEN AND YANQIU WANG T wakly coupld formulation givs mor frdom in coosing t discrtizations for t Stoks sid and t Darcy sid sparatly T work in [21, 22, 23, 32] ar basd on t wakly coupld formulation Rsarc on t strongly coupld formulation as bn focusd on dvloping unifid discrtization for t coupld problm Tat is, t Stoks sid and t Darcy sid ar discrtizd using t sam finit lmnt Tis approac can simplify t numrical implmntation, of cours only if t unifid discrtization is not significantly mor complicatd tan commonly usd discrtizations for t Darcy and t Stoks problms On ssntial difficulty in coosing t unifid discrtization is tat, t Stoks sid vlocity is in H 1 wil t Darcy sid vlocity is only in Hdiv Commonly usd stabl finit lmnts for t Stoks quation do not work for t Darcy quation, and vic vrsa Spcial tcniqus usually nd to b mployd In [3], a conforming, unifid finit lmnt as bn proposd for t strongly coupld mixd formulation Howvr, it is constructd only on rctangular grids, and rquirs spcial tratmnt of t nodal dgrs of frdom along t intrfac According to t autors knowldg, t lmnt proposd in [3] is probably t only xisting conforming and unifid lmnt Otr rsarcrs av rsortd to lss rstrictiv discrtizations suc as t non-conforming unifid approac [31] or t discontinuous Galrkin DG approac [30, 38, 39] Du to its discontinuous natur, som DG discrtizations for t coupld Darcy-Stoks problm may brak t strong coupling in t discrt lvl [38, 39], as ty impos t normal continuity across t intrfac via intrior pnaltis W ar intrstd in t Hdiv conforming DG approac [30] wic prsrvs t strong coupling vn in t discrt lvl T ida of t Hdiv conforming DG approac is to us Hdiv conforming lmnts, suc as t Raviart-Tomas [37] lmnts and t Brzzi-Douglas-Marini [9] lmnts, to discrtiz t ntir coupld problm Suc lmnts av normal continuity but not tangntial continuity on ms dgs/facs, and tus is not conforming for t Stoks sid T solution is to us t intrior pnalty mtods and impos t tangntial continuity on t Stoks sid wakly, via dg/fac intgrals W point out tat t Hdiv conforming intrior pnalty mtod for t Stoks quation as bn wll-studid in [15, 16, 47] In [30], t autors usd tis approac to dvlop a discrtization for t coupld Darcy-Stoks problm, wic strongly satisfis t normal continuity condition on t intrfac Enrgy norm a priori rror stimats is also provd Latr, an L 2 a priori rror stimation for tis approac is givn [24] T purpos of tis papr is to dvlop an a postriori rror stimator for t Hdiv conforming mtod proposd in [30] A postriori rror stimations av bn wll-stablisd for bot t mixd formulation of t Darcy flow [2, 8, 11, 34], tc, and t Stoks flow [1, 6, 13, 18, 20, 26, 35, 42, 43, 45, 46], tc, among wic [26, 45] covrs a postriori rror stimation for Hdiv conforming intrior pnalty mtods for Stoks quations Howvr, tr ar only a fw works xisting for t coupld Darcy-Stoks problm [5, 17], wr [5] concrns t wakly coupld mixd formulation wil [17] uss t primal formulation on t Darcy sid To our knowldg, tr is no a postriori rror stimation for t strongly coupld mixd formulation yt for t coupld Darcy-Stoks flow On immdiatly wants to ask, ow diffrnt can t a postriori rror stimation for t strongly coupld mixd formulation b, comparing wit stimations for t wakly coupld formulation or vn for t pur Darcy or pur Stoks quations?

3 A POSTERIORI ERROR ESTIMATE FOR COUPLE DARCY-STOKES EQUATIONS 3 Hr, t tcnical difficulty lis in t combination of a postriori rror stimations and t strongly imposd intrfac condition For t mixd formulation of pur Darcy problm, t asist way of prforming a postriori rror stimation is to us a Hlmoltz dcomposition [11], wil for t intrior pnalty mtod for t pur Stoks quation, on may want to dfin a continuous approximation to t discontinuous vlocity [26, 45] Wn coupling ts two compltly diffrnt tcniqus togtr, t normal continuity condition across t intrfac nds to b satisfid all t tim Spcial intrpolation oprators nd to b constructd to fulfill tis rquirmnt, and tr ar many tcnical complxitis tat nd to b clarifid T papr is organizd as follows For simplicity, only two-dimnsional problms ar considrd In Sction 2, t modl problm for t coupld Darcy-Stoks flow and its strongly coupld mixd formulation ar introducd, togtr wit svral notations T Hdiv conforming discrtization for t strongly coupld formulation will b prsntd in Sction 3 Tn, in Sction 4, an a postriori rror stimator is drivd T procss of driving actually also srvs as t proof for t global rliability of t stimator T global fficincy of t stimator is vrifid in Sction 5 Finally in Appndix A, w construct an important intrpolation oprator wic prsrvs t normal continuity on t intrfac wil satisfying crtain proprtis 2 Modl problm and notation W follow t modl dvlopd in [24, 30] Contnts of sctions 2 and 3 can b found in [24, 30] and otr rfrncs as will b statd For radr s convninc, w prsnt som dtails r T notations usd in tis papr ar sligtly diffrnt from tos in [24, 30], nc sctions 2 and 3 also srv t purpos of introducing t notations Considr t coupld Darcy-Stoks systm in a polygon Ω dividd into two nonovrlapping subdomains Ω S and Ω D, wic ar occupid by t Stoks fluid and porousmdium, rspctivly Forsimplicity, assumbotω S andω D arpolygonal DnottintrfacbtwntstwosubdomainsbyΓ SD DfinΓ S = Ω S \Γ SD and Γ D = Ω D \Γ SD Wn t associatd domain is clar from t contxt, us n to rprsnt t unit outward normal vctor and t t unit tangntial vctor suc tat n,t forms a rigt-and coordinat systm On Γ SD, dnot ˆn to b t unit normal vctor pointing from Ω S towards Ω D, andˆt accordingly, as sown in Figur 1 Figur 1 Domain of t coupld Darcy-Stoks problm Ω S t ΓSD Γ S Ω D n Γ D

4 4 WENBIN CHEN AND YANQIU WANG Considr t following coupld Darcy-Stoks problm, wr t flow is govrnd by t Stoks quation in Ω S and t Darcy s law in Ω D : 21 Tu, p = f in Ω S, K 1 u+ p = f in Ω D, u = g in Ω Hr u is t vlocity, p is t prssur, f and g ar givn vctor-valud and scalarvalud functions, rspctivly, in Ω T strss tnsor is dfind by Tu, p = 2νDu pi, wr ν > 0 is t fluid viscosity, Du = 1 2 u + ut is t strain tnsor, and I is t idntity matrix Finally, K is a symmtric and uniformly positiv tnsor dnoting t prmability tnsor dividd by t fluid viscosity For simplicity, assum K as smoot componnts and is also uniformly boundd from abov To distinguis btwn t Darcy and t Stoks sids wn ncssary, w somtims dnot u S = u ΩS, u D = u ΩS, and p S, p D in t sam fasion T boundary condition is st to b: u S = 0 on Γ S, 22 u D n = 0 on Γ D For simplicity, w assum tat Γ S On t intrfac Γ SD, w impos t consrvation of mass, t balanc of normal forcs, and t Bavrs-Josp-Saffman condition [7, 40] u S ˆn = u D ˆn, Tu S, p S ˆn ˆn = p D, Tu S, p S ˆn ˆt = µk 1/2 u S ˆt, wr µ > 0 is a variabl rlatd to t friction and sall b dtrmind xprimntally, K 1/2 is dfind using t standard ignvalu dcomposition W assum tat µ is smoot and uniformly boundd bot abov and away from zro It is not ard to s tat conditions 24 and 25 ar quivalnt to 26 Tu S, p S ˆn+p Dˆn+µK 1/2 u S ˆtˆt = 0 on Γ SD Du to t boundary condition 22, w clarly nd to assum t compatibility condition gdx = 0 In addition, t prssur is uniqu only up to a constant Ω Tus it is convnint to assum tat pdx = 0 Ω T mixd wak formulation and t xistnc of t wak solution of problm 21 as bn torougly discussd in [24, 32] Blow w brifly stat ts rsults First, w introduc svral notations For a on- or two-dimnsional polygonal domain K, dnot H s K, wr s R, to b t usual Sobolv spac, wit t norm s,k Wn s = 0, it coincids wit t squar intgrabl spac L 2 K and w usually supprss 0 in t subscript of t norm, tat is, K = 0,K Dnot, K and <, > K to b t L 2 innr-product and duality form, rspctivly, in K Wn K is on-dimnsional, t convntion is to us <, > K for bot t L 2 innr-product and duality form Wn K = Ω, w usually supprss t subscript K in, K Finally, all ts notations can b asily xtndd to vctor and tnsor spacs

5 A POSTERIORI ERROR ESTIMATE FOR COUPLE DARCY-STOKES EQUATIONS 5 W follow t convction tat a bold caractr dnots a vctor or vctor-valud function Dfin wit t norm Lt Γ K Dfin and Hdiv, K = {v L 2 K 2, v L 2 K}, v Hdiv,K = v 2 K + v 2 K 1/2 H 0,Γ div, K = {v Hdiv, K, v n = 0 on Γ}, H 1 0,ΓK = {v H 1 K, v = 0 on Γ} If Γ = K, simply dnot H 0, K div, K = H 0 div, K and H 1 0, K K = H1 0K W ar intrstd in t trac of functions in H 0,Γ div, K and H 1 0,Γ K on Γ = K\Γ It is wll-known tat for all v H 1 0,Γ K, w av v Γ H 1/2 00 Γ and for all v H 0,Γ div, K, w av v n Γ H 1/2 00 Γ [32] radrs may rfr to [33] for t dfinition and norm of H 1/2 00 Γ An important proprty of H 1/2 00 Γ, is tat, for any function in H 1/2 00 Γ, it can b xtndd by zro on K\ Γ and yilds a function in H 1/2 K Dfin V = {v H 0 div,ω v S H 1 Ω S 2 and v ΓS = 0} T spac V is a Hilbrt spac undr t norm v 2 1,Ω S + v 2 Hdiv,Ω D 1/2 Latr w will also introduc an quivalnt nrgy norm on V For convninc, dnot V S and V D to b t confinmnts of V on Ω S and Ω D rspctivly It is clar tat functions in V satisfy t strong coupling condition 23 on t intrfac Γ SD Furtrmor, v S ΓSD H 1/2 00 Γ SD 2 for all v V Dfin a bilinar form a, : V V R by wr au,v = a S u,v+a D u,v+a I u,v, a S u,v = 2νDu,Dv ΩS, a D u,v = K 1 u,v ΩD, a I u,v =< µk 1/2 u s ˆt,v s ˆt > ΓSD Dnot Q = L 2 0Ω, t man-valu fr subspac of L 2 Ω, and dfin a bilinar form b, : V Q R by bv,q = v,q Ω Now w can introduc t wak formulation of t Darcy-Stoks coupld problm: Find u, p V Q suc tat { au,v+bv,p = f,v for all v V, 27 bu,q = g,q for all q Q Clarly, Equation 27 can b writtn into Λu,p, v,q = Fv,q,

6 6 WENBIN CHEN AND YANQIU WANG wr Λu,p, v,q = au,v+bv,p+bu,q, Fv,q = f,v g,q It as bn sown [32] tat t wak formulation 27 is quivalnt to t boundary valu problm Radrs may also rfr to [25] for a dtaild discussion on t intrfac conditions on Γ SD To stat t wak xistnc rsults for problm 27, w first nd to introduc svral norms Dfin an nrgy norm on V by 1/2 v V = v 2 Ω S + µ 1/2 K 1/4 v S ˆt 2 Γ SD + K 1/2 v 2 Ω D + v Ω 2 It is not ard to s tat C 1 v V v 2 1,Ω S + v 2 Hdiv,Ω D 1/2 C 2 v V, wr C 1 and C 2 dpnd only on t sap of domain, µ, and K Dnot Q to b t L 2 norm on Ω and v,q v,q V Q = v 2 V + q 2 Q Tn, it is not ard to stablis t following Ladyznskaya-Babuška-Brzzi condition [24, 32]: sup v V av,v α v 2 V for all v Z, bv, q β q Q for all q Q, v V wr Z = {v V v = 0} Ts guarant tat Equation 27 admits a uniqu solution in V Q Furtrmor, it is wll-known tat conditions ar quivalnt to t following Babuška s form: [10] Λw,ξ, v,q 212 sup Cα,β w,ξ v,q V Q v, q for all w,ξ V Q Hr Cα,β is a constant dpnding on α and β Equation 27 is t strongly coupld formulation studid in [32] An altrnativ wak formulation for problm 21, t wakly coupld formulation, as also bn prsntd in [32] T wakly coupld formulation is dfind on t spac H 1 0,Γ S Ω S 2 H 0,ΓD div,ω D Q, and t intrfac condition 23 is tn wakly imposd by using a Lagrang multiplir In [32], Layton, Sciwck and Yotov av provd tat wn t porous mdium is ntirly nclosd in t fluid rgion, t wakly coupld formulation and t strongly coupld formulation ar quivalnt, and bot ar wll-posd Howvr, for gnral domains, it can only b provd tat t strongly coupld formulation 27 is wll-posd, wil t wll-posdnss of t wakly coupld formulation is unknown du to a tcnical difficulty of rstricting H 1/2 Ω D on Γ SD [32] Intrstd radrs may rfr to [32] for t dtails Mixd finit lmnt mtods introducd in [21, 22, 23, 32] ar all basd on t wakly coupld formulation T a postriori rror stimation givn in [5] is also basd on t wakly coupld formulation Diffrnt from t work in [5], r w propos

7 A POSTERIORI ERROR ESTIMATE FOR COUPLE DARCY-STOKES EQUATIONS 7 an a postriori rror stimation for t strongly coupld formulation 27 and its Hdiv conforming finit lmnt discrtization introducd in [24, 30] 3 Mixd finit lmnt discrtization Rcntly, an Hdiv conforming, unifid mixd mtod for 27, in wic bot t Stoks part and t Darcy part ar approximatd by t Raviart-Tomas RT lmnts, as bn proposd [30] Latr, optimal rror in L 2 norm for t vlocity of t Hdiv conforming formulation as bn provd in [24] Of cours, t RT lmnts ar not H 1 conforming on t Stoks sid T ida is to us t intrior pnalty discontinuous Galrkin Hdiv approac for t Stoks quation [15, 16, 47] in t Stoks rgion, wil using a usual mixd finit lmnt discrtization in t Darcy rgion Tr ar tr main advantags of tis approac First, t unifid finit lmnt spac may simplify t numrical simulation Scond, t normal continuity condition23 on t intrfac is strongly imposd on t discrt lvl Tird, t discrtization is strongly consrvativ [30, 47] For xampl, wn g = 0 in Ω S, tn t discrt vlocity is xactly divrgnc fr, instad of wakly divrgnc fr Nxt, w brifly prsnt tis mtod Lt T b a gomtrically conformal, sap-rgular ms on Ω W rquir tat T b alignd wit Γ SD For ac triangl T T, dnot by T its diamtr Lt b t maximum of all T Dnot T S and T D to b t mss in Ω S and Ω D, rspctivly Dnot by E t st of all dgs in T For ac dg E, dnot by its lngt Lt E SD b t st of all dgs in T Γ SD, and lt E S, ED b t st of all dgs in T Ω S Γ S, T Ω D Γ D, rspctivly W also dnot E0, S and E0, D to b t st of dgs intrior to Ω S and Ω D, rspctivly Lt O, P b oprators dfind on ac T T S or T D, but may not b wlldfind on t ntir Ω For xampl, t gradint oprator on t spac of discontinuous picwis polynomials on T W introduc t notation for discrt L 2 innr-products as following O, P T S = O, P T, T T S O, P T D = T T D O, P T Similarly, dfin <, > E S, <, > E S 0,, <, > E D, <, > E D 0,, and <, > E SD in t sam fasion Wit t aid of ts notations, w can also dnot msdpndnt brokn L 2 norms in a straigt-forward mannr For xampl, T S, 1/2 and T S E SD <, > 1/2 W spcially rmark tat t brokn norm E SD may vn contain T or in it For xampl, T v T S = T v, T v 1/2 T S 1/2 = 2 T v T 2, T T S 1/2 v E SD =< 1/2 v, 1/2 v > 1/2 E SD = E SD Suc notations sall gratly clan up t styl of tis papr 1 v 2 1/2

8 8 WENBIN CHEN AND YANQIU WANG Lt V H 0 div,ω and Q Q b a pair of t Raviart-Tomas [37] finit lmnt spacs, xcpt for t lowst-ordr on, dfind on T Tat is, V is t RT k spac, wit k 1, and Q is t discontinuous P k spac Notic r w xclud t RT 0 lmnt, sinc otrwis V V will b mpty Radrs may rfr to [10, 14] for mor dtails and proprtis of t RT lmnts For convninc, dnot V,S and V,D to b t confinmnt of V in Ω S and Ω D, rspctivly Similarly, for any v V, it can b split into v,s V,S and v,d V,D Dnot P t spac of H 1 conforming Lagrang finit lmnt spac on T consisting of picwis polynomials wit dgr lss tan or qual to k, and lt P,S, P,D b t confinmnts of P on Ω S and Ω D, rspctivly By t dfinition, w av P,S 2 V S V,S Finally, w point out tat all functions in V satisfy t strong coupling condition 23 on t intrfac Γ SD Dfin a discrt bilinar form a S, u,v = 2ν Du,Dv T S < {Du}n,[v] > E S < [u],{dv}n > E S + < σ [u],[v] > E S, wr { } and [ ] dnot t avrag and t jump on dgs, rspctivly, and σ > 0 is a paramtr of O1 On t boundary dg Γ S, { } and [ ] ar just t onsidd valus On ac dg, t dirction n is takn to b t sam as t dirction in [ ], tat is, if [v] v T1 v T2 wr T 1 and T 2 sar t dg, tn n points from T 1 to T 2 T notations in t dfinition of bilinar form a S, ar standard in t discontinuous Galrkin litratur and radrs may rfr to [15, 16, 30, 47] for mor dtails Now, dfin a u,v = a S, u,v+a D u,v+a I u,v W av t discrt problm [24, 30]: Find u, p V Q suc tat { a u,v+bv,p = f,v for all v V, 31 bu,q = g,q for all q Q Sinc V,S is not in H0,Γ 1 S Ω S 2, t discrt spac V is not a subspac of V Trfor it dos not inrit t norm of V Hr w sall dfin a discrt norm on V by v V = v 2 T S + 1/2 [v] 2 E + µ 1/2 K 1/4 v S S ˆt 2 E SD + K 1/2 v 2 Ω D + v 2 Ω 1/2 Altoug t norm V is not wll-dfind on V, w point out tat t discrt norm V is wll-dfind on V Indd, it is clar tat v V = v V for all v V, sinc t jump trm 1/2 [v] E S vaniss Latr w sall us tis proprty and build crtain discrt functions in V Tn for ts functions, on can asily sift from t discrt V norm to t continuous V norm

9 A POSTERIORI ERROR ESTIMATE FOR COUPLE DARCY-STOKES EQUATIONS 9 T spac Q is a subspac of Q and inrits its norm, wic is just t L 2 norm Dfin t norm in V +V Q by 1/2 v,q = v 2 V + q Q 2 Again, w av v,q = v,q for all v V and q Q Wll-posdnss and a priori rror stimats for 31 av bn givn in [24, 30]: Torm 31 Equation 31 as a uniqu solution u,p for σ larg noug, but not dpnding on Assum t solution u,p of 27 is in H s Ω 2 H s Ω wit 3/2 < s k +1,tn u u,p p C s 1 u s,ω + p s,ω, wr C is a positiv constant indpndnt of t ms siz 4 Rsidual basd a postriori stimation Tgoaloftissctionistodrivanapostriorirrorstimatorfortproblm 31 For simplicity of notation, w sall us to dnot lss tan or qual to up to a constant indpndnt of t ms siz, variabls, or otr paramtrs apparing in t inquality In tis sction, w will also frquntly us t following wll-known inquality: for any function ξ H 1 T wr T is a triangl wit an dg, t following stimat olds: 41 ξ 2 ξ 2 T + 2 T ξ 2 T To driv an a postriori rror stimator, w first dnot ε u = u u, ε p = p p, wr u,p is t solution to 27 and u,p is t solution to 31 T ida of driving a rliabl a postriori rror stimator is to find an uppr bound for ε u,ε p As pointd out in t prvious sction, t norms and ar idntical on V Q Tus w introduc a nw discrt function ũ V, wic is dfind from t discrt solution u and satisfy 42 u ũ V 1/2 [u ] E S T dfinition of ũ and t proof of Equation 42 will b givn in Appndix A Not tat ũ is not ncssarily in V T trm u ũ V is usually calld t nonconformity stimator in t a postriori rror stimation litratur Dnot ε u = u ũ V, tn ε u,ε p u ũ V + ε u,ε p = u ũ V + ε u,ε p Sinc u ũ V isbounddinequation42,wonlyndtostimat ε u,ε p By t Babuška s condition 212, ε u,ε p sup v,q V Q Λ ε u,ε p, v,q v, q

10 10 WENBIN CHEN AND YANQIU WANG Not tat Λ ε u,ε p, v,q = a ε u,v+bv,ε p +b ε u,q = f,v aũ,v bv,p + = Rs 1 v+rs 2 q g,q+ ũ,q Hr 43 Rs 2 q g ũ q g u + u ũ V q g u + 1/2 [u ] E S q Nxt, w concntrat on stimating Rs 1 v Lt v V b any function, by 31, w av Tus Rs 1, v f,v a u,v bv,p = 0 Rs 1 v = Rs 1 v Rs 1, v = f,v v aũ,v a u,v bv v,p = f,v v ΩD a D ũ,v a D u,v + v v,p ΩD +f,v v ΩS a S ũ,v a S, u,v + v v,p ΩS a I ũ,v a I u,v = R D +R S +R I, Nxt, w sall driv uppr bounds for R D, R S and R I on by on Clarly, t coic of v is ssntial to t stimation W would lik v to satisfy t following conditions: 1 v is in V V ; 2 v is a good approximation of v; 3 v sould lad to a straigt-forward a postriori rror stimation, wic allows us to follow wll-known tcniqus from t a postriori rror stimations of pur Darcy and pur Stoks problms To satisfy t tird condition, w must first invstigat a postriori rror stimators for t pur Darcy and t pur Stoks quations For t Stoks quations, w follow t proof in [26] wr v is cosn to b t Clémnt intrpolation of v For t Darcy quation, w follow t proof in [11] wr v nds to b dfind using a Hlmoltz dcomposition Now t difficulty is, ow to coupl ts two diffrnt typ of dfinitions wil nsuring tat v V V? Not tat v must satisfy t strong coupling condition 23 across t intrfac Γ SD

11 A POSTERIORI ERROR ESTIMATE FOR COUPLE DARCY-STOKES EQUATIONS Dfining v Givn v V, w dfin t Stoks sid approximation v,s and t Darcy sid approximation v,d sparatly Tn v will b t combination of v,s and v,d as long as ty satisfy 44 v,s ˆn = v,d ˆn on Γ SD On t Stoks sid, t approximation can b don dirctly by a Clémnt typ intrpolationontop,s 2 V S V,S HrwpicktScott-Zangintrpolation [41], sinc it also prsrvs t non-omognous boundary condition T ida of t Scott-Zang intrpolation is to first assign, for ac Lagrang intrpolation point usd as dgrs of frdom for P,S, an associatd intgration rgion s Figur 2 Tn, dfin t intrpolatd valu at ac Lagrang point by tsting t function wit t dual basis in t associatd intgration rgion For Lagrang points intrior to a triangl T T S, t associat intgration rgion is t triangl T For Lagrang points lying on dgs, t associat intgration rgion is cosn to b an dg Not for points wr svral dgs mt, t coic may not b uniqu In ordr to prsrv t boundary condition, t associatd intgration rgion for Lagrang points lying on Ω S nd to b cosn as a boundary dg on Ω S W spcially not tat, at t nd points Γ S Γ SD, t associatd intgration rgion nds to b cosn on Γ S, in ordr to nsur t intrpolatd valu at ts points ar qual to zro s Figur 2 Dnot I to b t Scott- Zang intrpolation mntiond abov tat maps H 1 Ω S to P,S, prsrving t omognous boundary condition on Γ S On Γ SD, I maps t valu at t nd pointsofγ SD intozro, wilproducstintrpolatdvaluonγ SD usingonlyt function valu on Γ SD Indd, I ΓSD can b viwd as a wll-dfind intrpolation fromh 1/2 00 Γ SDtoP ΓSD, bysimplysttingtintrpolationatnd-pointsofγ SD to b zro Notic tat H 1/2 00 Γ SD can b xtndd by zro on itr Γ S or Γ D, w ar abl to asily mak transition from Ω S to Ω D Tat is, for ξ H0,Γ 1 D Ω D and consquntly ξ ΓSD H 1/2 00 Γ SD, t intrpolation I ΓSD ξ is also wll-dfind on Γ SD Furtrmor, similar to t proof in [41], on can sow tat for any E SD 45 I I ΓSD ξ 2 T S D 2 T ξ 2 T, wr S D is t st of triangls in T D tat av a non-mpty intrsction wit In t rst of t papr, wn tr is no ambiguity, w will just dnot I ΓSD by I Clarly I I 2 will map V S into P,S 2 V S V,S It is also known [41] tat I is a projction In otr words, I ϕ S = ϕ S for all ϕ S in P,S 2 V S Dfin v,s = I v S Sinc I is a linar oprator, w av v,s ˆn = I v S ˆn = I v S ˆn on Γ SD On t Darcy sid, t intrpolation will b dfind using a Hlmoltz dcomposition Tat is, w first split v D = w +curlη,, wr curlη = η x 2 η, x 1

12 12 WENBIN CHEN AND YANQIU WANG Figur 2 Associatd intgration rgion for diffrnt typ of Lagrang points Eac Lagrang point is dnotd by a black dot, and t associatd intgration rgion is dnotd by itr a sadd triangl or a bold dg For Lagrang points on Γ S, Γ SD, or t intrsction of Γ S and Γ SD, tr ar spcial ruls for coosing t associatd intgration rgion Lagrang points inω S Lagrang points on Γ S Ω S Lagrang points on Γ SD Γ S Ω S Ω S Ω S Ω S Γ S Γ SD Γ SD and w satisfis w = v D, w ΓD = 0, w ΓSD = v S ΓSD Hr, condition 47 is imposd to nsur w n ΓD = 0 and w ˆn ΓSD = v S ˆn Tn w n ΩD = v D n ΩD and consquntly t compatibility condition Ω D wdx = Ω D v D dx = Ω D v D nds = Ω D w nds is satisfid Of cours on nds to mak sur suc a dcomposition is wll-dfind and w, η av crtain rgularity rsults Indd, sinc v S ΓSD H 1/2 00 Γ SD 2, according to [4], tr xists suc a w H 1 Ω D 2 satisfying 46 and 47 Furtrmor, w av 48 w 1,ΩD v V Now v D w ΩD H 0 div,ω D is divrgnc-fr Tus tr xists a potntial function η H 1 0Ω D suc tat curlη = v D w, and 49 η 1,ΩD curlη 0,ΩD = v D w 0,ΩD v V Now w can start to dfin v,d First, w nd an intrpolation oprator from H 1 Ω D 2 to V,D, wic must map w ˆn to I w ˆn on Γ SD Rcall tat a usual nodal valu intrpolation Π : H 1 Ω D 2 V,D associatd wit t dgrs of frdom of t RT k lmnts [10] will map w ˆn to P w ˆn on Γ SD, wr P is t L 2 projction onto V,D ˆn ΓSD = P ΓSD Hnc, w dfin a nw intrpolation Π : H 1 Ω D 2 V,D suc tat it is t sam as Π on all otr dgrs of frdom xcpt for tos associatd wit w ˆn ΓSD On ts dgrs of frdom, dfin Π by 410 Π w ˆns r ds = I w ˆns r ds for all E SD and 0 r k

13 A POSTERIORI ERROR ESTIMATE FOR COUPLE DARCY-STOKES EQUATIONS 13 Of cours, on otr dgrs of frdom, Π inrits t proprtis of Π, spcially t following ons w Π w nq ds = 0 for all E D and q Q,D, 411 w Π w q dx = 0 for all T T D and q Q,D T Combin Proposition III36 in [10] and approximation proprty of t Scott-Zang intrpolation in [41], tn us t scaling argumnt, Inquality 45, and t proprty of t L 2 projction, w av for all T T D, 412 w Π w 0,T w Π w 0,T + Π w Π w 0,T T w T + 1/2 T I P w ˆn T ΓSD 1/2 2 T w 2 T T S DT wr S D T is t st of all triangls in T D tat as a non-mpty intrsction wit T Γ SD Diffrnt from Π, wic satisfis Π w = Q w s [10], wr Q is t L 2 projction onto Q,D, Π dos not satisfy t sam rlation Instad, for all q Q,D, by t dfinition of Π and its proprtis , w av 413 w Π w,q ΩD = < w Π w n,q > T w Π w, q T T T D Finally, dfin = < w Π w ˆn,q > E SD = < I I v S ˆn,q > E SD v,d = Π w +curlη, wr η is t Clémnt intrpolation of η into t continuous picwis P k+1 polynomials on T D tat prsrvs t omognous boundary condition on Ω D Of cours on can also cos η to b t Scott-Zang intrpolation By t proprtis of t Raviart-Tomas lmnts [10], it is asy to s tat v,d V,D Furtrmor, v,s and v,d satisfy t intrfac coupling condition 44 and nc v V V By t approximation proprtis of t Scott-Zang intrpolation I and t Clémnt intrpolation, w av v S v,s 2 T + 2 T v S v,s 2 1,T 2 T v 2 1,T, 414 T T S T T S η η 2 T + 2 T η η 2 1,T 2 T η 2 1,T T T D T T D Now w ar rady to driv uppr bounds, or quivalntly t a postriori rror stimators, for R D, R S and R I

14 14 WENBIN CHEN AND YANQIU WANG 42 Driving t Darcy stimator By t dfinition of a D,, Equation 413, and t Scwarz inquality, R D = f,v v ΩD a D ũ,v a D u,v + v v,p ΩD = f,v v ΩD K 1 ũ u,v ΩD K 1 u,v v ΩD + v v,p ΩD = f K 1 u,v v ΩD K 1 ũ u,v ΩD < I I v S ˆn,p,D > E SD f K 1 u,v v ΩD + u ũ v ΩD Nxt, notic tat < I I v S ˆn,p,D > E SD f K 1 u,v v ΩD = f K 1 u,w Π w ΩD +f K 1 u,curlη η ΩD, wr by 411, 412 and 48, f K 1 u,w Π w ΩD inf p Q,D T f K 1 u p T D w 1,ΩD inf p Q,D T f K 1 u p T D v V, and by 41, 49, intgration by parts, t boundary condition of η, and 414 f K 1 u,curlη η T curlf K 1 u T D + 1/2 [f K 1 u ] t E D 0, η 1,ΩD T curlf K 1 u T D + 1/2 [f K 1 u ] t E D 0, v V Hr curl is dfind for any vctor-valud function ξ = ξ 1,ξ 2 t by curlξ = ξ 1 x 2 + ξ 2 x 1 Combining t abov and using 42, w av R D inf T f K 1 u p T D p Q + T curlf K 1 u T D,D + 1/2 [f K 1 u ] t E D 0, + 1/2 [u ] E S v V < I I v S ˆn,p,D > E SD

15 A POSTERIORI ERROR ESTIMATE FOR COUPLE DARCY-STOKES EQUATIONS Driving t Stoks stimator Not tat [v ] = 0 on all E S By using t dfinition of a S,, a,s,, and T, w av R S = f,v v ΩS a S ũ,v a S, u,v + v v,p ΩS = f,v v ΩS +2νDu ũ,dv T S 2νDu,Dv T a S S, u,v p I, v v T S = f,v v ΩS +2νDu ũ,dv T S Tu,p,Dv v T S 2ν < [u ],{Dv }n > E S f,v v ΩS Tu,p, v v T S + u ũ V v V 2ν < [u ],{Dv }n > E S In t abov w av usd t algbraic rlation tat for any symmtric tnsor τ and domain K, τ, v v K = τ,dv v K Using intgration by parts and 41, 414, f,v v ΩS Tu,p, v v T S =f + Tu,p,v v T S < Tu,p n,v v > T T T S T f + Tu,p T S + 1/2 [Tu,p ]n E S 0, v V < Tu,p ˆn,I I v S > E SD Combin all t abov and using 42, w av R S T f + Tu,p T S + 1/2 [Tu,p ]n E S 0, + 1/2 [u ] E S v V < Tu,S,p,S ˆn, I I v S > E SD 44 Driving t intrfac stimator By t dfinition of a I, and using t Scwarz inquality, inqualitis 41, 42, w av R I = a I ũ,v+a I u,v = < µk 1/2 ũ,s u,s ˆt,v S ˆt > E SD < µk 1/2 u,s ˆt,v S v,s ˆt > E SD ũ u V v V < µk 1/2 u,s ˆt,I I v S ˆt > E SD 1/2 [u ] E S v V < µk 1/2 u,s ˆt,I I v S ˆt > E SD

16 16 WENBIN CHEN AND YANQIU WANG 45 Estimator for t coupld problm Finally, by adding R D, R S and R I togtr, w av Rs 1 v inf T f K 1 u p T D p Q + T curlf K 1 u T D,D + 1/2 [f K 1 u ] t E D 0, + 1/2 [u ] E S + T f + Tu,p T S + 1/2 [Tu,p ]n E S 0, 1/2 v V < I I v S ˆn,p,D > E SD < µk 1/2 u,s ˆt,I I v S ˆt > E SD Tn, using 41 and 414, < I I v S ˆn,p,D > E SD + < µk 1/2 u,s ˆt,I I v S ˆt > E SD < Tu,S,p,S ˆn,I I v S > E SD + < Tu,S,p,S ˆn,I I v S > E SD = < Tu,S,p,S ˆn+p,Dˆn+µK 1/2 u,s ˆtˆt,I I v S > E SD Tu,S,p,S ˆn+p,Dˆn+µK 1/2 u,s ˆtˆt v V 1/2 Combining t stimation for Rs 1 v wit t stimation 43 for Rs 2 q, and stting p = p W can now construct an a postriori rror stimator for Problm 31 Lt f T and f b t L 2 projction of f on a triangl T and an dg, rspctivly onto t spac of kt ordr polynomials W dfin t a postriori rror stimator for t coupld Darcy-Stoks quation as following: 1 for T T S ηt,s 2 = 2 T f T + Tu,p 2 T + 1 [Tu,p ]n 2 2 E0, S T + 1 [u,s ] 2 + [u,s ] 2 + g u 2 2 T, E S 0, T 1 T Γ S 1 2 for T T D ηt,d 2 = 2 T f T K 1 u p 2 T + 2 T curlf T K 1 u 2 T + 1 [f 2 K 1 u ] t 2 + g u 2 T, E0, D T 3 for E SD E SD η 2,SD = Tu,S,p,S ˆn+p,Dˆn+µK 1/2 u,s ˆtˆt 2 Tn t global a postriori rror stimator is η 2 = ηt,s 2 + ηt,d 2 + T T S T T D E SD η 2,SD In practic, on may distribut t valu of η,sd by crtain formula on t two triangls saring dg, wr on triangl is in Ω S and anotr in Ω D Tis sall giv a functioning adaptiv rfinmnt stratgy Of cours on can also dsign mor spcific rfinmnt stratgy tat uss η,sd dirctly Hr w do not mov furtr

17 A POSTERIORI ERROR ESTIMATE FOR COUPLE DARCY-STOKES EQUATIONS 17 into t adaptiv rfinmnt stratgis, sinc w ar only intrstd in t global uppr and lowr bounds for η To conclud tis sction, in t abov w av constructd and provd t rliability of t a postrior stimator η, tat is Torm 41 Lt ε u, ε p and η b dfind as in tis sction, tn ε u,ε p η +Rf, wr Rf,g is t igr ordr oscillation trm Rf = T f f T T + T curlf f T T D + 1/2 f f t E D 0, 5 Efficincy of t a postriori rror stimator T a postriori rror stimator is considr fficint if it also satisfis 51 η ε u,ε p +Rf In tis sction, w sall prov tis By xamining η T,S and η T,D, w immdiatly raliz tat all trms ar itr ntirly intrior to t Darcy sid or to t Stoks sid In otr words, wn using t standard tcniqu of dfining bubbl functions, t support of ac bubbl function is containd itr in Ω S or Ω D Tus, to prov 52 η 2 T,S + η 2 T,D ε u,ε p 2 +Rf2, T T S T T D it suffics to us only t Darcy quation or only t Stoks quation T proof will b xactly t sam as t proof for pur Darcy and pur Stoks quations Radr can rfr to [11, 26, 45] for dtails Now w only nd to prov t uppr bound for E SD vry similar to t proof of Lmma 45 in [5] Blow ar givn t dtails η,sd 2 T proof is For ac E SD, dfin an dg bubbl function φ wic as support only in t two triangls saring Lt T S and T D b t triangls in T S and T D rspctivly, tat contain dg, and L : P k P k T S b an xtnsion suc tat Lq = q for all q P k On may rfr to [44] for t dfinition of φ, L and t proof of t following proprtis: For any polynomial q wit dgr at most m, tr xist positiv constants d m, D m and E m, dpnding only on m, suc tat d m q 2 q 2 φ ds D m q 2, Lqφ T S E m 1/2 q Dnot L = L 2 wic maps P k 2 to P k T S 2,

18 18 WENBIN CHEN AND YANQIU WANG Dnot χ = Tu,S,p,S ˆn+p,Dˆn+µK 1/2 u,s ˆtˆt on E SD Tn using 26 and 53, χ 2 χ 2 φ ds = χ φ Tu,S,p,S Tu S,p S ˆn+p,D p D ˆn +µk 1/2 u,s u S ˆtˆt ds χ φ Tu,S,p,S Tu S,p S ˆnds + χ p,d p D + µk 1/2 u,s u S ˆt Tn, using t support of φ, t invrs inquality and Inquality 54, χ φ Tu,S,p,S Tu S,p S ˆnds = Lχ φ Tu,S,p,S Tu S,p S dx T S + Lχ φ Tu,S,p,S Tu S,p S dx T S 1/2 χ 1 Tu,S,p,S Tu S,p S T S + Tu,S,p,S +f T S T S 1/2 χ ε u T S + ε p T S +η T S,S + T S f f T T S Using 41, w av p D p,d 1/2 p D p,d T D + 1/2 p D p,d T D = 1/2 ε p T D + 1/2 f K 1 u p,d T D 1/2 ε p T D + 1/2 f K 1 u p,d T D + 1/2 K 1 u u T D 1/2 ε p TD +η TD,D + f f TD TD + ε u TD Combining t abov, using 52, t dfinition of and Rf, w av Torm 51 T a postriori rror stimator η satisfis 51 Rmark 52 In bot Torm 41 and 51, t constant containd in may dpnd on σ, t stabilization paramtr in t dfinition of t bilinar form a S,, Howvr, sinc σ is of O1 and dos not dpnd on, its ffct on t stability and fficincy of t a postriori rror stimator η is rstrictd Appndix A Dfinition and proprtis of ũ V V Givn u V, r w dfin ũ V satisfying 42 Not tat ũ is not ncssarily in V It is a commonly usd tcniqu to introduc suc a ũ in a postriori rror stimations for nonconforming or discontinuous Galrkin mtods

19 A POSTERIORI ERROR ESTIMATE FOR COUPLE DARCY-STOKES EQUATIONS 19 Radrs may rfr to [12] and rfrnc trin for similar usag In [12], ũ is constructd using t Hlmoltz dcomposition Hr w can not borrow tir rsults dirctly, for two rasons First, w nd an stimation of u ũ in t V norm wil t construction in [12] only provids a brokn H 1 smi-norm stimation Scond, spcial tratmnt nds to b takn in ordr to nsur tat ũ satisfis t intrfac condition 23 strongly Noticing tat in ac T T, u is a polynomial wit dgr lss tan or qual to k + 1 Dnot P k+1 T,S to b t H 1 conforming discrt functional spac wic consists of picwis polynomials of dgr up to k+1 on ac T T,S W dfin ũ as following as partly illustratd in Figur 3: 1 ũ,s P k+1 T,S 2 is dfind by stting its valus on all k + 1st ordr Lagrang intrpolation points in T T S At Lagrang points intrior to any T T S, its valu is inritd from t valu of u,s At Lagrang points on Γ S, including Γ S Γ SD, t valu is st to b zro At Lagrang points locatd on dgs in E0, S ESD but not on Γ S, dfin t valu of ũ,s to b t valu of u,s from a prscribd triangl among all triangls in T S saring tis Lagrang point Not tat by suc a dfinition, t valus at Lagrang points on Γ SD ar st by using only t Stoks sid solution u,s 2 Now ũ,s as bn dfind Nxt, dfin ũ,d in t Hdiv conforming RT k+1 spacont,d bycopyingtvalusofu,d onalldgrsoffrdom xcpt for tos associatd wit E SD, namly, t dgrs of frdom dfind by ũ,d ns r ds for all E SD and 0 r k +1 At ts dgrs of frdom, to mak sur tat ũ,s ˆn = ũ,d ˆn on Γ SD, w dfin ũ,d ˆns r ds = ũ,s ˆns r ds Clarly, ũ dfind as abov is in V, but not V W av t following lmma: Lmma A1 For all T T S, A1 u,s ũ,s 2 T + 2 T u,s ũ,s 2 T E S T [u,s ] 2, wr E ST dnots t st of dgs in ES wo av non-mpty intrsctions wit T W spcially point out tat, bnfitd from t dfinition of ũ,s, E S T dos not contain dgs tat li on E SD Proof T proof follows from a routin scaling argumnt and t fact tat all norms on finit dimnsional spacs ar quivalnt To tis nd, w obsrv tat u,s ũ,s 2 T + 2 T u,s ũ,s 2 T 2 T u,s ũ,s x j 2, x j G k T wr G k T is t st of all k + 1st ordr Lagrang intrpolation points in T and dnots t Euclidan norm of a vctor It follows from t dfinition of ũ tat u,s ũ,s vaniss at all intrnal Lagrang points in T W only nd to xamin t valu of u,s ũ,s at Lagrang points on T Tr ar svral diffrnt cass as illustratd in Figur 3

20 20 WENBIN CHEN AND YANQIU WANG Figur 3 Stting t valus of ũ,s at diffrnt typ of Lagrang points on dgs For ac Lagrang point, t sadd triangl mans it is t dsignatd triangl tat dfins t valu of ũ,s on tis Lagrang point On Lagrang points on Γ S, including t intrsction of Γ S and Γ SD, t valu of ũ,s is simply st to b zro Ω S Ω S Γ S Ω S Ω S Ω D Γ SD Ω S Γ S Ω S Ω D Γ SD At Lagrang points on dgs in E0, S ESD but not on Γ S, tr ar two possibilitis: 1 x j is in t intrior of an dg ; 2 x j is a vrtx of T In t first cas, w s tat u,s ũ,s x j is itr 0 or [u,s ] x j, wr [ ] dnots t jump on dg, on t two triangls saring dg Furtrmor, if x j lis in t intrior of an dg on Γ SD, tn u,s ũ,s x j = 0 In t scond cas, w can us t triangl inquality to travrs troug all dgs E0, S tat as x j as on nd point, wic w sall dnot as E Sx j, and to obtain A2 u,s ũ,s x j [u,s ] x j E Sxj Notic tat E Sx j dos not contain dgs in E SD Finally, considr Lagrang points on Γ S Clarly for all x j in t intrior of dg Γ S, u,s ũ,s x j = u,s x j = [u,s ] x j For x j at t nd of dg Γ S, again by travrsing troug all dgs E0, S, w av Inquality A2 Combining t abov analysis, w av for all T T S u,s ũ,s x j 2 [u,s ] x j 2, x j G k T E ST x j G k wr G k dnots t corrsponding Lagrang points on dg Tn, using t routin scaling argumnt on dgs, inquality A1 follows immdiatly Using Lmma A1, inqualitis 41 and A1, w av µ 1/2 K 1/4 u,s ũ,s ˆt 2 E SD 1 T u,s ũ,s 2 T + T u,s ũ,s 2 T T T S [u,s ] 2 E S 1/2 [u,s ] 2 E S

21 A POSTERIORI ERROR ESTIMATE FOR COUPLE DARCY-STOKES EQUATIONS 21 Nxt, considr t Darcy sid Clarly u,d ũ,d is only non-zro on triangls wo as at last an dg on Γ SD Using t dfinition of ũ, t scaling argumnt, t normal dirction continuity on Γ SD, inqualitis 41 and A1, w av on suc triangls u,d ũ,d 2 T + 2 T u,d ũ,d 2 T T Γ SD u,d ũ,d ˆn 2 = T Γ SD u,s ũ,s ˆn 2 E S T [u,s ] 2 Hr sinc T lis on t Darcy sid, E ST mans t st of dgs in ES wo as non-mpty intrsction wit all triangls in T S tat sars an dg wit T Combining t abov and using t fact tat [ũ ] = 0 on all E S, tis complts t proof of 42 Rfrncs 1 M Ainswort and JT Odn, A Postriori Error Estimators for Stoks and Osn s Equations, SIAM J Numr Anal, , A Alonso, Error stimators for a mixd mtod, Numr Mat, , T Arbogast and DS Brunson, A computational mtod for approximating a Darcy-Stoks systm govrning a vuggy porous mdium, Computational Goscincs, , DN Arnold, LR Scott and M Voglius, Rgular invrsion of t divrgnc oprator wit Diriclt boundary conditions on a polygon, Ann Scuola Norm Sup Pisa Cl Sci 4, , I Babuška and G N Gatica, A rsidual-basd a postriori rror stimator for t Stoks- Darcy coupld problm, SIAM J Numr Anal , R Bank and B Wlfrt, A Postriori Error Estimats for t Stoks Problm, SIAM J Numr Anal, , GS Bavrs and DD Josp, Boundary conditions at a naturally prmabl wall, J Fluid Mc, , D Brass and R Vrfurt, A postriori rror stimators for t Raviart-Tomas lmnt, SIAM J Numr Anal, , F Brzzi, J Douglas, and L Marini, Two familis of mixd finit lmnts for scond ordr lliptic problm, Numr Mat, , F Brzzi and M Fortin, Mixd and Hybrid Finit Elmnts, Springr-Vrlag, Nw York, C Carstnsn, A postriori rror stimat for t mixd finit lmnt mtod, Mat Comp, , C Carstnsn and J Hu, A unifying tory of a postriori rror control for nonconforming finit lmnt mtods, Numr Mat, , C Carstnsn, T Gudi, M Jnsn, A Unifying Tory of A Postriori Control for Discontinuous Galrkin FEM, Numr Mat, , Z Cn, Finit Elmnt Mtods and Tir Applications, Springr-Vrlag, Brlin Hidlbrg, B Cockburn, G Kanscat, and D Scötzau, A locally consrvativ LDG mtod for t incomprssibl Navir-Stoks quations, Mat Comput, , B Cockburn, G Kanscat, and D Scötzau, A not on discontinuous Galrkin divrgncfr solutions of t Navir-Stoks quations, J Sci Comput, , M Cui and N Yan, A postriori rror stimat for t Stoks-Darcy systm, Mat Mt Appl Sci, , E Dari, R G Durán and C Padra, Error stimators for nonconforming finit lmnt approximations of t Stoks problm, Mat Comp, ,

22 22 WENBIN CHEN AND YANQIU WANG 19 M Discacciati and A Quartroni, Navir-Stoks/Darcy Coupling: Modling, Analysis, and Numrical Approximation, Rv Mat Complut, , W Dorflr and M Ainswort, Rliabl a postriori rror control for non-conforming finit lmnt approximation of Stoks flow, Mat Comp, , J Galvis and M Sarkis, Non-matcing mortar discrtization analysis for t coupling Stoks- Darcy quations, Elcton Trans Numr Anal, , GN Gatica, S Mddai and R Oyarzúa, A conforming mixd finit-lmnt mtod for t coupling of fluid flow wit porous mdia flow, IMA J Numr Anal, , GN Gatica, R Oyarzúa and F-J Sayas, Convrgnc of a family of Galrkin discrtizations for t Stoks-Darcy coupld problm, Numr Mt Part Diff Eq, , V Girault, G Kanscat and B Riviér, Error analysis for a monolitic discrtization of coupld Darcy and Stoks problms, prprint 25 V Girault and B Riviér, DG approximation of coupld Navir-Stoks and Darcy quations by Bavr-Josp-Saffman intrfac condition, SIAM J Numr Anal, , A Hannukainn, R Stnbrg and M Voralik, A unifid framwork for a postriori rror stimation for t Stoks problm, Numr Mat, submittd 27 W Jägr and A Miklić, On t boundary conditions at t contact intrfac btwn a porous mdium and a fr fluid, Ann Scuola Norm Sup Pisa Cl Sci, , W Jägr and A Miklić, On t intrfac boundary condition of Bavrs, Josp and Saffman, SIAM J Appl Mat, , W Jägr, A Miklić and N Nuss, Asymptotic analysis of t laminar viscous flow ovr a porous bd, SIAM J Sci Comput, , G Kanscat and B Rivièr, A strongly consrvativ finit lmnt mtod for t coupling of Stoks and Darcy flow, J Comput Pys, , T Karpr, K-A Mardal and R Wintr, Unifid Finit Elmnt Discrtizations of Coupld DarcyCStoks Flow Numr Mt Part Diff Eq, , WJ Layton, F Sciwck and I Yotov, Coupling fluid flow wit porous mdia flow, SIAM J Numr Anal, , J L Lions and E Magns, Non-omognous Boundary Valu Problms and Applications, Vol 1, Springr-Vrlag, Nw York-Hidlbrg, C Lovadina and R Stnbrg, Enrgy norm a postriori rror stimats for mixd finit lmnt mtods, Mat Comp, , F Nobil, A postriori rror stimats for t finit lmnt approximation of t Stoks problm, TICAM Rport 03-13, April LE Payn and B Straugan, Analysis of t boundary condition at t intrfac btwn a viscous fluid and a porous mdium and rlatd modlling qustions, J Mat Purs Appl, , PA Raviart and JM Tomas, A mixd finit lmnt mtod for scond ordr lliptic problms, Matmatical aspcts of t finit lmnt mtod, Eds I Galligani and E Magns, Lctur nots in Matmatics, Vol 606, Springr-Vrlag, B Riviér, Analysis of a Discontinuous Finit Elmnt Mtod for t Coupld Stoks and Darcy Problms, J Sci Comp, , B Riviér and I Yotov, Locally consrvativ coupling of Stoks and Darcy flows, SIAM J Numr Anal, , PG Saffman, On t boundary condition at t intrfac of a porous mdium, Stud Appl Mat, , L R Scott and S Zang Finit lmnt intrpolation of nonsmoot function satisfying boundary conditions, Mat Comp, , R Vrfurt, A postriori rror stimators for t Stoks quations, Numr Mat, , R Vrfurt, A postriori rror stimators for t Stoks quations II Non-conforming discrtizations, Numr Mat, , R Vrfurt, A rviw of a postriori rror stimation and adaptiv ms-rfinmnt tcniqus Tubnr Skriptn zur Numrik BG Willy-Tubnr, Stuttgart, J Wang, Y Wang and X Y, A postriori rror stimation for an intrior pnalty typ mtod mploying Hdiv Elmnts for t Stoks quations, SIAM J Sci Comp, ,

23 A POSTERIORI ERROR ESTIMATE FOR COUPLE DARCY-STOKES EQUATIONS J Wang, Y Wang and X Y, A postriori rror stimat for stabilizd finit lmnt mtods for t Stoks quations, Int J Numr Anal Modl, , J Wang and X Y, Nw finit lmnt mtods in computational fluid dynamics by Hdiv lmnts, SIAM J Numr Anal, , Scool of Matmatics, Fudan Univrsity, Sangai, Cina addrss: wbcn@fudanducn Dpartmnt of Matmatics, Oklaoma Stat Univrsity, Stillwatr, OK, USA addrss: yqwang@matokstatdu