CONVERGENCE RESULTS FOR THE VECTOR PENALTY-PROJECTION AND TWO-STEP ARTIFICIAL COMPRESSIBILITY METHODS. Philippe Angot.

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1 DISCRETE AND CONTINUOUS doi:.3934/dcdsb DYNAMICAL SYSTEMS SERIES B Volum 7, Numbr 5, July pp CONVERGENCE RESULTS FOR THE VECTOR PENALTY-PROJECTION AND TWO-STEP ARTIFICIAL COMPRESSIBILITY METHODS Philipp Angot Aix-Marsill Univrsité Laboratoir d Analys, Topologi, Probabilités - CNRS UMR7353 Cntr d Mathématiqus t Informatiqu 3453 Marsill cdx 3, Franc Pirr Fabri Univrsité d Bordaux & IPB Institut Mathématiqus d Bordaux - CNRS UMR55 ENSEIRB-MATMECA, Talnc, Franc Communicatd by Rogr Tmam) Abstract. In this papr, w propos and analyz a nw artificial comprssibility splitting mthod which is issud from th rcnt vctor pnalty-projction mthod for th numrical solution of unstady incomprssibl viscous flows introducd in [], [] and [3]. This mthod may b viwd as an hybrid two-stp prdiction-corrction mthod combining an artificial comprssibility mthod and an augmntd Lagrangian mthod without innr itration. Th prturbd systm can b viwd as a nw approximation to th incomprssibl Navir- Stoks quations. In th main rsult, w stablish th convrgnc of solutions to th wak solutions of th Navir-Stoks quations whn th pnalty paramtr tnds to zro.. Introduction and stting of th problm. Th artificial comprssibility mthod was introducd by Chorin [6] and Tmam [7] for th solution of th unstady incomprssibl Stoks or Navir-Stoks quations; s also [] for th thortical analysis. Thn, som othr numrical schms to fficintly comput th solutions of Navir-Stoks problms can b viwd as discrtizations of prturbd systms of th typ of pnalization [4] or psudo-comprssibility. This is th cas with th famous projction mthods from Chorin [7] and Tmam [8, 9] and thir variants [], s.g. [5]. Hr, w prsnt a nw approximation mthod for th Navir-Stoks quations modling incomprssibl viscous flows in a boundd rgular opn st ndowd with Dirichlt boundary conditions on Γ = Lipschitz-continuous). With a givn sourc trm f, th Navir-Stoks systm rads: Mathmatics Subjct Classification. Primary: 35Q3, 76D5, 76N, 35A35; Scondary: 65M, 65N. Ky words and phrass. Artificial comprssibility, Navir-Stoks quations, vctor pnaltyprojction, psudo-comprssibility, pnalty mthod. 383

2 384 PHILIPPE ANGOT AND PIERRE FABRIE whr R dnots th Rynolds numbr. v t + v )v v + p = f, R div v =, v) = v, v Γ =, According to th idntity, ϕ = curl curl ϕ div ϕ, w considr th following approximat mthod to obtain a solution of th abov Navir-Stoks systm, with th paramtrs r, γ > and, ε > ṽ ε t + v ε )ṽ ε + div v ε)ṽ ε + R curl curl ṽ ε R div ṽ ε r div ṽ ε + p ε = f v ε t + v ε ) v ε + div v ε) v ε + R curl curl v ε R div v ε r div v ε εr div v ε + div ṽ ε ) = v ε = ṽ ε + v ε γ p ε t + γp ε + ε div v ε + r div ṽ ε =. W associat to th prvious systm th following boundary conditions and initial data, ṽ ε ) = v, v ε ) =, p ε ) = p, ṽ ε ν Γ =, v ε ν Γ =, curl ṽ ε ) ν Γ =, curl v ε ) ν Γ =, whr ν dnots th outward unit normal vctor on Γ. To vanish, at th limit procss, th two tangntial componnt of th vlocity filds, ṽ ε ν and v ε ν, w us a pnalization mthod which will b dtaild blow. This mthod is clos to th artificial comprssibility mthod of Chorin [6] and Tmam [7], but prsnts on important diffrnc. It is a two-stp splitting mthod. Th first quation of th prvious systm givs a prdictd vlocity ṽ ε and th scond on is th approximat projction of ṽ ε on th fr-divrgnc vctor filds. This quation may b sn as an approximat mthod to solv th wll-posd problm s appndix A) : div v ε = div ṽ ε, curl v ε =, v ε.ν Γ =. Rmark. This approximat mthod is issud from th Vctor Pnalty-Projction VPP r,ε ) mthods for th numrical solution of unstady incomprssibl viscous flows introducd in [] and [3]. A fast vrsion of ths mthods, th so-calld VPP ε ) mthod, is rcntly proposd also for th numrical solution of th nonhomognous Navir-Stoks quations in [, 4]. It is shown to b vry fficint to comput multiphas flows, i.. fast, chap, and robust whatvr th dnsity, viscosity or prmability jumps.

3 THE VECTOR PENALTY-PROJECTION METHOD 385 Evn for r =, th rsulting mthod which corrsponds to a two-stp psudocomprssibility mthod, is diffrnt from th original artificial comprssibility mthod of Chorin [6] and Tmam [7, ]. Th nw important point is th pnalty trm ε div v ε + div ṽ ε ) that appars in th vlocity corrction stp which allows us a dirct stimat on th divrgnc of th vlocity. Morovr, this systm is quit asy to solv and prsnts good stability proprtis, s [,, 3]. Th vlocity v ε and th prssur p ε satisfy th quations: v ε t + v ε )v ε + div v ) ε) v ε + curl curl v ε div v ε R R r div v ε div v ε + p ε = f εr γ p ε t + γ p ε + ε div v ε + r div ṽ ε =, v ε ) = v, p ε ) = p, v ε ν =, curl v ε ) ν Γ =. Γ Th vanishing of th tangntial componnt at th limit procss, is fullfilld by a pnalization mthod, which implis that this boundary condition is satisfid at th ordr ε for th approximat solution... Notations. Lt b a rgular boundd and simply-connctd opn st of R d, for d = or 3. W not H s ) th classical Sobolv spac, and H s th associatd norm. Th norm of a function in L p ) is dnotd by L p, and if B is a Banach spac, w dnot by. Lp,B th norm in L p ], T [; B). L p ) = L p )) d H div ) = {v L )) d, div v L )} H = {v L )) d, div v =, v ν Γ = } H ν) = {v H )) d, v ν Γ = } G = {v L )) d, q H ), v = q}.. Mathmatical rcalls. Proposition. Lt b a rgular boundd and connctd opn st of R d. Thn, w hav th following proprtis: L ) = H G, Kr curl ) = G. Morovr, thr xists on constant C > dpnding only on such that: u H = u L + u L C u L + div u L + curl u L ), u H ν ). Bsids, if w suppos that th opn st is simply-connctd, thr xist two constants λ and λ dpnding only on such that: and w hav: u L λ div u L + curl ) u L, u H ν ), u L + u L λ div u L + curl u L ), u H ν ) Kr curl ) H = {}. )

4 386 PHILIPPE ANGOT AND PIERRE FABRIE Proof. All ths rsults may b found in [9] and [8]. For a Banach spac E w introduc th Nikolskii spac dfind for q < +, < σ < : N σ q ], T [; E) = { f L q ], T [; E), ndowd with th following norm: f N σ q = f q L q ],T [;E) + sup <h<t } f + h) f ) L sup q ],T h[;e) <h<t h σ < +, ) q ) q h σ f + h) f ) L q ],T h[;e). Lt us rcall th following proprty; s for xampl [5, pag 5]. Proposition. Lt H b an Hilbrt spac and f a function givn in L ], T [; H) such that, for som < σ <, R τ σ F f)τ) H dτ C, whr f dnots th xtnsion by of th function f outsid [, T ]. N σ ], T [; H) and w hav Thn f f N σ M σ + C), whr M σ is a constant dpnding only on σ. W now rcall th important compactnss thorm, s for xampl [6] Thorm.. Aubin-Lions-Simon Lt B, B, B thr Banach spacs with B B B with continuous imbdding. Suppos morovr that th injction of B in B is compact. Thn, for all q + and < σ <, th imbdding L q ], T [; B ) N σ q ], T [; B ) L q ], T [; B ) is compact.. Main rsult. W associat to th prvious approximat systm, th variational problm whr th tangntial componnts of th vlocitis ṽ ε and v ε ar pnalizd. This problm is studid in th nxt sction.

5 Find ṽ ε, v ε, p ε ) in THE VECTOR PENALTY-PROJECTION METHOD 387 L ], T [; L )) L ], T [; H ν))) L ], T [; L )) satisfying in D ], T [), ṽ ε v t ϕ dω + ε )ṽ ε + ) div v ε) ṽ ε ϕ dω + curl ṽ ε curl ϕ dω + div ṽ ε div ϕ dω R R + r div ṽ ε div ϕ dω p ε div ϕ dω + ṽ ε ν) ϕ ν) dσ = f ϕ dx, ε Γ v ε v t ψ dω + ε ) v ε + ) div v ε) v ε ψ dω + curl v ε curl ψ dω + div v ε div ψ dω R R Γ + r div v ε div ψ dω + εr + v ε ν) ψ ν) dσ =, ε div ṽ ε + div v ε ) div ψ dω ) v ε = ṽ ε + v ε, p ε γ t π dω + γ p ε π dω + π div v ε dω + r π div ṽ ε dω =, ε ϕ, ψ, π) H ν)) L ), ṽ ε ) = v, v ε ) =, p ε ) = p. Thn th vlocity v ε and th prssur p ε satisfy in D ], T [), v ε v t ϕ dω + ε )v ε + ) div v ε) v ε ϕ dω + curl v ε curl ϕ dω + div v ε div ϕ dω R R +r div v ε div ϕ dω + div v ε div ϕ dω εr p ε div ϕ dω + v ε ν) ϕ ν) dσ ε Γ = f ϕ dω, ϕ H ν), v ε ) = v 3) Rmark. In ordr to stablish th strong convrgnc of th squnc v ε ) ε> whn ε, w us in Sction 4 th Lray s orthogonal dcomposition in th

6 388 PHILIPPE ANGOT AND PIERRE FABRIE boundd domain. Th curl-fr componnt vanishs with th pnalty trm introducd by our mthod, whras th divrgnc-fr componnt strongly convrgs thanks to an stimat of a fractional drivativ in tim, s []. Howvr, this rquirs to considr vlocity filds having only thir normal componnt which is zro on th boundary. Sinc at th limit procss, w aim at solving th Navir- Stoks problm with homognous Dirichlt boundary condition, w also pnaliz th tangntial part of th vlocity filds. W prov in sction 3 th following rsults. Lmma.. Lt us suppos that f blongs to L ], T [; L )). Thn, thr xists at last a solution to th systm ). This solution is uniqu in two spac dimnsion. For th dimnsion d 3, this solution satisfis th following nrgy inquality: d r ε ṽε L dt + ε v ε L + v ε L + γε p ε ) L + γε p ε L + r ε curl ṽ ε L R + ε curl v ε L R + curl v ε L R + R div v ε L + r ε R div ṽ ε L + ε R div v ε L + εr div ṽ ε L + εr div v ε L + εr div v ε L + r ṽ ε ν) L Γ) + v ε ν) L Γ) + ε v ε ν) L Γ) λ R + rε) f L For th dimnsion d =, on has th following nrgy quality: d r ε ṽε L dt + v ε L + γε p ε ) r ε L + curl ṽ ε L R + curl v ε L R + γ ε p ε L + r ε div ṽ ε L R + div v ε L R + εr div ṽ ε L + div v ε L εr + ṽ ε ν) L Γ + ε v ε ν) L Γ) = r ε f ṽ ε dω + f v ε dω. This rsult is quit classical and w only giv th sktch of proof in th sction 3. In fact, w can prcis th prvious nrgy inquality if w suppos that th data f blongs to L R + ; L )). This shows th absolut stability of th approximat mthod. Thorm.. Suppos that th data f satisfis f L R +, L )), thn, thr xists a constant α indpndnt of th data, such that r ε ṽε t) L + ε v εt) L + v εt) L + γε p εt) L ) αt + r ε) v L + γ ε p L ) + λ R α + r ε) f L, L, t R +. Th main goal of this papr is to prov th following convrgnc thorm:

7 THE VECTOR PENALTY-PROJECTION METHOD 389 Thorm.3. For d 3, thr xists a subsqunc v εk, p εk ) k solution of 3) that convrgs to a solution v, p) to th systm of Navir-Stoks quations with homognous Dirichlt boundary conditions. For d =, th solution v, p) is uniqu, and for all squncs ε k, v εk, p εk ) εk convrgs to v, p). Morovr, for all squncs ε k, v εk ) k convrgs strongly to v in L, T ; H ν)). W now giv an intrprtation of th prssur and prcis its convrgnc. Lt us dfin q ε = p ε + ε εr + r ) div v ε. Th scalar function q ε appars to b th ffctiv approximat prssur, and w hav Thorm.4. Th function q εk satisfis if d = 3, q εk convrgs wakly to p in H, T ) ) ) 3 if d =, q εk convrgs strongly to p in H, T ) ) ) Ths convrgnc rsults for both vlocity and prssur ar provd in Sction Enrgy stimats. W first stablish th following xistnc rsult. Proposition 3. For v ε, p givn in L ) L ), thr xists at last a solution of th systm ) satisfying for d = 3: ṽ ε L ], T [; L ) L ], T [; H ν)), ṽ ε t L 4 3 ], T [; H ν )) ) v ε L ], T [; L ) L ], T [; H ν)), v ε t L 4 3 ], T [; H ν )) ) p ε L ], T [; L )), p ε t L ], T [; L )) ṽ ε ) = v ε o in H ν)), v ε ) = in H ν)), p ε ) = p in L ). If d =, th uniqu solution of ) satisfis th following rgularity rsults: ṽ ε L ], T [; L ) L ], T [; H ν)), ṽ ε t L ], T [; H ν)) )) + L ], T [; L 3 )) v ε L ], T [; L ) L ], T [; H ν)), v ε t L ], T [; H ν)) )) + L 4 3 ], T [; L 4 3 )) p ε L ], T [; L )), p ε t L ], T [; L )) ṽ ε ) = v ε o in L ), v ε ) = in L ), p ε ) = p in L ). Rmark 3. In th thr-dimnsional cas, th qualitis ar valid in th trac sns. ṽ ε ) = v ε o in H ν)), v ε ) = in H ν)),

8 39 PHILIPPE ANGOT AND PIERRE FABRIE Proof. For fixd paramtrs ε >, r and γ >, w build approximat solutions by a classical Galrkin procss. Lt us introduc th slf-adjoint oprator A = curl curl div dfind on th domain H ν) H )) d. Thn, for th approximation of th two filds of vlocity, w us as spcial basis th ignfunctions of this oprator associatd with th following boundary conditions: u ν =, Γ curl u) ν Γ =. For th prssur, on can us as spcial basis th ignfunctions of th slf-adjoint oprator A = with domain H ) associatd to th Numann boundary conditions. This approximat finit dimnsional systm is thn a classical ordinary diffrntial quation which has a uniqu solution. Nxt, to prform th limit w us th sam stratgy as for th classical Navir-Stoks quations i.. a priori stimats and compactnss rsults using an stimat on th tmporal drivativ, s for xampl [3], [],[5]. Now, w will focus our attntion on th stimats on th tim drivativ according to th dimnsion d. Lt us bgin with th thr-dimnsional cas. W hav to stimat th two nonlinar trms v ε w, w div v ε, with ithr w = ṽ ε or w = v ε and th prssur p ε. Suppos first that d = 3. By Sobolv mbdding, th two nonlinar trms of th form v ε w and w div v ε blong to L 4 3, T ; H ν) ) sinc w hav, for xampl for all ϕ H ν) : v ε )w ϕ dω + div v ε ) w ϕ dω C ) v L 3 w H ν ϕ L 6 + w L 3 v ε H ν ϕ L 6 ) C v ε L v ε H w ν H + w ν L w H v ε H ϕ ν ν H ν. Th bounds of th linar trms ar straightforward and w hav curl ṽ ε curl ϕ dω + div ṽ ε div ϕ dω + r div ṽ ε div ϕ dω C ṽ ε H ν ϕ H ν p ε div ϕ dω C p ε L ϕ H ν ) div ṽε + div v ε div ϕ dω C ṽ ε H + v ) ν ε H ϕ H ν ν. and, with standard trac thorms v ε ν) ϕ ν) dω C v ε H ν ϕ H ν. Thus it follows, from quation ) that Γ ṽ ε t L 4 3 ], T [; H ν )) ), p ε t L ], T [; L )) v ε t L 4 3 ], T [; H ν )) )

9 THE VECTOR PENALTY-PROJECTION METHOD 39 Ths stimats show that th vlocitis ṽ ε, v ε ) ar qual almost vrywhr to continuous functions with valus in H ν)). Bsids, th prssur p ε is qual almost vrywhr to a continuous function with valu in L ). For th two-dimnsional cas, th situation is quit diffrnt. W obsrv first that th vlocity filds v ε and ṽ ε blong to L 4 ], T [; L 4 )), so that: vε )ṽ ε + div vε )ṽε L 4 3 ], T [; L 4 3 )), vε ) v ε + div vε ) vε L 4 3 ], T [; L 4 3 )). So it follows from th quation ) that ṽ ε t L ], T [; H ν)) ) + L ], T [; L 3 )) v ε t L ], T [; H ν)) ) + L ], T [; L 3 )) p ε t L ], T [; L )) W now obsrv that th two vlocity filds ṽ ε and v ε blong to which is th dual spac of L 4 ], T [; L 4 )) L ], T [; H ν)) ) L ], T [; H ν)) ) + L 4 3 ], T [; L 4 3 )). Thus th functions ṽ ε, v ε ) ar qual almost vrywhr to continuous functions with valus in L ). This nds th proof of proposition Stability. In th cas of thr-dimnsional vctor spacs, w do not hav an quality for th consrvation of th nrgy, w hav only an inquality. Nvrthlss for two-dimnsional vctor spacs, th wak solutions satisfy th nrgy quality. Proof. Through classical computations on obtains, with quations ) and 3): d dt ṽ ε L + ṽ ε R curl L + div ṽ ε L R + r div ṽ ε L + ε ṽ ε ν) L Γ) 4) p ε div ṽ ε dω = f ṽ ε dω, d dt v ε L + v ε R curl L + div v ε L R ε L + ε v ε ν) L Γ) + div v ε div v ε dω =, εr 5) d dt v ε L + v ε R curl L + div v ε L R div v ε L εr + r div v ε L + ε v ε ν) L Γ) p ε div v ε dω = f v ε dω, 6) γ ε d dt p ε L + γ ε p ε L + p ε div v ε dω + r ε p ε div ṽ ε dω =. 7)

10 39 PHILIPPE ANGOT AND PIERRE FABRIE Multiplying 4) by r ε and 5) by ε and summing with 6) and 7), on obtains: d dt r ε ṽε L + ε v ε L + v ε L + γε p ε L ) + r ε R curl ṽ ε L + ε R curl v ε L + curl v ε L R + γ ε p ε L + r ε div ṽ ε L R + ε div v ε L R + div v ε L R + r ṽ ε ν) L Γ) + v ε ν) L Γ) + ε v ε ν) L Γ) + εr div ṽ ε L + rε div v ε L + div v ε L εr = div v ε div v ε dω + r ε f ṽ ε dω + f v ε dω, R Lt us now giv som bounds of th right-hand sid trms. Th trm v ε, div v ε ) is boundd by div v ε R div L ε R + ε div v ε L R. According to th stimat of th L norm in H ν) givn by quation ), w bound th sourc trms in th following way: f v d ω f L v L λ f L div v L + curl v L ) R div v L + curl v L ) + R λ f L. Using ths bounds, w gt from th prvious quation th following fundamntal stimat: d r ε ṽε L dt + ε v ε L + v ε L + γε p ε ) L + γε p ε L + r ε curl ṽ ε L R + ε curl v ε L R + curl v ε L R + R div v ε L + r ε R div ṽ ε L + ε R div v ε L + εr div ṽ ε L + εr div v ε L + εr div v ε L + r ṽ ε ν) L Γ) + v ε ν) L Γ) + ε v ε ν) L Γ) λ R + ε r) f L. 8) Aftr intgration in tim, w dduc from th prvious stimat that thr xists a continuous function g dfind on [, T ] such that: for all t >,

11 THE VECTOR PENALTY-PROJECTION METHOD 393 ) r ε ṽ ε t) + ε v ε t) + v ε t) L + γε p εt) L t + γ ε p ε s) L ds + rε t curl ṽ ε s) L R ds + ε t R with + R + ε R + r + ε + εr t t t t t curl v ε s) L ds + rε R div v ε s) L ds + R t ṽ ε s) ν) L Γ) ds + t t div ṽ ε s) L ds div v ε s) L ds v ε s) ν) L Γ) ds t v ε s) ν) L Γ) ds + εr div ṽ ε s) L ds div v ε s) L ds + εr t div v ε s) Lds gt), curl v ε s) L ds 9) gt) = r ε ṽ ε ) + ε v ε ) + v ε ) L + γε p ε) L ) + λr + rε) t fs) L ds. This inquality is th ky point to stablish th convrgnc rsult. To improv th convrgnc rsult in th two-dimnsional cas, w us th following nrgy quality drivd as abov without using 5). d dt r ε ṽε L + v ε L + γε p ε ) r ε L + curl ṽ ε L R + curl v ε L R + γ ε p ε L + r ε div ṽ ε L R + div v ε L R + εr div ṽ ε L + div v ε L εr + r ṽ ε ν) L Γ) + ε v ε ν) L Γ) = r ε f ṽ ε dω + f v ε dω. ) This concluds th proof of lmma.. Rmark 4. Th nrgy stimat 9) still holds with a diffrnt function g without assuming that is simply-connctd by using th Gronwall inquality to dal with th two right-hand sid trms: r ε f ṽ ε dω + f v ε dω. Hnc, all th convrgnc rsults stablishd in this papr rmain tru for a smooth boundd and connctd domain. Howvr, th uniform stability proprty in Thorm. rquirs th hypothsis that is simply-connctd.

12 394 PHILIPPE ANGOT AND PIERRE FABRIE 3.. Uniform stability for th approximat solution. In this sction, w dal with th stability of th proposd approximation mthod. W notic that this proprty is valid for all solutions satisfying th nrgy inquality 8), as it is th cas whn thy ar built by a finit dimnsional approximation mthod such as th Galrkin mthod for xampl. Proof. Lt us writ χ ε t) = ε r ṽ ε t) L + ε v εt) L + v εt) L + γε p εt) L. W not λ > th smallst ignvalu of th slf-adjoint oprator A = curl curl div with th domain D = H ν) H )) d, ) λ and w introduc α = min R,. Classically, th inquality 8) lads to th diffrntial inquality d dt χ εt) + αχ ε t) λ R + r ε) ft) L, which implis th following uniform bound χ ε t) αt χ ε ) + λ R α + r ε) f L,L. This concluds th proof of thorm.. 4. Convrgnc analysis and compactnss rsults. 4.. Compactnss rsults for th vlocity. Lt us introduc th Lray projction w ε of a vlocity fild v ε t) H ν) dfind as follows v ε = w ε + q ε, div w ε =, w ε ν Γ =, q ε ν Γ =, q ε dω =. By th stimat 9), w s that th irrotational part of v ε gos to zro with ε. Thus it rmains to bring to th for th bhavior of th fr divrgnc part w ε and to obtain an stimat on a fractional tim drivativ of this trm. W dtail th diffrnt stps of this stratgy. From th rgularity of th Lray projctor s R. Tmam [, pag 8]), on has: w ε L,L c v ε L,L, w ε L,H c v ε L,H. ) Morovr, w can asily prov th following lmma. Lmma 4.. Thr xists two constants dpnding only on T and such that: q ε L,H c ε, q ε L,L c. )

13 THE VECTOR PENALTY-PROJECTION METHOD 395 Proof. Th function q ε blongs to H ) and satisfis This implis using th stimat 9) q ε t) = div v ε t), q ε t) ν Γ =. q ε t) L,L = div v ε L,L C ε. Bsids, w hav q ε q ε dω = q ε q ε dω, so that, with Poincaré-Numann inquality, w gt q ε t) L,L C q εt) L,L q ε L,L, C ε q ε t) L,L. Th rgularity proprtis of th Numann problm giv q ε L,H C q ε L,H C q ε L,L C ε. 3) Morovr, by orthogonality of th Lray projctor in L, on has This concluds th proof of th lmma 4.. q ε L,L v ε L,L C. 4) So by intrpolation and using stimats 3)-4), w hav provd th rsult blow. Corollary. Th function q ε satisfis: q ε strongly convrgs to in L p ], T [; L )) ), p, p < +. Now w hav to writ th quation satisfid by w ε. As th Lray projction is orthogonal in L ), this quation rads ϕ H ν), div ϕ =, w ε ϕ dω + t + R = v ε )v ε + ) div v ε)v ε ϕ dω curl v ε curl ϕ dω + v ε ν) ϕ ν) dγ ε Γ f ϕ dω in L, T ). Now w introduc th xtnsion by of w ε rsp. v ε ) outsid [, T ] dnotd, only in this part, by w ε rsp. ṽ ε ) and w tak th Fourir transform in tim of th quation 5) to obtain ϕ H ν), div ϕ =, iτ F w ε )τ) ϕ dω + + R = F ṽ ε )ṽ ε + ) div ṽ ε)ṽ ε τ) ϕ dω Fṽ ε ν)τ) ϕ ν) dγ curl Fṽ ε )τ) curl ϕ dω + ε Γ F f)τ) ϕ dω + v ε ) ϕ dω iτt π π v ε T ) ϕ dω. 5)

14 396 PHILIPPE ANGOT AND PIERRE FABRIE Following Boyr-Fabri [5, pag 53], w tak ϕ = F w ε )τ) as tst function in th prvious quation to obtain for all τ R: iτ F w ε )τ) ) dω = Fṽε )ṽ ε τ) F wε )τ) dω ) Fdiv ṽε )ṽ ε τ) F wε )τ) dω curl Fṽ ε )τ) curl F w ε )τ) dω R Fṽ ε ν)τ) F w ε )τ) ν) dγ ε Γ + F f)τ) F w ε )τ) dω + π v ε ) F w ε )τ) dω iτt π v ε T ) F w ε )τ) dω. As w look for an stimat indpndnt of ε, w hav to pay a spcial attntion to th imaginary part of th pnalty trm: A ε = ε Γ Fṽ ε ν)τ) F w ε )τ) ν) dγ. 6) By writing w ε = v ε q ε, w hav: ε Fṽ ε ν)τ) F w ε )τ) ν) dγ = Fṽ ε )τ) ν dγ Γ ε Γ F qε )τ) ν ) Fṽ ε )τ) ν) ) dγ. ε Γ So, th imaginary part of A ε is boundd as follows: ε Γ F qε )τ) ν ) Fṽ ε )τ) ν) ) dγ ε F q ε)τ) L Γ) Fṽ ε)τ) ν L Γ), 7) C ε F q ε)τ) H Fṽ ε )τ) ν L Γ). From stimats ) and 9), w hav q ε L,H C ε, v ε ν L,L Γ) C ε. So thr xists a function f 4 ε τ) L R) boundd indpndntly on ε such that: ε Γ F qε )τ) ν ) Fṽ ε )τ) ν) ) dγ f ε 4 τ) 8)

15 THE VECTOR PENALTY-PROJECTION METHOD 397 Now w can driv th stimat of τ F w ε )τ) dω, and w hav τ F w ε )τ) ) dω Fṽε )ṽ ε τ) F wε )τ)dω + ) Fdiv ṽε )ṽ ε τ) F wε )τ)dω + R curl Fṽ ε )τ) curl F w ε )τ)dω + ε F qε )τ) ν ) Fṽ ε )τ) ν) ) dγ Γ + Ff)τ) F w ε )τ) dω + v ε ) F w ε )τ)dω π + v ε T ) F w ε )τ)dω π f ε τ) + f ε τ) + f 3 ε τ) + f 4 ε τ) + f 5 ε τ) + f 6 ε τ) + f 7 ε τ). W now stimat ach trm of th right-hand sid of th prvious inquality for d 3. ) Trm fε = Fṽε )ṽ ε τ) F wε )τ)dω According to th nrgy stimat 9), th function v ε is boundd in L, T ; H ν)) and hnc, by Sobolv injction, it is boundd in L 6 5, T ; L 6 )). So by Hausdorff-Young thorm, W also hav th inquality: 9) F w ε )) ε is boundd in L 6 R; L 6 )). ) ṽ ε ṽ ε L 6 5 v ε L v ε L 3 C v ε 3 L v ε L, which implis, according to 9), that v ε v ε is boundd in L 4 3, T ; L 6 5 )), and ncssarily in L 6 5, T ; L 6 5 )). So, by Hausdorff-Young thorm, th family of functions ) Fṽ ε )ṽ ε is boundd in L 6 R; L 6 5 )). Thn with Höldr inquality, Trm fε = Th sam argumnts show that Trm fε 3 = R fε ) ε is boundd in L 3 R). ) ) Fdiv ṽε )ṽ ε τ) F wε )τ)dω fε ) ε is boundd in L 3 R). ) curl Fṽ ε )τ) curl F w ε )τ)dω According to th rgularity of th Lray projction rcalld abov and stimat 9), on has: f 3 ε ) ε is boundd in L R). 3)

16 398 PHILIPPE ANGOT AND PIERRE FABRIE Trms fε 4 = ε Γ F qε )τ) ν ) Fṽ ε )τ) ν) ) dγ As w hav sn by th stimat 8), Trm fε 5 = fε 4 ) ε is boundd in L R). 4) F f)τ) F w ε )τ) dω By hypothsis, f is a givn function in L R; L )) and from stimat 9), w gt that ṽ ε is boundd in L R; L )), so as w ε is th Lray projction of ṽ ε, th function w ε is also boundd in L R; L )). Finally, w obtain Trms f 6 ε = π fε 5 ) ε is boundd in L R). 5) v ε ) F w ε )τ)dω and f ε 7 = π v εt ) F w ε )τ)dω Ths two trms com from th Dirac masur whn w driv discontinuous functions. Lt us considr f 7 ε. f 7 ε τ) π v ε T ) L F w ε )τ) L. According to 9), this trm is boundd in L R). Morovr, th st of functions F wε )τ) ) ε is boundd in L R; L )) so, f 7 ε ) ε is boundd in L R). W trat in th sam way th trm f 6 ε, and thus: f 6 ε ) ε and f 7 ε ) ε ar boundd in L R) 6) W ar now abl to show that th st of functions w ε ) ε is boundd in an appropriat Nikolskii spac. For all γ <, thr xists a constant d such that: τ γ d + τ ) + τ γ, so that, τ γ F w ε )τ) L d F w ε )τ) L + τ ) + τ γ F w ε)τ) L Lt us dnot f ε τ) = F w ε )τ) L, which blongs to L R), th prvious inquality rads with 9): τ γ F w ε )τ) L d fε d ) τ) + + τ γ fε τ) + fε τ) + fε 3 τ) + fε 4 τ) + fε 5 τ) + fε 6 τ) + fε 7 τ) hτ) If w suppos that th function τ + τ γ blongs to L R) L R), thn th function τ hτ) blongs to L R). This condition is satisfid for γ ] 3, [. So w hav provd :

17 THE VECTOR PENALTY-PROJECTION METHOD 399 Lmma 4.. Lt us suppos that σ ], 6 [, thn thr xists a constant C such that τ σ Fw ε )τ) Ldτ C. 7) R Thn, from lmma 4. and 4. w dduc th following ky rsult: Thorm 4.3. Thr xists a squnc ε k ) k which convrgs to zro and a function v L ], T [; L )) satisfying div v = such that: v εk ) k v in L ], T [; L )) strongly. Proof. Th function v ε is th sum of two trms q ε and w ε. From corollary th first trm convrgs strongly to in L ], T [; L )). Now, from Aubin-Lions- Simon Thorm, it follows from lmma 4., that thr xists a squnc ε k ) k such that: w εk ) k v in L ], T [; L )) strongly. Morovr, sinc div w εk =, w hav div v =. 4.. Convrgnc of th mthod. W first giv a gnral convrgnc thorm for a subsqunc solution of th approximat schm 3), to a wak solution of th initial Navir-Stoks problm, in th cas d 3. For th two-dimnsional cas, sinc th wak solution of th Navir-Stoks quation is uniqu, th whol squnc of approximat solution v ε convrgs to v. Morovr, in this cas, w prov that th convrgnc is strong Th gnral cas d 3. Proof. Lt θ an lmnt of C, T ), satisfying θt ) =, and ϕ a fr-divrgnc vctor fild in H ) ) d H )) d. An intgration by parts givs from th quation 3) T T v ε ϕ dω θ τ) dτ + + T R = T curl v ε curl ϕ dω θτ) dτ f ϕ dω θτ) dτ + v ε )v ε + div v ε) v ε ) ϕ dω θτ) dτ v ϕ dω θ). According to stimat 9), thr xists a squnc ε k such that v εk v in L, T ; L )) strongly, div v εk div v = in L, T ; L )) strongly, curl v εk curl v in L, T ; L )) wakly. Sinc u L + div u L + curl u L ) is a norm quivalnt to th H -norm on H ν), w hav 8) v εk v in L, T ; L )) wakly. and so, w can tak th limit on th trm v ε )v ε as v εk v εk v v in L, T ; L )) ) div vεk vεk in L, T ; L )).

18 4 PHILIPPE ANGOT AND PIERRE FABRIE From stimat 9), w hav for th tangntial tracs t and sinc for any function v in H ) ) d, w obtain that v εk v ε ν)τ) L Γ) dτ ε gt), v L Γ) C v L v H, v in L ], T [; L Γ)) d) strongly. This implis v ν) Γ =, and so, sinc by construction v ν) Γ =, v blongs to H ) ) d Finally, at th limit procss w obtain T T v ϕ dω θ τ) dτ + v )v ϕ dω θτ) dτ + T T curl v curl ϕ dω θτ) dτ = f ϕ dω θτ) dτ + v ϕ dω θ), R div v =, v Γ =. From th idntity v : ϕ dω = curl v curl ϕ dω + div v div ϕ dω, which is valid for all functions v, ϕ) H )) d H )) d, th prvious quality shows that th limit function v satisfis th classical Navir-Stoks quations in a wak sns Th spcial cas d =. Th ky point to stablish th strong convrgnc in th two-dimnsional cas, lis on an ida of R. Tmam []. It is basd on th fact that in this cas, th solution of th approximat problm and th solution of th Navir-Stoks quation vrify th quality of nrgy. Th ida is to bring to th for an nrgy quation satisfid by th diffrnc btwn th approximat solution and th xact solution. Proof. W first obsrv that, according to th quality = curl curl div th classical wak solution v of th Navir-Stoks quation satisfis for any tst function in H )) d, with fr-divrgnc: v t ϕ dω + v )v ) ϕ dω + R curl v curl ϕ dω = f ϕ dω. Th quation satisfid by th rror v ε v = u ε with a fr-divrgnc tst function ϕ in H ) ) d H ν ). u ε u t ϕ dω + ε )u ε + ) div u ε) u ε ϕ dω + v )u ε ϕ dω + u ε )v + ) div u ε) v ϕ dω + curl u ε curl ϕ dω =. R

19 THE VECTOR PENALTY-PROJECTION METHOD 4 Aftr intgration in tim, this quation givs t u ε t) ϕ dω + t + v )u ε ϕ dω dτ + + t R u ε )u ε + div u ε) u ε ) ϕ dω dτ t curl u ε curl ϕ dω dτ = u ε )v + ) div u ε) v ϕ dω dτ u ε ) ϕ dω. Taking th limit whn ε gos to, on obtains with th convrgncs proprtis statd in th prvious sction: Lmma 4.4. ϕ H )) d, div ϕ =, lim u ε t) ϕ dω =. 9) ε Following R. Tmam [], w introduc χ ε t) = + r ε R ε r ṽε t) L + γε p εt) L + v ε v)t) L ) t t curl ṽ ε τ) L dτ + γε p ε τ) L dτ + t curl v ε v)τ) L R dτ + t div v ε v)τ) L R dτ +r + εr R t t ṽ ε ν)τ) L Γ) dτ + ε t div ṽ ε τ) L dτ + εr t v ε ν)τ) L Γ) dτ div v ε τ) L dτ. Hr v is th uniqu solution of th two-dimnsional Navir-Stoks quations. Du to th nrgy quality ), χ ε t) satisfis χ ε t) = ε r ṽε ) L + v ε) L + γ ε p ε) ) t L + rε fτ) ṽ ε τ)dω dτ t + fτ) v ε τ)dω dτ v ε t) vt)dω t curl v ε τ) curl vτ)dω dτ R t R div v ε τ)div vτ)dω dτ + vt) L + t curl vτ) L R dτ + t div vτ) L R dτ.

20 4 PHILIPPE ANGOT AND PIERRE FABRIE By wak convrgnc in L ], t[; H )) of th squnc v ε ) ε and from th lmma 4.4, w obsrv that : t lim ε v ε) L v ε t) vt)dω + fτ) v ε τ) dω dτ t curl v ε τ) curl vτ)dω dτ t ) div v ε τ) div vτ)dω dτ R R = t v) L + t div vτ) L R dτ. fτ) vτ) dω dτ vt) L t curl vτ) L R dτ In th two-dimnsional cas, th uniqu solution of th Navir-Stoks quation satisfis th following nrgy quality vt) L + R t Morovr, from stimat ), curl uτ) L dτ + t div uτ) L R dτ = t v) L + t lim ε ε fτ) vτ) dω dτ. fτ) ṽ ε τ) dω dτ =. So, w hav provd that lim χ εt) =. ε In othr words, w hav stablishd th following rsult v ε v in C [, T ]; L )) L ], T [; H ν)) v C [, T ]; H) L ], T [; H )) This concluds th proof of thorm.3. W ar now abl to prcis th convrgnc for th ffctiv prssur and stablish th thorm.4. Proof. Lt us writ th quation satisfid by th vlocity v ε and th prssur p ε in th distribution sns. W hav v ε t + v ε )v ε + div v ε)v ε + curl curl v ε + p ε + ε + r ) ) div v ε = f. R εr Introducing th ffctiv prssur this quation rads q ε = f q ε = p ε + ε εr + r ) div v ε, vε t + v ε )v ε + div v ε)v ε + ) curl curl v ε R and th proof of Thorm.4 follows from th prvious stps.

21 THE VECTOR PENALTY-PROJECTION METHOD Appndix. Lt us considr th following problm: Proposition 4. For a fixd ε > and a coupl of functions f, g) givn in L ) L ), thr xists a uniqu solution v ε {w H div ), w ν Γ = } solution of: Morovr, if u, u ) H div ) H is solution of εv ε div v ε = f + εg. 3) div u = f, curl u =, u ν Γ =, u ν Γ =, and w hav th following stimat div u =, curl u = curl g, v ε u u H ε u + u g L. Proof. Stp : Existnc of v ε Lt us not H div, ) = {v L ), div v L, v ν Γ = }. Th xistnc of a uniqu solution to th quation 3) is obtaind by a straightforward application of th Lax-Milgram thorm with th bilinar form dfind on H div, ) by εu, v) + div u, div v), and th right-hand sid: f, div v) + εg, v). Stp : xistnc of u and u Th xistnc of u satisfying 3) coms from th rsolution of th following Numann problm q = f, q ν Γ =, and w st u = q, with q H )/R ) H ). Th xistnc of u H is th consqunc of th Lray projction applid to g by writing g = u + p, div u =, u ν Γ =. Now, writting v ε = u + u + u ε, w gt that u ε H div, satisfis and w hav th stimat εu ε div u ε = εu εu + εg, curl u ε =, ε u ε L + div u ε L ε u + u g L u ε L. W obsrv that, according to [8], H ν) ndowd with th norm div v + curl v ) is qual to {w H )) d, w ν γ }. So th prvious stimat givs 3) ε u ε L + div u ε L + curl u ε L ε u + u g L u ε H ν Cε u + u g L div uε L + curl u ε ) L,

22 44 PHILIPPE ANGOT AND PIERRE FABRIE which implis using Young inquality and th proof of proposition 4 follows. Rmark 5. ε u ε L + div u ε L ε u + u g L,. Th function u blongs to H ν), which is not th cas for v ε or u, without som additional rgularity hypothss on th function g. Nvrthlss, th function u ε = v ε u u blongs to H ν).. In th cas whr g =, th function v ε blongs to H ν). REFERENCES [] Ph. Angot, J.-P. Caltagiron and P. Fabri, Vctor pnalty-projction mthod for th solution of unstady incomprssibl flows, in Finit Volums for Complx Applications V ds. R. Eymard and J.-M. Hérard), ISTE, London, 8), [] Ph. Angot, J.-P. Caltagiron and P. Fabri, A spctacular vctor pnalty-projction mthod for Darcy and Navir-Stoks problms, in Finit Volums for Complx Applications VI ds J. Fořt, t al.), Intrnational Symposium FVCA6 in Pragu, Jun 6-, Springr Procdings in Mathmatics, 4, Vol., Springr-Vrlag, Brlin, ), [3] Ph. Angot, J.-P. Caltagiron and P. Fabri, A nw fast mthod to comput saddl-points in constraind optimization and applications, Applid Mathmatics Lttrs, 5 ), [4] Ph. Angot, J.-P. Caltagiron and P. Fabri, A fast vctor pnalty-projction mthod for incomprssibl non-homognous or multiphas Navir-stoks problms, Applid Mathmatics Lttrs,, in prss. [5] F. Boyr and P. Fabri, Élémnts d Analys pour l Étud d qulqus Modèls d Écoulmnts d Fluids Visquux Incomprssibls, Mathématiqus & Applications, 5, Springr-Vrlag, 6. [6] A. J. Chorin, A numrical mthod for solving incomprssibl viscous flow problms, J. Comput. Phys., 967), 6. [7] A. J. Chorin, Numrical solution of th Navir-Stoks quations, Math. Comput., 968), [8] C. Foias and R. Tmam, Rmarqus sur ls équations d Navir-Stoks stationnairs t ls phénomèns succssifs d bifurcation, Annali dlla Scuolo Normal Suprior di Pisa, Class di Scinz 4), 5 978), [9] V. Girault and P.-A. Raviart, Finit Elmnt Mthods for th Navir-Stoks Equations. Thory and Algorithms, Springr Sris in Comput. Math., 5, Springr-Vrlag, Brlin, 986. [] J.-L. Gurmond, P. D. Minv and J. Shn, An ovrviw of projction mthods for incomprssibl flows, Comput. Mth. Appl. Mch. Engrg., 95 6), [] O. A. Ladyzhnskaya, Th Mathmatical Thory of Viscous Incomprssibl Flow, nd dition, Mathmatics and its Applications, Vol., Gordon and Brach, Scinc Publishrs, Nw York-London-Paris, 969. [] J. Lray, Essai sur ls mouvmnts plans d un liquid visquux qu limitnt ds parois, J. Math. Purs Appl., 3 934), [3] J.-L. Lions, Qulqus Méthods d Résolution ds Problèms aux Limits Non Linéairs, Dunod & Gauthir-Villars, Paris, 969. [4] J. Shn, On rror stimats of th pnalty mthod for unstady Navir-Stoks quations, SIAM J. Numr. Anal., 3 995), [5] J. Shn, On a nw psudocomprssibility mthod for th incomprssibl Navir-Stoks quations, Appl. Numr. Math., 996), 7 9. [6] J. Simon, Compact sts in th spac L p, T ; B), Ann. Mat. Pura Appl. 4), ), [7] R. Tmam, Un méthod d approximation d la solution ds équations d Navir-Stoks, Bull. Soc. Math. Franc, ), 5 5. [8] R. Tmam, Sur l approximation d la solution ds équations d Navir-Stoks par la méthod ds pas fractionnairs. I, Arch. Ration. Mch. Anal., 3 969), [9] R. Tmam, Sur l approximation d la solution ds équations d Navir-Stoks par la méthod ds pas fractionnairs. II, Arch. Ration. Mch. Anal., ),

23 THE VECTOR PENALTY-PROJECTION METHOD 45 [] R. Tmam, Navir-Stoks Equations. Thory and Numrical Analysis, 3 rd dition, Studis in Mathmatics and its Applications,, North-Holland Publishing Co., Amstrdam, 984. Rcivd July ; rvisd Dcmbr. addrss: angot@cmi.univ-mrs.fr addrss: pirr.fabri@math.u-bordaux.fr