Christine Bernardi 1, Tomás Chacón Rebollo 1, 2,Frédéric Hecht 1 and Zoubida Mghazli 3

Transcription

1 ESAIM: MAN 4 008) DOI: /man: ESAIM: Matmatical Modlling and Numrical Analysis MORTAR FINITE ELEMENT DISCRETIZATION OF A MODEL COUPLING DARCY AND STOES EQUATIONS Cristin Brnardi 1, Tomás Cacón Rbollo 1,,Frédéric Hct 1 and Zoubida Mgazli 3 Abstract. As a first draft of a modl for a rivr flowing on a omognous porous ground, w considr a systm wr t Darcy and Stoks quations ar coupld via appropriat matcing conditions on t intrfac. W propos a discrtization of tis problm wic combins t mortar mtod wit standard finit lmnts, in ordr to andl sparatly t flow insid and outsid t porous mdium. W prov a priori and a postriori rror stimats for t rsulting discrt problm. Som numrical xprimnts confirm t intrst of t discrtization. Matmatics Subjct Classification. 65N30, 65N55, 76M10. Rcivd January 8, 007. Rvisd Octobr 3, 007. Publisd onlin April 1st, Introduction W first dscrib t modl w intnd to work wit. Lt Ω b a rctangl in dimnsion d = or a rctangular paralllpipd in dimnsion d = 3. W assum tat it is dividd witout ovrlap) into two connctd opn sts Ω P and Ω F wit Lipscitz-continuous boundaris, wr t indics P and F stand for porous and fluid, rspctivly. T fluid tat w considr is viscous and incomprssibl. So in t porous mdium, wic is assumd to b rigid and saturatd wit t fluid, w considr t following quations, du to Darcy, { α u + grad p = f in Ω P, 1.1) div u =0 inω P. In Ω F, t flow of tis sam fluid is govrnd by t Stoks quations { ν u + grad p = f in Ω F, div u =0 inω F. 1.) ywords and prass. Mortar mtod, finit lmnts, Darcy quations, Stoks quations. 1 Laboratoir Jacqus-Louis Lions, C.N.R.S. & Univrsité Pirr t Mari Curi, B.C. 187, 4 plac Jussiu, 755 Paris Cdx 05, Franc. brnardi@ann.jussiu.fr; ct@ann.jussiu.fr Dpartamnto d Ecuacions Difrncials y Análisis Numrico, Univrsidad d Svilla, Tarfia s/n, 4101 Svilla, Spain. cacon@numr.us.s 3 Équip d Ingéniri Matématiqu, LIRNE, Faculté ds Scincs, Univrsité Ibn Tofail, B.P. 133, énitra, Morocco. mgazli zoubida@yaoo.com Articl publisd by EDP Scincs c EDP Scincs, SMAI 008

2 376 C. BERNARDI ET AL. Figur 1. An xampl of tr-dimnsional domain Ω. T unknowns bot in 1.1) and 1.) ar t vlocity u and t prssur p of t fluid. T paramtrs ν and α ar positiv constants, rprsnting t viscosity of t fluid and t ratio of tis viscosity to t prmability of t mdium, rspctivly. T porous mdium is supposd to b omognous, so tat w tak α constant on t wol subdomain Ω P w rfr to [1] and[1] for andling t somwat mor ralistic cas wr α is picwis constant in a diffrnt framwork). Not also tat t dformation tnsor is rplacd by t gradint oprator in 1.) for matmatical simplicity tis simplification is standard in gopysics, s.g. [9], Sct. 1..3). Concrning t boundary conditions, as illustratd in Figur 1 d = 3) and also in Figur d = ) wit mor dtails, w dnot by Γ a t uppr dg d =)orfacd = 3) of Ω, wr t indx a mans in contact wit t atmospr. Lt Γ ap b t intrsction Γ a Ω P and Γ af t intrsction Γ a Ω F not tat Γ ap can b mpty in som practical situations). W st: Γ P = Ω Ω P ) \ Γ ap and Γ F = Ω Ω F ) \ Γ af. Lt n stand for t unit outward normal vctor to Ω on Ω andalsotoω P on Ω P. Wprovidtprvious partial diffrntial quations 1.1) and 1.) wit t conditions u n = k on Γ P and p = p a on Γ ap, 1.3) and u = g on Γ F and ν n u p n = t a on Γ af. 1.4) Not tat ts conditions ar of Diriclt typ on Ω \ Γ a, wil t condition on Γ ap only mans tat t prssur, r qual to p a, dpnds on t atmospric prssur. T condition on Γ af mans tat t variations of t fr surfac at t top of t flow ar nglctd in t modl. Tus t a mainly dpnds on t atmospric prssur and t wind on t rivr. Tis is standard in gopysics, s.g. [9], Sction 1.4; not owvr tat, wn t flux Γ F g n)τ )dτ + Γ P kτ )dτ is too larg, tis boundary condition is not compatibl wit t pysics of t problm. To conclud, lt Γ dnot t intrfac Ω P Ω F. On Γ w considr t matcing conditions u ΩP n = u ΩF n and p ΩP n = ν n u ΩF p ΩF n on Γ. 1.5)

3 MORTAR FINITE ELEMENTS FOR DARCY AND STOES EQUATIONS 377 Γ af Γ ap Γ af Ω F Γ F Ω F Γ F Γ P Γ P Γ P Γ P Ω P Ω P Γ P Γ P Figur. Two xampls of two-dimnsional domains Ω. Indd, from a pysical point of viw, consrvation of mass nforcs continuity of t normal vlocitis at t intrfac. Similarly, consrvation of momntum nforcs continuity of t normal strsss. Suc intrfac conditions ar studid for instanc in [1,8] and[17], Sction 4.5. Not tat a lot of rcnt works dal wit t coupling of Darcy and Stoks quations in diffrnt framworks, nc wit otr typs of matcing conditions suc as t Bavr-Josp-Saffman conditions, s [10,16,1,3,8,35] and t rfrncs trin. Systm 1.1) 1.5) is only a first draft of a modl for t laminar flow of a rivr ovr a porous rock suc as limston, owvr it sms tat its discrtization as not bn considrd bfor. Of cours, in mor ralistic modls, t Stoks quations must b rplacd by t Navir-Stoks quations for instanc wn t rivr mts obstacls) and t Darcy quations must b rplacd by mor complx modls as proposd in [33] salso[4] or [0]). Howvr w ar intrstd wit tis systm. W first writ an quivalnt variational formulation of it and prov tat it admits a uniqu solution. T discrtization tat w propos rlis on t mortar lmnt mtod, a domain dcomposition tcniqu introducd in [7] salso[11] for t nw trnds). Indd it sms convnint to us a subdomain for t fluid and anotr on for t porous mdium. Morovr, owing to t flxibility of t mortar mtod, indpndnt mss can b usd on t diffrnt parts of t domain. On ac subdomain, w considr a finit lmnt discrtization, rlying on standard finit lmnts bot for t Stoks problm t lmnt first introducd in [] and analyzd in [6]) and t Darcy quations t Raviart-Tomas lmnt [34]). Ts coics can b justifid as follows: T Raviart-Tomas lmnt is t simplst and lss xpnsiv lmnt wic is conforming in t domain of t divrgnc oprator, so tat w us it on Ω P. It is usually associatd wit picwis constant prssurs, in ordr tat t inf-sup condition linking t two spacs on Ω P is optimal wr optimal mans wit a constant indpndnt of t discrtization paramtr ). Tus, for simplicity, picwis constant prssurs ar usd on t wol domain. T Brnardi-Raugl lmnt is t lss xpnsiv lmnt wic, wn associatd wit t spac of picwis constant prssurs, lads to an optimal inf-sup condition on Ω F. W construct a discrt problm and w cck tat it as a uniqu solution. W tn prov optimal apriori and a postriori uppr bounds for t rror, dspit t lack of conformity of t mortar mtod. Tanks to t rror indicators issud from t a postriori analysis, w ar in a position to prform ms adaptivity indpndntly in t porous and fluid domain. W dscrib t adaptivity stratgy tat w us. Nxt w prsnt numrical xprimnts. T rsults ar in good agrmnt wit t rror stimats, so ty justify our coic of discrtization.

4 378 C. BERNARDI ET AL. T outlin of t papr is as follows. In Sction, w writ t variational formulation of t problm and prov its wll-posdnss. Sction 3 is dvotd to t dscription of t discrt problm and to t proof of its wll-posdnss. W prov t aprioriand a postriori stimats in Sctions 4 and 5, rspctivly. T adaptivity stratgy and numrical xprimnts ar prsntd in Sction 6.. Analysis of t modl W first intnd to writ a variational formulation of systm 1.1) 1.5). From now on, for ac domain O in R d wit a Lipscitz-continuous boundary, w us t full scal of Sobolv spacs H s O) andh0o), s s 0, tir trac spacs on O and tir dual spacs. W dnot by C O) t spac of rstrictions to O of indfinitly diffrntiabl functions on R d and by DO) its subspac mad of functions wit a compact support in O. Lt also Hdiv, Ω) dnot t spac of functions v in L Ω) d suc tat div v blongs to L Ω), quippd wit t norm v Hdiv,Ω) = v L Ω) d + div v L Ω) ) 1..1) W rcall t Stoks formula, valid for smoot noug functions v and q, div v)x) qx)dx + vx) grad q)x)dx = v n)τ )qτ )dτ. Ω Ω Ω Sinc C Ω) d is dns in Hdiv, Ω) [4], Captr I, Torm.4, w driv from tis formula tat t normal trac oprator: v v n is dfind and continuous from Hdiv, Ω) into H 1 Ω). Tis lads to dfin H 0 div, Ω) = { } v Hdiv, Ω); v n =0on Ω..) Tn DΩ) d is dns in H 0 div, Ω) [4], Captr I, Torm.6, and bot Hdiv, Ω) and H 0 div, Ω) ar Hilbrt spacs for t scalar product associatd wit t norm dfind in.1). Rmark.1. Lt Γ b any part of Ω wit positiv masur. W rfr to [30], Captr 1, Sction 11, for t dfinition of H 1 00 Γ ) as t spac of functions in H 1 Γ ) suc tat tir xtnsion by zro blongs to H 1 Ω). T normal trac on Γ of a function v in Hdiv, Ω) maks sns in H 1 00 Γ ), owing to t following formula q H 1 00 Γ ), v n)τ )qτ )dτ = Γ Ω div v)x) qx)dx + Ω vx) grad q)x)dx, wr q is any lifting in H 1 Ω) of t xtnsion by zro of q to Ω clarly t intgral in t lft-and sid of t prvious quality rprsnts a duality pairing). Not morovr tat H 1 Γ ) is imbddd in H 1 00 Γ ). W now introduc t variational spacs { XΩ) = v Hdiv, Ω); v ΩF H 1 Ω F ) d}, } X 0 Ω) = {v XΩ); v n =0 onγ P and v = 0 on Γ F..3) Bot of tm ar quippd wit t norm v XΩ) = v Hdiv,Ω P ) + v H 1 Ω F ) ) 1,.4)

5 MORTAR FINITE ELEMENTS FOR DARCY AND STOES EQUATIONS 379 and ar Hilbrt spacs for t corrsponding scalar product. W also considr t bilinar forms au, v) =a P u, v)+a F u, v), wit a P u, v) =α ux) vx)dx, Ω P a F u, v) =ν grad u)x) :grad v)x)dx,.5) Ω F bv,q)= div v)x)qx)dx. Ω It is radily cckd tat t first tr forms ar continuous on XΩ) XΩ), wil t last on is continuous on XΩ) L Ω). T variational problm tat w considr now rads: Find u,p) in XΩ) L Ω) suc tat and tat u n = k on Γ P and u = g on Γ F,.6) v X 0 Ω), au, v)+bv,p)=lv), q L Ω), bu,q)=0,.7) wr t linar form L ) is dfind by Lv) = fx) vx)dx Ω v n)τ )p a τ )dτ + Γ ap vτ ) t a τ )dτ. Γ af.8) Not tat, in tis dfinition, w av usd intgrals for t sak of clarity, owvr ty ar most oftn rplacd by duality pairings. Indd, from now on, w mak t following assumption on t fiv data k H 1 ΓP ), g H 1 ΓF ) d, f X 0 Ω), p a H 1 00 Γ ap ), t a H 1 ΓaF ) d,.9) wr H 1 Γ P )andh 1 Γ af ) stand for t dual spacs of H 1 Γ P )andh 1 Γ af ), rspctivly. Wit tis coic, t boundary conditions.6) maks sns s Rm..1) and t form L ) is continuous on X 0 Ω). Standard argumnts lad to t quivalnc of problms 1.1) 1.5) and.6).7). Proposition.. Any smoot noug pair of functions u,p) is a solution of problm.6).7) if and only if it is a solution of problm 1.1) 1.5). To prov t wll-posdnss of problm.6).7), w first construct a lifting of t boundary conditions.6). Lmma.3. Tr xists a divrgnc-fr function u b in XΩ) wic satisfis u b n = k on Γ P and u b = g on Γ F,.10) and ) u b XΩ) c k H 1 + g ΓP ) H 1..11) ΓF )d

6 380 C. BERNARDI ET AL. Proof. It is prformd in tr stps. 1) Lt g b an xtnsion of g into H 1 Ω F ) d. W introduc a fixd smoot vctor fild ϕ wit support in Γ and st g Ω = g F g n)τ )dτ ϕ. Γ ϕ n)τ )dτ So t function g blongs to H 1 Ω F ) d and satisfis g n)τ )dτ =0 and g 1 c g H ΩF ) Ω d H 1 Γ F ) d. F Tus, t Stoks problm { ν ubf + grad p bf = 0 in Ω F, div u bf =0 inω F,.1) u bf = g on Ω F, as a solution in H 1 Ω F ) L Ω F ), wic is uniqu up to an additiv constant on t prssur [4], Captr I, Torm 5.1. Morovr, tanks to t prvious inquality, tis solution satisfis ) W now dnot by Y Ω P )tspac u bf H 1 Ω F ) c g d H 1 Γ F ) d..13) Y Ω P )= { µ H 1 Ω P ); µ =0 onγ ap }. Wn Γ ap as a positiv masur, w considr t problm: Find λ in Y Ω P ) suc tat µ Y Ω P ), grad λ)x) grad µ)x) = kτ )µτ )dτ + u bf n)τ )µτ )dτ..14) Ω P Γ P Γ Tis problm as a uniqu solution. Morovr t function u bp = grad λ is divrgnc-fr on Ω P as follows by taking µ in DΩ) in t prvious problm) and satisfis and u bp n = k on Γ P and u bp n = u bf n on Γ,.15) u bp Hdiv,ΩP ) c k H 1 Γ P ) + g H 1 Γ F ) d )..16) 3) Wn Γ ap as a zro masur, it follows from t dfinition of Γ ap and Γ af tat Γ af as a positiv masur. Tus, w introduc a furtr function g in H 1 Γ) d suc tat g n)τ )dτ = kτ )dτ, Γ Γ P and tr xists a function g in H 1 Ω F ) d qual to g on Γ F and to g on Γ not tat tis rquirs som compatibility conditions btwn g and g on Γ F Γ wn tis last st is not mpty). By adding to g a constant tims a fixd smoot function now wit support in Γ af, w construct a function g in H 1 Ω F )wic satisfis g n)τ )dτ =0. Ω F

7 MORTAR FINITE ELEMENTS FOR DARCY AND STOES EQUATIONS 381 Tn t Stoks problm.1) wit tis modifid function g still admits a solution, and tis solution satisfis ) u bf H 1 Ω F ) d c k H 1 + g Γ P ) H 1..17) Γ F ) d Nxt, sinc t function qual to k on Γ P and to u bf n = g n on Γ as a null intgral on Ω P, problm.14) admits a solution λ, uniqu up to an additiv constant not tat Y Ω P ) now coincids wit H 1 Ω P )). T function u bp = grad λ is divrgnc-fr on Ω P and still satisfis.15) and.16). To conclud, w obsrv from itr.13) or.17) and.16) tat t function u b qual to u bp on Ω P and to u bf on Ω F satisfis all t dsird proprtis. Rmark.4. Not tat t first assumption in.9) could b rplacd by t wakr on k H 1 00 Γ P ), s Rmark.1. Howvr, t prvious proof dos not work wit only tis assumption wn, for instanc, Γ P Γ is not mpty, s.14). So w do not andl tis modifid assumption sinc w av no dirct application for it. To go furtr, w st: u 0 = u u b,wru b is t function xibitd in Lmma.3. W obsrv tat problm.6).7) admits a solution if t following problm as on: Find u 0,p) in X 0 Ω) L Ω) suc tat It is radily cckd tat t krnl v X 0 Ω), au 0, v)+bv,p)= au b, v)+lv), q L Ω), bu 0,q)=0..18) V Ω) = { } v X 0 Ω); q L Ω), bv,q)=0,.19) coincids wit t spac of functions in X 0 Ω) wic ar divrgnc-fr on Ω. W first cck t llipticity of t form a, ) onv Ω). Lmma.5. Assum tat i) itr Γ F as a positiv masur in Ω F, ii) or t normal vctor nx) runs troug a basis of R d wn x runs troug Γ. Tr xists a constant α > 0 suc tat t following llipticity proprty olds Proof. Lt us obsrv tat, for all v in V Ω), v V Ω), av, v) α v XΩ)..0) av, v) min{α, ν} v L Ω P ) d + v H 1 Ω F ) d ),.1) and v XΩ) = v L Ω P ) d + v H 1 Ω F ) d + v L Ω F ) d ) 1..) Lt now v b a function in V Ω) suc tat v L Ω P ) d and v H 1 Ω F ) d ar qual to zro. Tus, v is zro on Ω P and is qual to a constant c on Ω F. Wn assumption i) olds, it follows from t dfinition of X 0 Ω) tat tis constant is zro. Wn assumption ii) olds, sinc v is zro on Ω P, c n is zro on Γ and, sinc n runs troug a basis of R d, c is zro. Tn v is zro on Ω. Tanks to t Ptr-Tartar lmma [4], Captr I, Torm.1, it follows from tis proprty,.) and t compactnss of t imbdding of H 1 Ω F )intol Ω F )tat v V Ω), v L Ω P ) d + v H 1 Ω F ) d ) 1 c v XΩ).

8 38 C. BERNARDI ET AL. Tis, combind wit.1), givs t dsird llipticity proprty. Lmma.6. Tr xists a constant β>0 suc tat t following inf-sup condition olds q L Ω), sup v X 0Ω) bv,q) v XΩ) β q L Ω)..3) Proof. Lt Ω + b a rctangl d = ) or a rctangular paralllpipd d = 3) suc tat Γ + =Γ a Ω + is containd in t intrior of Γ a and as a positiv masur. Tn, t function q + dfind by { q on Ω, q + = 1 masω +) Ω qx)dx on Ω +, blongs to L Ω Ω + ) and as a null intgral on tis domain. It tus follows from t standard inf-sup condition, s [4], Captr I, Corollary.4, tat tr xists a function v + in H 1 0 Ω Γ + Ω + ) d suc tat div v + = q + and v + H 1 Ω Γ + Ω +) d c q + L Ω Γ + Ω +). Taking v qual to t rstriction of v + to Ω wic obviously blongs to X 0 Ω)) lads to t dsird inf-sup condition. W ar now in a position to prov t main rsult of tis sction. Not tat, du to t mixd boundary conditions, no furtr assumption on t flux of t data is ndd for t xistnc of a solution. Torm.7. If t assumptions of Lmma.5 old, for any data k, g, f,p a, t a ) satisfying.9), problm.6).7) as a uniqu solution u,p) in XΩ) L Ω). Morovr tis solution satisfis ) u XΩ) + p L Ω) c k H 1 + g ΓP ) H 1 + f X ΓF 0Ω) + p a )d H 100ΓaP + t a ) H 1..4) ΓaF )d Proof. It follows from Lmmas.5 and.6, s [4], Captr I, Torm 4.1, tat problm.18) as a uniqu solution u 0,p)inX 0 Ω) L Ω) and tat tis solution satisfis ) u 0 XΩ) + p L Ω) c u b XΩ) + f X0Ω) + p a H 100ΓaP + t a ) H 1..5) ΓaF )d Tn, t pair u = u 0 + u b,p) is a solution of problm.6).7), and stimat.4) is a consqunc of.5) and.11). On t otr and, lt u 1,p 1 )andu,p ) b two solutions of problm.6).7). Tn, t diffrnc u 1 u,p 1 p ) is a solution of problm.18) wit data u b, f, p a and t a qual to zro. Tus, it follows from.5) tat it is zro. So t solution of problm.6).7) is uniqu. From now on, w assum tat t non rstrictiv assumptions of Lmma.5 old. W conclud wit som rgularity proprtis of t solution u,p). Proposition.8. Lt us assum tat t fiv data satisfy k H 1 ΓP ), g H 3 ΓF ) d, f H 1 Ω) d, p a H 3 ΓaP ), t a H 1 ΓaF ) d..6) Tn, t rstriction u ΩP,p ΩP ) of t solution u,p) of problm.6).7) to Ω P blongs to t spac H sp Ω P ) d H sp +1 Ω P ) for a ral numbr s P > 0 givn by s P =1/4 if Ω P is a polygon d =); s P =1/ if Γ ap is mpty or if Ω P is a polygon or a polydron and tr xists a convx nigbourood in Ω P of Γ P Γ) Γ ap ;

9 MORTAR FINITE ELEMENTS FOR DARCY AND STOES EQUATIONS 383 s P < 1 if Γ ap is mpty and Ω P is a convx polygon or polydron or as a C 1,1 -boundary. T rstriction u ΩF,p ΩF ) of t solution u,p) of problm.6).7) to Ω F blongs to t spac H sf +1 Ω F ) d H sf Ω F ) for a ral numbr s F > 0 givn by s F =1/4 if Ω F is a polygon d =); s F =1/ if Γ F is mpty or if Ω F is a polygon d =) and tr xists a convx nigbourood in Ω F of Γ af Γ) Γ F ; s F < 1 if Γ F is mpty and Ω F is a convx polygon or polydron or as a C 1,1 -boundary. Proof. W cck succssivly t two assrtions. 1) T function p ΩP is a solution of t Poisson quation wit mixd boundary conditions p = div f in Ω P, p = p a on Γ ap, n p = f n αk on Γ P, n p = f n α u ΩF n on Γ. Morovr, sinc u ΩF blongs to H 1 Ω) d, its normal trac u ΩF n blongs to H 1 Γ). T dsird rgularity of p ΩP is asily drivd from [6], Torms...3 and 3..1., or [19], Sction 3, tanks to appropriat Sobolv imbddings. T rgularity of u ΩP tn follows from t first lin in 1.1). ) T pair u ΩF,p ΩF ) is a solution of t Stoks problm wit mixd boundary conditions ν u + grad p = f in Ω F, div u =0 inω F, u = g on Γ F, ν n u p n = t a on Γ af, ν n u p n = p ΩP n on Γ. It can also b notd from part 1) of t proof tat p ΩP n blongs at last to H 1 Γ) d. So t dsird rsults follow from [3]. Assumption.6) is too strong for most rsults of Proposition.8, and w only mak it for simplicity. Morovr t norms of u ΩP,p ΩP )inh sp Ω P ) d H sp +1 Ω P )andofu ΩF,p ΩF )inh sf +1 Ω F ) d H sf Ω F ) ar boundd as a function of wakr norms of t data. Not also tat compatibility conditions on t data at t intrsctions of diffrnt parts of t boundaris sould b mad to obtain igr rgularity, i.. to brak t rstrictions s P < 1ands F < 1. Similar rsults old in otr situations tat w do not considr in tis work for instanc, wn Γ af is mpty). 3. T discrt problm and its wll-posdnss T mortar finit lmnt discrtization rlis on t partition of Ω into Ω P and Ω F. Indd, vn if som furtr partitions could b introducd to andl anisotropic domains for instanc, w do not considr tm in tis work. Lt T P ) P and T F ) F b rgular familis of triangulations of Ω P and Ω F, rspctivly, by closd triangls d = ) or ttradra d = 3), in t usual sns tat: For ac P, Ω P is t union of all lmnts of T P and, for ac F, Ω F is t union of all lmnts of T F. T intrsction of two diffrnt lmnts of T P, if not mpty, is a vrtx or a wol dg or a wol fac of bot of tm, and t sam proprty olds for t intrsction of two diffrnt lmnts of T F. T ratio of t diamtr of any lmnt of T P F or of T to t diamtr of its inscribd circl or spr is smallr tan a constant σ indpndnt of P and F.

10 384 C. BERNARDI ET AL. As usual, P stands for t maximum of t diamtrs of t lmnts of T P and F for t maximum of t diamtrs of t lmnts of T F.Fromnowon,c, c,... dnot for gnric constants tat may vary from on lin to t nxt but ar always indpndnt of P and F. W mak t furtr standard and non rstrictiv assumptions. Assumption 3.1. T intrsction of ac lmnt of T P wit itr Γ ap or Γ P or Γ, if not mpty, is a vrtx or a wol dg or a wol fac of. T intrsction of ac lmnt of T F wit itr Γ af or Γ F or Γ, if not mpty, is a vrtx or a wol dg or a wol fac of. It must b notd tat, up to now, no assumption is mad on t intrsction of t lmnts of T P and T F. So t Γ, T P F,andt Γ, T, form two indpndnt triangulations of Γ, tat w dnot by E P,Γ and E F,Γ, rspctivly. Howvr, w ar ld to mak anotr assumption. Assumption 3.. For any lmnt of T F, t numbr of lmnts of T P suc tat as a positiv d 1)-masur is boundd indpndntly of, P and F. W now dfin t local discrt spacs. For t rasons alrady xplaind in t introduction, t spac of discrt vlocitis in Ω P is constructd from t Raviart-Tomas finit lmnt [34], wic lads to t following dfinition X P = { v Hdiv, Ω P ); T P, v P RT ) }, 3.1) wr P RT ) stands for t spac of rstrictions to of polynomials of t form a + b x, a R d and b R. W also introduc t spac X0 P = { v X P } ; v n =0 onγ P. 3.) Similarly, on Ω F, w considr t spac rlatd to t Brnardi-Raugl lmnt [6], i.. X F = { v H 1 Ω F ) d ; T F, v P BR ) }, 3.3) wr P BR ) stands for t spac spannd by t rstrictions to of affin functions on R d wit valus in R d and t d+1 normal bubbl functions ψ n for ac dg d =)orfacd =3) of, ψ dnots t bubbl function on qual to t product of t barycntric coordinats associatd wit t ndpoints or vrtics of and n stands for t unit outward normal vctor on ). W also nd t spac X F 0 = { v X F ; v = 0 on Γ F }. 3.4) Lt now dnot t discrtization paramtr, r qual to t pair P, F ), and lt T stand for t union of T P and T F. W dfin t discrt spac of prssurs as { } M = q L Ω); T, q P 0 ), 3.5) wr P 0 ) is t spac of constant functions on. Rmark 3.3. Otr coics of finit lmnts ar possibl. Indd, t Raviart-Tomas lmnt is t simplst div-conforming lmnt and t Brnardi-Raugl lmnt is t lss xpnsiv H 1 -conforming finit lmnt for t Stoks problm. In dimnsion d =, picwis quadratic vlocitis can also b usd on Ω F and in dimnsion d =3,P BR ) can b rplacd by t spac spannd by affin functions and t ψ,uptot powr d. T sklton of t dcomposition is now t intrfac Γ. As standard for t mortar lmnt mtod, s [7] and [11], t construction of t global spac of vlocitis rlis on t fact tat t matcing conditions ar nforcd via t ortogonality to functions dfind on T P or T F. Sinc ts matcing conditions only dal wit t normal trac of t vlocity, w av dcidd to mak t coic proposd in [1], Sction 3, wic is

11 MORTAR FINITE ELEMENTS FOR DARCY AND STOES EQUATIONS 385 mor naturally associatd wit functions dfind on T P, i.. w dfin t spac W = { ϕ L Γ); E P,Γ,ϕ P 0 ) }, 3.6) wit obvious dfinition for P 0 ). T global spacs of vlocitis ar tn t spacs X and X 0 of functions v suc tat tir rstrictions v ΩP to Ω P blong to X P and XP 0, rspctivly; tir rstrictions v ΩF to Ω F blong to X F and XF 0, rspctivly; t following matcing conditions old on Γ ϕ W, v ΩP v ΩF ) n ) τ )ϕ τ )dτ =0, 3.7) Γ wr τ stands for t tangntial coordinats) on Γ. Not tat ts conditions ar not sufficint to nforc t continuity of v n troug Γ, so tat t discrtization is nonconforming: For instanc, X is not containd in Hdiv, Ω). Howvr, t spacs X and X 0 ar still quippd wit t norm XΩ). Rmark 3.4. In t implmntation of t discrt problm, t matcing conditions 3.7) ar andld via t introduction of a Lagrang multiplir, as usual for t mortar mtod. W rfr to [5] fortfirstanalysisof tis algoritm and to [11], Sction 4, for anotr way of trating ts conditions. To discrtiz t ssntial boundary conditions tat appar in.6), w now dfin t approximations of t data k and g tat w us in tis work. W dnot by k t picwis constant approximation of k dfind by T P / mas Γ P ) > 0, k ΓP = 1 mas Γ P ) Γ P kτ )dτ. 3.8) Not tat tis coic rquirs tat k blongs to H σ Ω), σ< 1. W also introduc an approximation of g: Wn assuming tat g is continuous on Γ F wic is sligtly strongr tan t ypotsis mad in.9)), t function g blongs to t trac spac of X F ; for ac in T F,isqualtoga) at ac ndpoint or vrtx a of Γ F ; and satisfis g n)τ )dτ = g n)τ )dτ. Γ F Γ F Indd, ts conditions dfin k and g in a uniqu way, as follows from [34], Rmark 3, and [6], Lmma II.1. W ar now in a position to writ t discrt problm, wic is constructd by t Galrkin mtod from.7). It rads: Find u,p ) in X M suc tat and tat wr t bilinar form b, ) is dfind by u n = k on Γ P and u = g on Γ F, 3.9) v X 0, au, v )+ bv,p )=Lv ), q M, bu,q )=0, 3.10) bv,q)= Ω P div v ΩP )x)qx)dx Ω F div v ΩF )x)qx)dx. 3.11)

12 386 C. BERNARDI ET AL. T introduction of tis modifid form is du to t nonconformity of t discrtization, and it is radily cckd tat it coincids wit b, ) onhdiv, Ω) L Ω). As in t continuous cas, to prov t wll-posdnss of problm 3.9) 3.10), w first construct a lifting of t boundary conditions 3.9). It rquirs t Raviart-Tomas oprator Π RT Sction 1.3, for its tr-dimnsional analogu: For any smoot noug function v on Ω P,Π RT and satisfis on all dgs d = ) or facs d =3) of lmnts of T P,,s[34], Sction 3, and also [31], v blongs to XP Π RT v n)τ )dτ = v n)τ )dτ. 3.1) T fact tat ts quations dfin t oprator Π RT in a uniqu way and its main proprtis ar provd in [34], Torm 3, in t two-dimnsional cas. Morovr, tis oprator prsrvs t nullity of t normal trac on Γ P tis rquirs Assumption 3.1). Similarly, w introduc anotr oprator tat w call Brnardi-Raugl oprator and dnot by Π BR vrtx a of t lmnts of T F : For any continuous function v on Ω F,Π BR v blongs to XF,isqualtova) atany F and satisfis on all dgs d = ) or facs d =3) of lmnts of T, Π BR v n)τ )dτ = v n)τ )dτ. 3.13) Tis dfins t oprator Π BR in a uniqu way, s [6], Lmma II.1. W now stablis som proprtis of t oprator Π RT. W rfr to [5], Appndix, for tir proof in t two-dimnsional cas and for quadrilatral finit lmnts and to [15], Sction III.3, for additional rsults. It rquirs t Piola transform A, dfind as follows, s [4], Captr III, formula 4.63): For any lmnt of T P,dnotingbyF on of t affin mappings wic maps t rfrnc triangl or ttradron ˆ onto and by B t Jacobian matrix of F, w associat wit any vctor fild ˆv dfind on ˆ t vctor fild v = A ˆv dfind on by t formula A ˆv) F = 1 dt B B ˆv. 3.14) W rcall two proprtis of tis transform, valid for all smoot noug functions v and ϕ div v) F = v n)τ )ϕτ )dτ = 1 dt B ˆ div A 1 v), 3.15) A 1 v ˆn)ˆτ )ϕ F )ˆτ )dˆτ, 3.16) wr n and ˆn stand for t unit outward normal vctors to and ˆ, rspctivly. W also introduc t basis functions associatd wit t spac X P :IfEP dnots t st of dgs d = ) or facs d =3)oflmnts of T P, w associat t function ϕ in X P suc tat ϕ n)τ )dτ =1 and E P,, ϕ n)τ )dτ = ) T ϕ, E P,formabasisofXP. Morovr, it is radily cckd tat ac ϕ n is picwis constant, 1 qual to mas) on and to zro on all. Lmma 3.5. T following proprty olds for any in T P and any v in Hdiv, Ω P ), div Π RT v L ) div v L ). 3.18)

13 MORTAR FINITE ELEMENTS FOR DARCY AND STOES EQUATIONS 387 T following proprty olds for any in T P and any v in Hdiv, Ω P ) H s Ω P ) d, 0 <s< 1, Π RT v L ) d c v L ) d + s v H s ) d + div v L )). 3.19) Proof. W cck succssivly t two assrtions of t lmma. 1) Sinc t divrgnc of all functions in X P P is constant on ac lmnt of T,wav div Π RT It follows from t dfinition 3.1) of Π RT div Π RT v L ) =divπrt v) div Π RT v)x)dx =divπ RT v) Π RT v n)τ )dτ. tat v L ) =divπrt v) v n)τ )dτ = div Π RT v)divv)x)dx, so tat using a Caucy-Scwarz inquality yilds 3.18). ) Dnoting by E t st of dgs d = ) or facs d =3)of, w av from 3.1) so tat Π RT v) = E Π RT v L ) d E v n)τ )dτ ) ϕ, v n)τ )dτ ϕ L ) d. 3.0) Wn stting ê = F 1 ), it follows from 3.16) and 3.17) tat t function ϕ = A 1 ϕ is suc tat ϕ ˆn)ˆτ )dˆτ =1 and ê E ˆ, ê ê, ϕ ˆn)ˆτ )dˆτ =0, ê ê so tat ϕ L ˆ) d is boundd indpndntly of. Tus, standard argumnts rlying on 3.14) giv ϕ L ) d c1 d. 3.1) On t otr and, dnoting by χ t function qual to 1 on andto0on \, by χ t function χ F and by χ a lifting of χ to ˆ, w av from 3.16) v n)τ )dτ = A 1 v ˆn)ˆτ ) χ ˆτ )dˆτ = ˆ A 1 v)ˆx) grad χ )ˆx)dˆx + ˆ ˆ div A 1 v)) ˆx) χ ˆx)dˆx. Not owvr tat, sinc χ only blongs to H r ˆ) for all r< 1,grad χ )ˆx) only blongs to H r 1 ˆ) and tat t first intgral in t scond lin of t prvious quation must b rplacd by a duality pairing. Tn, coosing r suc tat 1 r = s yilds v n)τ )dτ c A 1 v H s ˆ) + div A 1 d v) ˆ)) L.

14 388 C. BERNARDI ET AL. Standard argumnts rlying on 3.14), 3.15) and t us of intrinsic norm and sminorm on H s ˆ), s for instanc [3], Sction 7.43, giv v n)τ )dτ c d 1 ) v L ) d + d +s 1 v H s ) d + d div v L ). 3.) Insrting 3.1) and 3.) into 3.0) lads to 3.19). W now brifly prov analogous rsults for t oprator Π BR. Lmma 3.6. T following proprty olds for any ral numbr s 0, 0 s 0 1, for any in T F <s, v E and any v in H s Ω F ) d, d Π BR v H s 0 ) d cs s0 v H s ) d. 3.3) Proof. Lt I dnot t Lagrang intrpolation oprator wit valus in picwis affin functions. It follows from t dfinition of Π BR tat, if E dnots t st of dgs d = ) or facs d =3)of, Π BR v) =I v) + ) v I v) n τ )dτ ψ ψ n. τ )dτ W rcall t usual stimat, for 0 r 0 1and d <r, Applying tis stimat wit r 0 = s 0 yilds v I v H r 0 ) d c r r0 v H r ) d. 3.4) v I v H s 0 ) d On t otr and, w driv from standard argumnts tat ψ n H s 0 ) d c d s0, ψ τ )dτ c d 1 c s s0 v H s ) d. 3.5) Combining tis wit 3.) and tr applications of 3.4) givs for ac in E ) v I v) n τ )dτ ψ τ )dτ ψ n H s 0 ) d c s s0 v H s ) d. Tis inquality and 3.5) yild t dsird stimat. To go furtr, w nd t following rsult wic is a consqunc of Assumption 3.. Lmma 3.7. For ac, lt λ dnot t maximal ratio /, wr runs troug T F, runs troug T P and as a positiv d 1)-masur. Tn, all λ ar smallr tan a constant λ indpndnt of. Proof. Lt b any lmnt of T F wic as an dg d =)orafacd =3)containd in Γ. Assumption 3. yilds tat is containd in t union of dgs or facs i,1 i I, oflmnts i of T P,wrI is boundd indpndntly of and. So,wav mas) I mas i ). i=1.

15 MORTAR FINITE ELEMENTS FOR DARCY AND STOES EQUATIONS 389 On t otr and, mas) isquivalntto d 1 and ac mas i)isquivalntto d 1 i, wit quivalnc constants only dpnding on t rgularity paramtr σ; wn i and j ar adjacnt, i.. sar a vrtx in dimnsion d = or an dg in dimnsion d =3,t ratio i / j is boundd by constants only dpnding on σ; for all i and j, tr xists a pat linking i to j, only going from an to an adjacnt and crossing at most c lmnts, wrc is boundd as a function of I. Combining all tis yilds t dsird rsult. Lmma 3.8. If t data k, g) blong to H σp Γ P ) H σf Γ F ) d, σ P > 1 and σ F > d 1, tr xists a function u b in X wic satisfis and u b n = k on Γ P and u b = g on Γ F, 3.6) u b XΩ) c k H σ P ΓP ) + g H σ F ΓF ) d ). 3.7) Proof. W us onc mor t function u b xibitd in Lmma.3 and, sinc it is constructd from t solutions of problms.1) and.14), w obsrv from [6], Sction 7.3.3, or [19], Corollary 7, tat, sinc Ω P and Ω F ar polygons or polydra, tr xist ral numbrs s P,0<s P <σ P + 1,ands F, d <s F <σ F + 1, suc tat t pair u b ΩP, u b ΩF ) blongs to H sp Ω P ) d H sf Ω F ) d and satisfis u b H s P ΩP ) d + u b H s F ΩF ) d c k H σ P ΓP ) + g H σ F ΓF ) d ). 3.8) T construction of t function u b is now prformd in two stps. 1) W first introduc t function w 1 suc tat w 1 Ω P =Π RT u b Ω P, w 1 Ω F =Π BR u b Ω F. It follows from Lmmas 3.5 and 3.6 tat, sinc u b is divrgnc-fr on Ω P, Morovr, owing to t dfinitions of Π RT ) Rcalling tat E P,Γ in Γ, w considr t function w w 1 Hdiv,ΩP ) + w 1 H1 Ω F ) d c u b H s P ΩP ) d + u b H s F ΩF ) d ). 3.9) and Π BR, t function w1 dnots t st of dgs d = ) or facs d =3)oflmntsofT P dfind by w Ω P = E P,Γ satisfis t boundary conditions 3.6). wic ar containd w 1 ΩF w 1 Ω P ) n ) ) τ )dτ ϕ, w Ω F = 0 wr t functions ϕ ar dfind in 3.17). W obsrv from t coic of w tat t function u b = w 1 +w satisfis t matcing conditions 3.7), nc blongs to X. Owing to t proprtis of t functions ϕ, w n vaniss on Γ P,sotatu b satisfis 3.6). Morovr, it follows from 3.15) and 3.1) tat, if dnots t triangl of T P tat contains, ϕ Hdiv,) c d. 3.30) Nxt, owing to t dfinition of w 1,wav w 1 ΩF w 1 Ω P ) n ) τ )dτ = ub w 1 Ω F ) n ) τ )dτ.

16 390 C. BERNARDI ET AL. Applying 3.) yilds ub w 1 Ω F ) n ) τ )dτ c κ d 1 κ u b Π BR u b L κ) d + d +s 1 κ u b Π BR u b Hs κ) d + d κ u b Π BR u b H1 κ) d ), wr t prvious summation is takn on all t κ in T F suc tat κ as a positiv masur. W us Lmma 3.6 to bound t norms on t κ. Combining all tis wit 3.30) yilds w 1 ΩF w 1 Ω P ) n ) τ )dτ ϕ Hdiv,) c d d +sf 1 κ u b H s F κ) d. Not also tat t ratio d κ / d is boundd by λ d, nc by a constant indpndnt of, s Lmma 3.7. Tis givs w sf 1 Hdiv,Ω P ) cf u b H s F ΩF ) d. 3.31) Finally, stimat 3.7) is drivd from 3.8), 3.9) and 3.31). W prov a furtr rsult wic is ndd in Sction 4. It rquirs t following paramtrs. Notation 3.9. T paramtrs λ P and λ F ar dfind as follows: i) λ P is positiv in t gnral cas, qual to 1/4 ifω P is a polygon d =),qualto1/ ifγ ap is mpty or if tr xists a convx nigbourood in Ω P of Γ P Γ) Γ ap and < 1ifΓ ap is mpty and Ω P is a convx polygon or polydron; ii) λ F is qual to 1/ in t gnral cas and to 1 if Ω F is convx. Corollary If t assumptions of Lmma 3.8 ar satisfid, t following stimats old btwn t function u b introducd in Lmma.3 and t function u b introducd in Lmma 3.8 and u b u b XΩ) c bub,q ) sup q M q L Ω) κ ) min{σp + 1,λP } P + min{σf 1,λF } k H ) σp F Γ P ) + g H σf ΓF ), 3.3) d c min{σf 1,λF } ) F k H σ P ΓP ) + g H σ F ΓF ). 3.33) d Proof. Owing to t rgularity proprtis of problms.1) and.14), s [6], Sction 7.3.3, or [19], Corollary 3.7, stimat 3.8) olds wit { s P =min σ P + 1 },λ P and { s F =min σ F + 1 },λ F +1. Wit t notation of t prvious proof, sinc bot u b and Π RT u b ar divrgnc-fr on Ω P,wavt inquality u b u b XΩ) u b Π RT u b L Ω P ) d + u b Π BR u b H1 Ω F ) d + w Hdiv,ΩP ). 3.34) T approximation proprtis of t oprator Π RT ar asily drivd from t fact tat it prsrvs t constants on ac in T P, by applying 3.19) to t function v c for an appropriat constant c and using t approximation proprtis of tis constant. Ty rad, for 0 <r 1, v Π RT v L ) d cr v Hr ) d. 3.35)

17 MORTAR FINITE ELEMENTS FOR DARCY AND STOES EQUATIONS 391 So, using 3.35) to bound t first trm in t rigt-and sid of 3.34), 3.3) to bound t scond trm and 3.31) to bound t tird trm yilds 3.3). W also driv from t proprtis 3.1) and 3.13) of t oprators Π RT and Π BR tat, sinc u b is divrgnc-fr on Ω, w av for all q in M, 1 bw,q )= q w 1 n)τ )dτ = q u b n)τ )dτ =0, T P T F T P T F so tat bub,q )= bw,q ) w Hdiv,Ω P ) q L Ω), and w driv 3.33) from 3.31). In analogy wit Sction, w now st: u 0 = u u b,wru b is t function xibitd in Lmma 3.8. Tis lads to considr t problm: Find u 0,p ) in X 0 M suc tat W also introduc t discrt krnl v X 0, au 0, v )+ bv,p )= au b, v )+Lv ), q M, bu0,q )= bu b,q ). 3.36) V = { v X 0 ; q M, bv,q )=0 }. 3.37) It must b notd tat t functions in V ar divrgnc-fr only on Ω P. W now study t proprtis of t forms a, ) and b, ) on t discrt spacs. Lmma If Γ F as a positiv masur in Ω F, tr xists a constant α >0 suc tat t following llipticity proprty olds v V, av, v ) α v XΩ). 3.38) Proof. Sinc functions in V ar divrgnc-fr on Ω P, proprtis.1) and.) still old for all functions v in V. So, w now wis to cck tat v V, v L Ω F ) d c v H1 Ω F ) d. Wn Γ F as a positiv masur, tis inquality is a simpl consqunc of t Poincaré-Fridrics inquality and of t imbdding of X F 0 into t spac of functions in H1 Ω F ) vanising on Γ F. Rmark 3.1. Wn Γ F as a zro masur but t normal vctor nx) wnx runs troug Γ runs troug a basis of R d tis is t scond possibl assumption of Lm..5), it is radily cckd tat any lmnt of V suc tat av, v ) = 0 is qual to zro. Tus, using t quivalnc of norms on t finit-dimnsional spac V yilds tat tr xists a constant α positiv but dpnding on t triangulations T P and T F suc tat v V, av, v ) α v XΩ). 3.39) Howvr t standard argumnts to valuat t dpndnc of α wit rspct to P and F sm to fail r. Fortunatly, t assumption tat Γ F as a positiv masur in Ω F is not rstrictiv for t applications tat w wis to considr. W now prov t inf-sup condition on b, BR ). It rquirs t modifid Brnardi-Raugl oprator Π dfind as follows: if R dnots a Clémnt typ rgularization oprator wit valus in t spac of picwis affin functions wic vanis on Γ F s for instanc [9], Sct. IX.3, for a dtaild dfinition of suc an oprator), Π BR v) =R v) + ) v R v) n τ )dτ ψ ψ n. 3.40) τ )dτ E

18 39 C. BERNARDI ET AL. Lmma Tr xist two constants 0 > 0 and β >0 suc tat, itr wn bot Γ ap and Γ af av a positiv masur or for all 0, t following inf-sup condition olds q M, bv,q ) sup v X 0 v β q L Ω). 3.41) XΩ) W must prov tis lmma in t nxt tr situations: Wn bot Γ ap and Γ af av a positiv masur, wn Γ ap as a zro masur and wn Γ af as a zro masur. Howvr, w skip t proof in t tird situation sinc it is lss ralistic tan t scond on s Fig. 1) and t argumnts ar xactly t sam. Proof. Cas wr Γ ap and Γ af av a positiv masur. In tis situation, it follows from xactly t sam argumnts as in t proof of Lmma.6 tat, for any function q in M, tr xists a function v P in H 1 Ω P ) d, vanising on Γ P andalsoonγsuctat div v P = q on Ω P and v P H1 Ω P ) d c q L Ω P ), 3.4) and also a function v F in H 1 Ω F ) d, vanising on Γ F Γ suc tat div v F = q on Ω F and v F H 1 Ω F ) d c q L Ω F ). 3.43) W now dfin v ΩP =Π RT v BR P, v ΩF = Π v F. Only for tis proof, w mak t furtr assumption tat t oprator R taks its valus in t spac of BR picwis affin functions wic also vanis on Γ, so tat Π v F vaniss on Γ F Γ. On t otr and, it is radily cckd tat all functions v in P RT ) arsuctatv n is constant on ac dg d =)orfac d =3)of, sotatπ RT v P n vaniss on Γ P Γ. Ts two proprtis yild tat t function v satisfis tat matcing conditions 3.7), nc blongs to X 0.Walsoav bv,q )= q v n)τ )dτ q v n)τ )dτ. T P So it follows from t dfinition of t oprators Π RT bv,q )= q T P Combining tis wit 3.4) and 3.43) yilds W also dduc from Lmma 3.5 tat and Π BR T F tat v P n)τ )dτ T F q = div v P )x)q x)dx div v F )x)q x)dx. Ω P Ω F v F n)τ )dτ bv,q )= q L Ω). 3.44) v Hdiv,ΩP ) c v P H1 Ω P ) d, wnc, from 3.4), v Hdiv,ΩP ) c q L Ω P ). 3.45)

19 MORTAR FINITE ELEMENTS FOR DARCY AND STOES EQUATIONS 393 T sam argumnts as in t proof of Lmma 3.6, wit 3.4) rplacd by s [9], Cap. IX, T. 3.11) wr is t union of lmnts κ of T F v R v H s 0 ) d c 1 s0 v H 1 ) d, suc tat κ is not mpty, lad to v H1 Ω F ) d c v F H1 Ω F ) d, wnc, owing to 3.43), v H 1 Ω F ) d c q L Ω F ). 3.46) T dsird inf-sup condition now follows from 3.44), 3.45) and 3.46). Proof. Cas wr Γ ap as a zro masur. Lt ϕ Γ b a smoot vctor fild wit support containd in t intrior of Γ suc tat ϕ Γ n)τ )dτ =1. Γ W dfin ϕ Γ in t following way: On Ω F, ϕ Γ is affin on all lmnts of T F and is qual to ϕ Γa) atall vrtics a of ts lmnts tat blong to Γ and to zro at all otr vrtics; on Ω P,wst ϕ Γ Ω P = ) ϕ Γ Ω F n)τ )dτ ϕ. E P,Γ Tus, it is radily cckd tat ϕ blongs to Γ X0 and morovr tat, wn is small noug, ϕ n)τ )dτ 1 Γ 3.47) Γ For a wil, w st b P v,q)= div v ΩP )x)qx)dx, Ω P b F v,q) div v ΩF )x)qx)dx. Ω F Nxt, w procd in two stps. 1) On Ω P, w us t dcomposition q ΩP = q + q, wit q = 1 masω P ) Ω P q x)dx. Indd, tr xists a stabl function v in H0 1 Ω P ) d suc tat div v = q ; tn, t function ṽ =Π RT v blongs to X P H 0div, Ω P ) and satisfis b P ṽ, q )= q L Ω P ) and ṽ Hdiv,ΩP ) c q L Ω P ). 3.48) On t otr and, it is radily cckd by intgration by parts and also from 3.47) tat t function v = q masω P ) Γ ϕ Γ n)τ )dτ ϕ Γ, satisfis b P v, q )= q L Ω P ) and v XΩ) c q L Ω P ). 3.49)

20 394 C. BERNARDI ET AL. Tus, applying t Boland and Nicolaids argumnt, s [13], wic rlis on t ortogonality proprtis b P ṽ, q )=0, q x)q x)dx =0, Ω P givs t xistnc of a constant µ indpndnt of suc tat t function v ΩP = ṽ + µ v satisfis b P v,q ) c q L Ω P ) and v Hdiv,ΩP ) c q L Ω P ). 3.50) ) It follows from t dfinition of ϕ Γ tat div v ) ΩF is constant on ac lmnt of T F. Tus, Lmma.6 yilds t xistnc of a function v in H 1 Ω F ) d, vanising on Γ F Γ, suc tat div v is qual to q +divµv ) and applying t modifid Brnardi-Raugl oprator Π BR v + µ v satisfis Π BR dfind in 3.40) to it yilds tat t function v ΩF = b F v,q )= q L Ω F ) and v H 1 Ω F ) d c q L Ω F ) + v XΩ) ). 3.51) To conclud, w obsrv tat t function v blongs to X 0. T dsird inf-sup condition is tn drivd from 3.50), 3.51) and 3.49). From now on, w assum tat is small noug for t inf-sup condition 3.41) to old. Indd, tis condition is only ndd wn Γ ap or Γ af as a zro masur and, wit t notation of t prvious proof, can b writtn, wn Γ ap as a zro masur for instanc, ϕ Γ ϕ Γ L 1 Γ) 1 So, sinc ϕ Γ is vry smoot, it is not at all rstrictiv. Owing to t prvious lmmas, w ar now in a position to prov t main rsult of tis sction. Torm Assum tat Γ F as a positiv masur in Ω F. Tn, for any data k, g, f,p a, t a ) satisfying k H σp Γ P ), g H σf Γ F ) d, f L Ω) d, p a H 1 00 Γ ap ), t a H 1 ΓaF ) d, 3.5) for som ral numbrs σ P > 1 and σ F > d 1, problm 3.9) 3.10) as a uniqu solution u,p ) in X M. Morovr tis solution satisfis u XΩ) + p L Ω) c k H σ P ΓP ) + g H σ F ΓF ) + f d L Ω) d + p a H 1 00 Γ ap ) + t a H 1 Γ af ) d ). 3.53) Proof. W cck sparatly t xistnc and t uniqunss. 1) Lt u b dnot t function xibitd in Lmma 3.8. It follows from t llipticity proprty 3.38) and t inf-sup condition 3.41), s [4], Captr I, Torm 4.1, tat problm 3.36) as a uniqu solution u 0,p ) in X 0 M wic morovr satisfis ) u 0 XΩ) + p L Ω) c u b XΩ) + f L Ω) d + p a H 100ΓaP + t a ) H ) ΓaF )d Tn, t pair u = u 0 + u b,p ) is a solution of problm 3.9) 3.10), and stimat 3.53) is a dirct consqunc of 3.7) and 3.54). ) If all data k, g, f,p a, t a ) ar qual to zro, u,p ) is a solution of problm 3.36) wit t rigt-and sids of t two quations qual to zro. Tus, it follows from 3.38) and 3.41) tat it is qual to zro. So, t solution of problm 3.9) 3.10) is uniqu.

21 MORTAR FINITE ELEMENTS FOR DARCY AND STOES EQUATIONS 395 Rmark T rgularity assumptions tat ar mad on t data f in Torm 3.14 can asily b waknd: It suffics to nforc tat f ΩP blongs to t dual spac of functions on Hdiv, Ω P ) wit zro normal tracs on Γ P and f ΩF blongs to t dual spac of functions on H 1 Ω F ) d vanising on Γ F. Howvr w av no dirct application for tis wakr rgularity. 4. A PRIORI rror stimats W intnd to prov an rror stimat btwn t solution u,p) of problm.6).7) and t solution u,p ) of problm 3.9) 3.10). T main difficulty r is tat applying t intrpolation oprator I or t oprator Π BR to t solution u ΩF in ordr to rcovr t boundary condition g of t discrt problm) would rquir tat u ΩF is continuous on Ω F. In viw of Proposition.8, tis assumption is not likly, at last in dimnsion d = 3. So w prfr to follow anotr approac, basd on t triangl inquality u u XΩ) u b u b XΩ) + u 0 u 0 XΩ), 4.1) wr t functions u b and u b ar introducd in Lmmas.3 and 3.8, rspctivly. An stimat for t quantity u b u b XΩ) is stablisd in Corollary So w ar now intrstd in proving t following vrsion of t scond Strang s lmma for problms.18) and 3.36), t main difficulty bing du to t nonconformity of t mortar lmnt discrtization. Lmma 4.1. Assum tat Γ F as a positiv masur in Ω F. T following stimat olds btwn t solution u 0,p) of problm.18) and t solution u 0,p ) of problm 3.36) u 0 u 0 XΩ) c inf u 0 w XΩ) + inf p r L Ω) w V r M bub,q ) + u b u b XΩ) + sup + sup q M q L Ω) v X 0 4.) Γ v ΩP v ΩF ) n ) τ ) p ΩP τ )dτ ) v XΩ) Proof. It is dividd in tr stps. 1) Owing to t inf-sup condition 3.41), tr xists [4], Captr I, Lmma 4.1, a function ũ in X 0 suc tat q M, bũ,q )= bu 0,q ), and, by using t scond lin of 3.36), ũ XΩ) 1 bub,q ) β sup 4.3) q M q L Ω) Tn, t function ũ 0 = u 0 ũ blongs to V and satisfis v V, aũ 0, v )= au b, v ) aũ, v )+Lv ). 4.4) ) Wn multiplying t first lins of 1.1) and 1.) by a function v of V, intgrating by parts and summing t two rsulting quations, w obtain au, v )+ bv,p)=lv ) v ΩP v ΩF ) n ) τ ) p ΩP τ )dτ. Γ

22 396 C. BERNARDI ET AL. Tis last quation can b writtn quivalntly as v V, au 0, v )+ bv,p)= au b, v )+Lv ) v ΩP v ΩF ) n ) τ ) p ΩP τ )dτ. 4.5) 3) Lt now w and r b any lmnts of V and M, rspctivly. It follows from 4.4) and 4.5) tat Γ v V, aũ 0 w, v )=au 0 w, v )+au b u b, v ) aũ, v ) + bv,p r )+ v ΩP v ΩF ) n ) τ ) p ΩP τ )dτ. Sinc ũ 0 w blongs to V, w now us t llipticity proprty 3.38) of t form a, ) onv. combind wit svral Caucy-Scwarz inqualitis, tis yilds ũ 0 w XΩ) c u 0 w XΩ) + u b u b XΩ) + ũ XΩ) + p r L Ω) + sup v X 0 Γ Γ v ΩP v ΩF ) n ) ) τ ) p ΩP τ )dτ v XΩ) Wn Combining tis wit 4.3) and using a furtr triangl inquality lad to 4.). In t rigt-and sid of 4.), t first two trms rprsnt t approximation rror. T nxt two ons ar issud from t tratmnt of t Diriclt boundary conditions. T last trm rprsnts t consistncy rror and is du to t nonconformity of t discrtization. Lmma 4.. If t assumptions of Lmma 4.1 ar satisfid, t following stimat olds btwn t solution u 0,p) of problm.18) and t solution u 0,p ) of problm 3.36) p p L Ω) c inf u 0 w XΩ) + inf p r L w V r M Ω) bub,q ) + u b u b XΩ) + sup q M q L Ω) + sup v X 0 Γ v ΩP v ΩF ) n ) τ ) p ΩP τ )dτ v XΩ) Proof. T sam argumnts as in t prvious proof yild, for any function r in M, v X 0, bv,p r )=au 0 u 0, v )+au b u b, v ) + bv,p r )+ v ΩP v ΩF ) n ) τ ) p ΩP τ )dτ. Γ ) 4.6) So t dsird stimat follows from t inf-sup condition 3.41) combind wit svral Caucy-Scwarz inqualitis, stimat 4.) and a furtr triangl inquality. W now valuat t approximation rrors. T distanc of t prssur to t spac M is boundd in a compltly standard way, s [9], Cap. IX, T..1, for instanc: If p ΩP blongs to H 1 Ω P ) wic is always tru, s Prop..8) and p ΩF blongs to H sf Ω F ), 0 s F 1, inf p r L r M Ω) c P p H 1 Ω P ) + sf F p H s F Ω F )). 4.7)

23 MORTAR FINITE ELEMENTS FOR DARCY AND STOES EQUATIONS 397 To stimat t distanc of u to V, w first us an argumnt du to [4], Captr II, formula 1.16): Sinc u 0 blongs to V Ω), it follows from t inf-sup condition 3.41) tat inf u 0 w XΩ) c inf u 0 w XΩ). 4.8) w V w X 0 Lmma 4.3. T following stimat olds for any function u 0 in V Ω) suc tat u 0 ΩP blongs to H sp Ω P ) d, 0 <s P 1, andu 0 ΩF blongs to H sf +1 Ω F ) d, 0 s F 1, inf u 0 w XΩ) c sp w X P u 0 H s P ΩP ) + d sf F u ) 0 H s F +1 Ω F ). 4.9) d 0 Proof. T construction of t function w is prformd in two stps. 1) W first st w Ω P =Π RT u 0, w Ω F = R u 0, wr t Clémnt rgularization oprator R is introducd in Sction 3, s 3.40). Sinc bot u 0 and w ar divrgnc-fr on Ω P,wav u 0 w Hdiv,Ω P ) = u 0 w L Ω P ) d. Tn, rlying on t fact tat Π RT prsrvs t constants on ac in T P approximation proprtis of tis constant lads to and combining 3.19) wit t u 0 w Hdiv,Ω P ) c sp P u 0 H s P ΩP ) d. 4.10) On t otr and, w driv from t approximation proprtis of t oprator R,s[9], Cap. IX, T. 3.11, tat u 0 w H 1 Ω F ) d csf F u 0 H s F +1 Ω F ) d. 4.11) ) For t functions ϕ introducd in 3.17), w now st w Ω P = w Ω F w Ω P ) n ) ) τ )dτ ϕ, w Ω F = 0. E P,Γ T argumnts for valuating w XΩ) ar narly t sam as in t proof of Lmma 3.8: Combining 3.30) wit 3.1) and a Caucy-Scwarz inquality yilds w XΩ) c E P,Γ 1 u 0 w Ω F L ) W rfr to [9], Cap. IX, Cor. 3.1, for t following rsult: On ac lmnt of E F,Γ, u 0 R u 0 L ) c sf + 1 u 0 H s F +1 ) d, wr is t union of lmnts κ of T F suc tat κ is not mpty. Using tis stimat for all suc tat as a positiv masur lads to, owing to Lmma 3.7, w XΩ) c sf F u 0 H s F +1 Ω F ) d. 4.1) 1.

24 398 C. BERNARDI ET AL. To conclud, w not tat t function w = w +w blongs to X 0. Estimat 4.9) is tn drivd from 4.10), 4.11) and 4.1). Estimating t consistncy rror rquirs t ortogonal projction oprator from L Γ) onto W,tatw dnot by π Γ. Lmma 4.4. T following stimat olds for any function p in L Ω) suc tat p ΩP blongs to H sp +1 Ω P ), 0 s P 1, sup Γ v ΩP v ΩF ) n ) τ ) p ΩP τ )dτ sp +1 cp p v X 0 v H s P +1 Ω P ). 4.13) XΩ) Proof. It follows from t matcing conditions 3.7) tat, for ac in E P,Γ, v ΩP v ΩF ) n ) τ ) p ΩP τ )dτ = v ΩP v ΩF ) n ) τ )p ΩP π Γ p Ω P )τ )dτ. Γ Γ Morovr, sinc t normal trac of v ΩP on Γ blongs to W for any v in X 0,tisgivs v ΩP v ΩF ) n ) τ ) p ΩP τ )dτ = v ΩF n ) τ )p ΩP πp Γ ΩP )τ )dτ. Γ Γ Tis yilds Γ v ΩP v ΩF ) n ) τ ) p ΩP τ )dτ v XΩ) v ΩF 1 p H Γ) d Ω P π Γp Ω P H 1 Γ), v XΩ) wnc, by applying t trac torm on Γ, Γ v ΩP v ΩF ) n ) τ ) p ΩP τ )dτ c p ΩP π v p Γ ΩP H 1. Γ) XΩ) T standard duality argumnt p ΩP π Γ p Ω P H 1 Γ) = sup ϕ H 1 Γ) Γ p Ω P π Γp Ω P )τ )ϕ π Γ ϕ)τ )dτ, ϕ 1 H Γ) combind wit t approximation proprtis of t oprator π Γ,s[9], Cap. IX, T..1, lads to wnc t dsird rsult. p ΩP π Γ p Ω P H 1 Γ) csp +1 P p ΩP H s P + 1 Γ), T fiv trms in t rigt-and sid of 4.) and 4.6) ar boundd in 4.8) and Lmma 4.3, 4.7), Corollary 3.10 and Lmma 4.4, rspctivly. Wn combining tis wit 4.1) and using onc mor Corollary 3.10, w driv t apriorirror stimat. W rcall tat t paramtrs λ P and λ F av bn introducd in Notation 3.9.

25 MORTAR FINITE ELEMENTS FOR DARCY AND STOES EQUATIONS 399 Torm 4.5. Assum tat Γ F as a positiv masur in Ω F and morovr tat i) t data k, g) blong to H σp Γ P ) H σf Γ F ) d, σ P > 1 and σ F > d 1 ; ii) t solution u 0,p) of problm.18) is suc tat u 0 ΩP,p ΩP ) blongs to H sp Ω P ) d H 1 Ω P ), 0 <s P 1, andu 0 ΩF,p ΩF ) blongs to H sf +1 Ω F ) d H sf Ω F ), 0 s F 1. Tn t following apriorirror stimat olds btwn t solution u,p) of problm.6).7) and t solution u,p ) of problm 3.9) 3.10) u u XΩ) + p p L Ω) c sp P u0 H s P ΩP ) + p ) d H 1 Ω P ) + s F F u0 H s F +1 Ω F ) + p ) d H s F Ω F ) + min{σp + 1,λP } P + min{σf 1,λF } ) ) ) F k H σ P ΓP ) + g H σ F ΓF ). 4.14) d T statmnt of Torm 4.5 is ratr complx. Not anyow tat: In t cas of zro boundary conditions k and g, stimat 4.14) can b writtn mor simply as u u XΩ) + p p L Ω) c sp P u H s P ΩP ) + p ) d H 1 Ω P ) + s F F u H s F +1 Ω F ) + p d H s F Ω F )) ). 4.15) Tis last stimat is fully optimal: Indd, for a smoot solution u,p), t rror bavs lik P + F. In t gnral cas, t ordr of convrgnc dpnds on t paramtrs λ P and λ F. So t ordr 1 is only obtaind wn Γ ap is mpty and bot Ω P and Ω F ar convx, for smoot data and solutions. Wn t rgularity of u,p) is unknown, t ordr of convrgnc is givn by Proposition.8 and, for instanc, is always largr tan 1/4 indimnsiond =. Morovr, a diffrnt analysis rlying on t construction of an approximation of u in X satisfying t boundary conditions 3.9), wic rquirs t continuity of u ΩF ), yilds tat, tr also, for a smoot solution u,p), t rror bavs lik P + F. To conclud, it can b obsrvd tat, in all cass and for smoot noug data k, g), t convrgnc of t discrtization rsults from Torm A POSTERIORI rror stimats Som furtr notation ar ndd to dfin t rror indicators. For ac in T P,wdnot by E t st of dgs d = ) or facs d =3)of wic ar not containd in Ω P ; by E ap t st of dgs d = ) or facs d =3)of wic ar containd in Γ ap. For ac in T F,wdnot by E t st of dgs d = ) or facs d =3)of wic ar not containd in Ω F ; by E af t st of dgs d = ) or facs d =3)of wicarcontaindinγ af. For ac in any of t E and also in E P,Γ, w agr to dnot by [ ] t jump troug making its sign prcis is not ncssary). W also dnot by t lngt d =)ordiamtrd =3)of. W nd a furtr notation for som global sts: E ap is t st of dgs or facs of lmnts of T P wic ar containd in Γ ap ; E P P is t st of all otr dgs or facs of lmnts of T. Wit ac lmnt of T F and ac dg of, w associat t quantitis γ and γ qualto1if or, rspctivly, intrscts Γ \ Γ F and to zro otrwis. W introduc t spac Z of functions in L Ω) d suc tat tir rstrictions to ac in T P F or in T is constant. Similarly, w dnot by Z F t spac of functions in L Γ af ) d suc tat tir rstriction to ac in E af, TF, is constant. Indd, w considr an approximation f of f in Z and an approximation t a of t a in Z F. Finally, assuming tat t datum p a is continuous on Γ ap, w dfin p a as t function wic is affin on ac in E ap, TP, and qual to p aa) at all ndpoints d = ) or vrtics d =3)a of ts.