Convergence to Suitable Weak Solutions for a Finite Element Approximation of the Navier–Stokes Equations with Numerical Subgrid Scale Modeling

CONVERGENCE TO SUITABLE WEAK SOLUTIONS FOR A FINITE ELEMENT APPROXIMATION OF THE NAVIER-STOKES EQUATIONS WITH NUMERICAL SUBGRID SCALE MODELING SANTIAGO BADIA†AND JUAN VICENTE GUTI´ ERREZ-SANTACREU‡ Abs ac . In his wo k we p o e ha weak solu ions cons uc ed by a a ia ional mul iscale me hod a e sui able in he sense o Sche e . In o de o p o e his esul , we conside a subg id model ha en o ces o hogonali y be ween subg id and ini e elemen componen s. Fu he , he subg id componen mus be acked in ime. Since his ype o schemes in oduce p essu e s abiliza ion, we ha e p o ed he esul o equal-o de eloci y and p essu e ini e elemen spaces ha do no sa is y a disc e e in -sup condi ion. 2010 Ma hema ics Subjec Classi ica ion: 35Q30; 65N30; 76N10. Keywo ds: Na ie –S okes equa ions; Sui able weak solu ions; S abilized ini e elemen me hods, Subg id scales. Con en s 1. In oduc ion 1 2. S a emen o he p oblem 3 2.1. No a ion 3 2.2. The Na ie -S okes equa ions 4 3. Fini e elemen app oxima ion 6 3.1. Hypo heses 6 3.2. The disc e e p oblem 7 3.3. Disc e e ope a o s 8 4. Technical p elimina y esul s 9 5. A p io i ene gy es ima es 13 6. Con e gence owa ds weak and sui able weak solu ions 17 Appendix A. P oo o he in e se inequali ies (8) 21 Acknowledgmen 21 Re e ences 22 1. In oduc ion Incomp essible New onian luids a e go e ned by he Na ie -S okes equa ions. The exis ence o solu ions is known om he wo ks by Le ay [31] and Hop [27]. Howe e , uniqueness is s ill an open Da e: May 21, 2018. †Uni e si a Poli `ecnica de Ca alunya, Jo di Gi ona1-3, Edi ici C1, E-08034 Ba celona & Cen e In e nacional de M`e odes Num`e ics en Enginye ia, Pa c Medi e ani de la Tecnologia, Es e e Te ades 5, E-08860 Cas ellde els, Spain E- mail: [email protected]. SB was pa ially suppo ed by by he Eu opean Resea ch Council unde he FP7 P og am Ideas h ough he S a ing G an No. 258443 - COMFUS: Compu a ional Me hods o Fusion Technology and he FP7 NUMEXAS p ojec unde g an ag eemen 611636. SB g a e ully acknowledges he suppo ecei ed om he Ca alan Go e nmen h ough he ICREA Acad`emia Resea ch P og am. ‡Dp o. de Ma em´a ica Aplicada I, E. T. S. I. In o m´a ica, Uni e sidad de Se illa. A da. Reina Me cedes, s/n. E- 41012 Se illa, Spain. E-mail: juan [email protected]. JVGS was pa ially suppo ed by he Spanish g an No. MTM2015-69875-P om Minis e io de Econom´ıa y Compe i i idad wi h he pa icipa ion o FEDER. 1 2 S. BADIA AND J. V. GUTI´ ERREZ-SANTACREU ques ion. The loss o egula i y is ela ed o u bulence [24], and Le ay deno ed weak solu ions as u bulen solu ion. Sche e de ined he concep o sui able weak solu ions in [36] and p o ed a bound o he Haussdo dimension o he singula se o a weak sui able solu ion. This esul was la e imp o ed by Ca a elli, Kohn, and Ni enbe g [9], p o ing ha his dimension is smalle han 1. This is he sha pes egula i y esul so a . Sui able weak solu ions o he Na ie -S okes equa ions can be cons uc ed by egula iza ion (see, e.g., [33]). Mo e ecen ly, Gue mond p o ed ha in -sup s able ini e elemen (FE) app oxima ions ha ing a disc e e commu a o p ope y also con e ge o sui able weak solu ions, i s o pe iodic bounda y condi ions in he h ee-dimensional o us [22], and nex on gene al domains and no-slip bounda y condi ions [23]. The Fou ie me hod does no sa is y he equi ed assump ions, and i is s ill an open ques ion whe he i p o ides sui able solu ions. The Na ie -S okes equa ions ha e a dissipa i e s uc u e, due o he iscous e m. The sys em has a singula limi in he assymp o ic egime as he Reynolds (Re) numbe , which is he a io o ine ia o ces o iscous o ces, goes o in ini y. The singula limi and he ac ha he sys em is inde ini e complica e i s nume ical app oxima ion. The i s p ope y equi es o in oduce some kind o con ec ion s abiliza ion, whe eas he second p e en s he use o he same FE space o bo h he eloci y and p essu e unknowns, he disc e e sys em is uns able. A he con inuous le el, he nonlinea con ec i e e m ans e s ene gy om he la ges o he smalles scales, ill eaching he Kolmogo o scale, whe e ene gy is dissipa ed. In di ec nume ical simula ions (DNS) he mesh needs o be ine enough o cap u e he smalles scales in he low. How- e e , his app oach is unaccep able o indus ial u bulen lows, due o he limi s in compu a ional esou ces. In eal applica ions, unde - esol ed simula ions a e needed. The smalles scales ha can be cap u ed in hese simula ions a e a om he Kolmogo o scale and dissipa ion is negligible. Thus, one has o add so-called la ge eddy simula ion (LES) u bulen models ha add a i icial di usion mechanisms. The concep o sui abili y and he ac ha ene gy is dissipa ed a he mesh scale in a physically consis en way ha e been ela ed in [24]. O he wise, an ene gy pile-up occu s a he smalles g id scales, leading o ins abili ies. Con ec ion s abiliza ion and u bulence models a e s ongly ela ed. In his sense, many au ho s ha e conside ed so-called implici LES (ILES) me hods ha do no modi y he o iginal Na ie -S okes equa ions bu in oduce addi ional nume ical a i ac s when ca ying ou he disc e iza ion [7, 18]. In he ame o FE echniques, one app oach is o conside a ia ional mul iscale (VMS) me hods [28, 29]. The idea is o use a wo-scale decomposi ion o he o iginal p oblem and p o ide a nume ically mo i a ed closu e o he ine scale (see, e.g., [21]). A simila s abiliza ion p ocedu e can be used o he con ec i e e m and he p essu e e m, leading o me hods ha do no equi e o sa is y a disc e e in -sup condi ion. An al e na i e o adi ional esidual-based me hods is o conside subscales ha a e in some sense o hogonal o he FE space. This idea has been p oposed by Codina [12], whe e L2(Ω) o hogonali y was used. This me hod in ol es global p ojec ions, which has mo i a ed he use o local p ojec ions (see, e.g., [5, 2]). The ea men o he ime dimension in he subg id model has also been objec o ac i e esea ch. In pa icula , he use o dynamic subscales me hods ha ack he subg id scale in ime ha e been p oposed in [12]. E en hough DNS is imp ac ical in eal applica ions, i is be e unde s ood han s abilized o ILES schemes. The g oundb eaking wo ks by Gue mond ha e p o ed ha he FE Gale kin me hod leads o weak sui able solu ions in [22, 23]. Howe e , he ex ension o ILES me hods is no s aigh o wa d, due o he in oduc ion o addi ional e ms o he nume ical o mula ion. The analysis o hese me hods has usually been es ic ed o a p io i e o es ima es o smoo h enough solu ion (see, e.g., [11]). Residual- based VMS schemes a e no amenable o weak con e gence analysis, due o he p oli e a ion o e ms, e.g., including new eloci y-p essu e coupling e ms. Howe e , en o cing he modelled subg id scales o be o hogonal o he FE space and conside ing he dynamic o mula ion in [12], he au ho s ha e p o ed in [4] ha he esul ing scheme con e ges o weak ( u bulen ) solu ions o he Na ie -S okes equa ions. Fo he same scheme, long- e m s abili y es ima es and exis ence o a global a ac o ha e been p o ed in [3]. Fu he , a e y de ailed nume ical expe imen a ion o hese me hods o CONVERGENCE TO SUITABLE WEAK SOLUTIONS FOR A SUBGRID FEM MODEL 3 iso opic and wall-bounded u bulen lows can be ound in [13], p o ing ha hese subg id models ac as accu a e u bulence models. Theo e ical analyses suppo ing hese esul s can also be ound in [20]. In his wo k, we wan o analyze whe he VMS- ype FE ILES schemes con e ge o sui able weak solu ions in he sense o Sche e . We p o e ha subg id closu es ha a e o hogonal and dynamic con e ge in ac o sui able solu ions o equal o de FE pai s o he eloci y and p essu e unknowns. The ou line o he wo k is he ollowing. Fi s , we s a e he p oblem and in oduce he no a ion in Sec ion 2. The FE app oxima ion based on he VMS- ype ILES scheme is in oduced in Sec ion 3. Sec ion 4 includes some echnical esul s in ac ional Sobole spaces. Ene gy es ima es a e p o ed in Sec ion 5. Finally, he con e gence owa ds weak and sui able solu ions is p o ed in 6. 2. S a emen o he p oblem Th oughou his pape we ollow ai h ully he no a ion used in [26] and [23] so ha he eade can ace wi h ease he main di e ences be ween hese wo wo ks and he one p esen ed he ein. 2.1. No a ion. Le Ω be an open subse o R3. Fo p∈[1,∞], we deno e by Lp(Ω) he usual Lebesgue space, i.e., Lp(Ω) = { : Ω →R: Lebesgue-measu able,ZΩ | (x)|pdx<∞}, wi h he usual modi ica ion when p=∞. This space is a Banach space endowed wi h he no m k kLp(Ω) = (RΩ| (x)|pdx)1/p i p∈[1,∞) o k kL∞(Ω) = ess supx∈Ω| (x)|i p=∞. In pa icula , L2(Ω) is a Hilbe space. We shall use (u, ) = RΩu(x) (x)dx o i s inne p oduc and k · k o i s no m. Fo m∈N, we deno ed by Hm(Ω) he classical Sobole -Hilbe spaces, i.e., Hm(Ω) = { ∈L2(Ω) : ∂k ∈L2(Ω) ∀ |k| ≤ m} associa ed o he no m k kHm(Ω) = X 0≤|k|≤m k∂k k2 L2(Ω)  1 2 , whe e k= (k1, ..., kd)∈Ndis a mul i-index and |k|=Pd i=1 ki. Le D(Ω) be he space o in ini ely imes di e en iable unc ions wi h compac suppo in Ω, i.e. he space o es unc ions on Ω. Thus Hm 0(Ω) is de ined as he comple ion o D(Ω) wi h espec o he Hm(Ω)-no m. F ac ional-o de Hilbe -Sobole spaces a e de ined by he eal me hod o K-me hod o in e pola ion due o Pee e and Lions [1]. Thus, we conside wo spaces: Hs(Ω) = [L2(Ω), H1(Ω)]s, o s∈(0,1), and ˜ Hs 0(Ω) = [L2(Ω), H1 0(Ω)]s o s∈[0,1]. Mo eo e , o s∈(0,1), Hs 0(Ω) is he closu e o D(Ω) wi h espec o he Hs(Ω)-no m. No e ha he spaces Hs(Ω) and Hs 0(Ω) coincide o s∈[0,1 2], wi h uni o m no ms [25, Th 11.1], and he spaces Hs(Ω) and ˜ Hs 0(Ω) coincide wi h equi alen no ms [34] o s∈[0,1 2). We also conside Hs(Ω) = [H1(Ω), H2(Ω)]s o s∈(1,2] and ˜ Hs 0(Ω) = Hs(Ω) ∩H1 0(Ω) o s∈(1,2]. The dual space o D(Ω), he space o dis ibu ions, is deno ed by D′(Ω). Mo eo e , o s < 0, ˜ Hs(Ω) is he dual o ˜ H−s 0(Ω) and he space H−s 0(Ω) is he complexion o D(Ω) unde he no m k kH−s(Ω) = sup w∈D(Ω) {0} ( , w) kwkHs(Ω) , We use h·,·i o deno e he duali y pai ing. Fo s∈[0,1 2)∪(1 2,3 2), H−s(Ω) coincides wi h ˜ H−s 0(Ω). We will use bold aced le e s o spaces o ec o unc ions, e.g. L2(Ω) in place o L2(Ω)d. We will make use o he ollowing space o ec o ields: ϑ={ ∈D(Ω) : ∇ · = 0 in Ω}. Rela ed o he space ϑ, we conside he closu es in he L2(Ω) and H1(Ω)-no m, which a e cha ac e ized by H={u∈L2(Ω) : ∇ · u= 0 in Ω,u·n= 0 on ∂Ω}, V={u∈H1(Ω) : ∇ · u= 0 in Ω,u=0on ∂Ω}, 4 S. BADIA AND J. V. GUTI´ ERREZ-SANTACREU whe e nis he ou wa d no mal o Ω on ∂Ω. This cha ac e iza ion is ue o locally Lipschi z-con inuous domains (see [38, Theo ems 1.4 and 1.6] o a de ailed p oo ). Fu he mo e, L2 R=0(Ω) ( esp. H1 R=0(Ω)) is he space o ze o-a e age L2(Ω)- unc ions ( esp. ze o-a e age H1(Ω)- unc ions ). Thus, by he eal me hod o in e pola ion, Hs R=0(Ω) = [L2 R=0(Ω), H1 R=0(Ω)] o s∈(0,1) (see [25]). Le Xbe a Banach space. Thus, Lp(a, b;X) deno es he space o Bochne -measu able, X- alued unc ions on he in e al (0, T ) such ha RT 0k (s)kp Xds < ∞i 1 ≤p < ∞o ess sups∈(0,T )k (s)kX< ∞i p=∞. Mo eo e , W1,1(0, T ;X) is he space o unc ions ∈L1(0, T ;X) and d ds ∈L1(0, T ;X) such ha RT 0(k (s)kX+kd ds (s)kX) ds < ∞and W1,1 0(0, T ;X) is he closu e o D(0, T ;X) wi h espec o he W1,1(0, T ;X)-no m, wi h D(0, T ;X) being he space o in ini ely imes di e en iable unc ions de ined on (0, T ) ha ing alues in o Xwi h compac suppo in (0, T ). Addi ionally, he dual space o W1,1 0(0, T ;X) is deno ed by W−1,∞(0, T ;X′) p o ided ha Xis sepa able and e lexi e. The Fou ie ans o m o a unc ion ∈L1(R;X) is deno ed by F (ξ) := Z+∞ −∞ e−2πi ·ξ ( )d . Le Hbe a Hilbe space and le S′(R;H) be he space o empe ed dis ibu ions aking alue in H. Thus, o γ∈R, one de ines Hγ(R;H) = { ∈ S′(R;H); ZR (1 + |ξ|)2γkF k2 Hdξ}, whe e His a Hilbe space. Addi ionally, he space Hγ(0, T ;H) is made up o empe ed dis ibu ions in S′(0, T ;H) wi h he no m k kHγ(0,T ;H)= in ∈S′(R;H)k kHγ(R;H), whe e is he ex ension o by ze o o (0, T ) belonging o S′(R;H). No e ha h oughou his pape we use he symbol C(wi h o wi hou subsc ip s) o ep esen gene ic posi i e cons an s which can ake di e en alues a di e en places. 2.2. The Na ie -S okes equa ions. The Na ie -S okes equa ions o he mo ion o a iscous, in- comp essible, New onian luid can be w i en as ∂ u−ν∆u+ (u· ∇)u+∇p= in Ω ×(0, T ), ∇ · u= 0 in Ω ×(0, T ),(1) wi h Ω being a bounded, h ee-dimensional domain and wi h 0 < T < +∞. He e u: Ω ×(0, T )→R3 ep esen s he incomp essible luid eloci y and p: Ω ×(0, T )→R ep esen s he luid p essu e. Mo eo e , is he ex e nal body o ce which ac s on he sys em, and ν > 0 is he kinema ic luid iscosi y. These equa ions a e supplemen ed by he no-slip bounda y condi ion u=0on ∂Ω×(0, T ),(2) and he ini ial condi ion u(0) = u0in Ω.(3) The i s au ho s dealing wi h he concep o weak solu ions o he Na ie -S okes equa ions we e Le ay [31] o he Cauchy p oblem in he whole space and la e Hop [27] o he ini ial-bounda y alue p oblem in bounded domains. Pa icula ly, weak solu ions we e called u bulen by Le ay due o he possible connec ion be ween he lack o egula i y o weak solu ions and u bulence. De ini ion 2.1. A unc ion uis said o be a weak solu ion o p oblem (1)-(2) i : u∈L∞(0, T ;H)∩L2(0, T ;V) (4) CONVERGENCE TO SUITABLE WEAK SOLUTIONS FOR A SUBGRID FEM MODEL 5 and −ZT 0 (u( ), ∂ ( )) d +ZT 0 h(u( )· ∇)u( ), ( )id +ZT 0 ν(∇u( ),∇ ( )) d = (u0, (0)) + ZT 0 h ( ), ( )id o all ∈W1,1(0, T ;V)wi h (T) = 0. Mo eo e , he ene gy inequali y 1 2ku( )k2+νZ 0 k∇u(s)k2ds≤1 2ku0k2+Z 0 h (s),u(s)ids(5) holds a. e. in [0, T ]. An equi alen de ini ion o weak solu ions in ol ing he p essu e e m is de ined as ollows. De ini ion 2.2. A pai (u, p)is said o be a weak solu ion o p oblem (1)-(2) i : u∈L∞(0, T ;H)∩L2(0, T ;V)and p∈W−1,∞(0, T, L2(Ω)/R) and ∂ u+ (u· ∇)u−ν∆u+∇p= in W−1,∞(0, T ;H−1(Ω)), u(0) = u0in H. Mo eo e , he ene gy inequali y 1 2ku( )k2+νZ 0 k∇u(s)k2ds≤1 2ku0k2+Z 0 h (s),u(s)ids holds a. e. in [0, T ]. We e e he eade o [16, Th. 1.3, Ch. V] o a p oo o he equi alence be ween De ini ions 2.1 and 2.2 wi h p∈ D′((0, T )×Ω), ha can easily be ex ended o p∈W−1,∞(0, T ;L2(Ω)/R), by using de Rham’s Lemma in [37, Lm. 2]. The wo p e ious de ini ions o weak solu ions can be p o ed o Ω being a bounded, Lipschi zian domain, and ∈L2(0, T ;H−1(Ω)) only. The weak solu ion ha will be p o ed in his pape equi es Ω o be, o ins ance, con ex, and ∈L2(0, T + 1; H−1(Ω)) ∩Lp(0, T + 1; Lq(Ω)), wi h p∈[1,2] and q∈[1,3 2] sa is ying 2 p+3 q= 4. De ini ion 2.3. A pai (u, p)is said o be a weak solu ion o p oblem (1)-(2) i : u∈L∞(0, T ;H)∩L2(0, T ;V)and p∈H− (0, T, H1−s R=0(Ω)) wi h s∈(1 2,7 10 ]and > ¯ =3 4−s 2, and (∂ u+ (u· ∇)u−ν∆u+∇p= in H− (0, T ;˜ H−s 0(Ω)), u(0) = u0in H. Mo eo e , he ene gy inequali y 1 2ku( )k2+νZ 0 k∇u(s)k2ds≤1 2ku0k2+Z 0 h (s),u(s)ids holds a. e. in [0, T ]. Sche e [36] in oduced he de ini ion o sui able weak solu ions so as o p o e a pa ial egula i y heo em. A e wa ds, Ca a elli, Kohn, and Ni enbe g [9] imp o ed Sche e ’s esul s, and F.-H. Lin [32] simpli ied he p oo s o he esul s in [9]. De ini ion 2.4. A weak solu ion (u, p)is said o be sui able i he local ene gy inequali y ∂ (1 2u2) + ∇ · ((1 2u2+p)u)−ν∆(1 2u2) + ν(∇u)2− ·u≤0 holds in D′((0, T )×Ω; R+). 6 S. BADIA AND J. V. GUTI´ ERREZ-SANTACREU 3. Fini e elemen app oxima ion 3.1. Hypo heses. Th oughou his pape we will assume he ollowing hypo heses: (H1) Le Ω be a connec ed, bounded, open subse o R3ha ing a polyhed al bounda y such ha he e exis ∈V∩H2(Ω) and p∈H1 R=0(Ω) sa is ying −∆ +∇p=gin Ω, ∇ · = 0 in Ω. (H2) Conside {Th}h>0 o be a shape- egula and quasi-uni o m amily o simplicial and con o ming meshes o Ω such ha Ω = ∪K∈ThKwi h h= maxK∈ThhKwhe e hK= diam K. (H3) Le {Wh}h>0and {Qh}h>0be wo amilies o ini e-elemen spaces associa ed wi h {Th}h>0 such ha Wh⊂H1 0(Ω) and Qh⊂H1 R=0(Ω). Mo eo e , he ini e-elemen spaces a e equi ed o sa is y he ollowing condi ions. Le πWh:L2(Ω) →Whand πQh:L2(Ω) →Qhbe he o hogonal p ojec ions on o Whand Qh, espec i ely, wi h espec o he L2(Ω)-inne p oduc . Fu he mo e, we deno e π⊥ Wh(·) := (·)−πWh(·) and π⊥ Qh(·) := (·)−πQh(·). (a) The e exis s a cons an Cin >0, independen o h, such ha , o all wh∈Wh, kwhkL∞(Ω) ≤Cin h−3 kkwhkLk(Ω) (6) and k∇whkLk(Ω) ≤Cin h−1kwhkLk(Ω) (7) o k∈[2,∞], kwhkH1(Ω) ≤Cin h−1+skwhk˜ Hs 0(Ω) (8) o each s∈[0,1], and kwhk˜ Hs 0(Ω) ≤Cin h−skwhkand kwhk ≤ Cin h−skwhk˜ Hs 0(Ω) (9) o s∈[0,1]. (b) The e exis s a cons an Cs (s)>0, independen o h, such ha , o s∈[0,3 2), kπWhwk˜ Hs 0(Ω) ≤Cs (s)kwk˜ Hs 0(Ω) o all w∈˜ Hs 0(Ω),(10) (c) The e exis s a cons an Cin >0, independen o h, such ha , o all land s, sa is ying 0≤l≤min{1, s}and l≤s≤2, he e holds kπ⊥ Whwk˜ Hl 0(Ω) ≤Cin hs−lkwk˜ Hs 0(Ω) o all w∈˜ Hs 0(Ω),(11) and kπ⊥ QhqkHl(Ω) ≤Cin hs−lkqkHs(Ω) o all q∈Hs R=0(Ω).(12) (d) The e exis s Ccom >0, independen o h, such ha , o 0 ≤l≤m≤1 and ϕ∈W2,∞ 0(Ω), kπ⊥ Wh(ϕwh)kHl(Ω) ≤Ch1+m−lkwhkHm(Ω)kϕkWm+1,∞ 0(Ω) o all wh∈Wh,(13) and kπ⊥ Qh(ϕqh)kHl(Ω) ≤Ch1+m−lkqhkHm(Ω)kϕkWm+1,∞ 0(Ω) o all qh∈Qh.(14) (H4) Le u0∈Vand ∈L2(0, T + 1; H−1(Ω)) ∩Lp(0, T + 1; Lq(Ω)), wi h p∈[1,2] and q∈[1,3 2] sa is ying 2 p+3 q= 4. Hypo hesis (H1) is ensu ed o domains ha ing a C1,1bounda y o being a con ex polygon (c . [30] o [19]) o polyhed on (c . [14] ), wi h con inuous dependence on . Hypo hesis (H3) is ex emely lexible and allows equal-o de ini e-elemen spaces o eloci y and p essu e. Fo ins ance, le Pk(K) be he se o piecewise polynomial unc ions o deg ee less han o CONVERGENCE TO SUITABLE WEAK SOLUTIONS FOR A SUBGRID FEM MODEL 7 equal o kon Kbeing a e ahed a. Thus he space o con inuous, piecewise polynomial unc ions o deg ee less han o equal o kon a mesh This deno ed as Xh= h∈C0(Ω) : h|K∈ Pk(K),∀K∈ Th, We choose he ollowing con inuous ini e-elemen spaces Wh=Xh∩H1 0(Ω) and Qh=Xh∩L2 R=0(Ω), o app oxima ing eloci y and p essu e, espec i ely. The shape- egula and quasi-uni o m p ope ies o {Th}h>0assumed in (H2) su ice o ensu e he p ope ies o (H3)(a). We ecommend he books [8, Sec. 4.5 ] and [15, Sec. 1.7] o a p oo o (6) and (7), Appendix A o a p oo o (8), and [17] o a p oo o (9). Mo eo e , he e o es ima es s a ed in (H3) make use o (H2) as well (see [26, Lm A.3, Rm 2.1] o a p oo ). The local app oxima ion p ope ies o he o hogonal p ojec ion ope a o s πWhand πQhgua an ee hypo hesis (H4). The eade is e e ed o [6]. Rema k 3.1. Le pand qbe as in (H4). We know om Sobole ’s embeddings ha ˜ Hs 0(Ω) is embedded in Lq′(Ω), whe e 1 q′+1 q= 1 and s= 3(1 q−1 2); hence Lq(Ω) is embedded in ˜ H−s 0(Ω). Mo eo e , H (R;H)is embedded in Lp′(R;H), whe e 1 p′+1 p= 1 and > ¯ =1 p−1 2wi h Hbeing a Hilbe space; hence Lp(R;H)is embedded in H− (R;H). Le be he ex ension o ou side [0, T ]as ze o. Then, by Hausdo -Young’s inequali y o he Fou ie ans o m, we ha e kF kH− (R;˜ H−s 0(Ω)) ≤CkF kLp′(R;Lq(Ω)) ≤Ck kLp(R;Lq(Ω)) =Ck kLp(0,T ;Lq(Ω)).(15) The e o e, ∈H− (0, T ;˜ H−s 0(Ω)) (16) As a e e ence o u he de elopmen , i is well o poin ou , he e, he condi ions o p,q,sand ¯ : (C) Le s= 3(1 q−1 2)and ¯ =1 p−1 2be de ined o pand qas in (H4). 3.2. The disc e e p oblem. Find uh∈H1(0, T ;Wh), ph∈L2(0, T ;Qh) and ˜ uh∈H1(0, T ;˜ Wh) such ha , o all ( h,˜ h, qh)∈Whט Wh×Qh, (∂ uh, h) + b(uh,uh, h) + ν(∇uh,∇ h) −(ph,∇ · h)−b(uh, h,˜ uh) = ( h, h),(17a) (uh,∇qh) + (˜ uh,∇qh) = 0,(17b) (∂ ˜ uh,˜ h) + b(uh,uh,˜ h) +τ−1(˜ uh,˜ h) + (∇ph,˜ h) = 0,(17c) uh(0) = u0h,(17d) whe e τ=1 Csν h2+CckuhkL∞(Ω) h =h2 Csν+CchkuhkL∞(Ω) , wi h Csand Ccbeing algo i hmic posi i e cons an s, and h∈Whis de ined by duali y as ( h,wh) = h ,whi, o all wh∈Wh. Le us de ine b(uh, h,wh) = hN (uh, h),whi, whe e N(uh, h) = (uh· ∇) h+1 2(∇ · uh) h. Le {ψi}i=1,...,nube a basis o Whand le {ψi}i=1,...,npbe a basis o Qh, whe e nuand npdeno e he space dimension o Whand Qh, espec i ely. Thus, one de ines ˜ Wh= span{π⊥ Wh(N(φi,φj)), π⊥ Wh(∇φk)}, and W⋆=Wh⊕˜ Wh. Mo eo e , one de ines V⋆={ ⋆∈W⋆: ( h,∇qh) + (˜ h,∇qh) = 0 o all qh∈Qh}. 8 S. BADIA AND J. V. GUTI´ ERREZ-SANTACREU which is a non-con o ming app oxima ion space o V. The ini ializa ion o he disc e e p oblem can be ob ained by he ollowing p ojec ion p oblem: ind u0h∈Vh,˜ u0h∈˜ Vhand ξh∈Qhsuch ha    (u0h, h)−(ξh,∇ · h) = (u0, h), o all h∈Vh, (˜ u0h,˜ ) + (∇ξh,˜ ) = (u0,˜ h), o all ˜ h∈˜ Vh, (∇ · u0h, qh)−(˜ u0h,∇qh) = 0, o all qh∈Qh. (18) 3.3. Disc e e ope a o s. This subsec ion is de o ed o in oducing he disc e e ope a o s ha a e used h ough ou his pape . Fi s ly, we will conside a con o ming and non-con o ming app oxima ion o he Laplace ope a o −∆ : ˜ H2 0(Ω) →L2(Ω). The non-con o ming app oxima ion is based on a s abilizing echnique. Conside −∆h:H1 0(Ω) →Wh o be he disc e e Laplacian ope a o de ined as: −(∆hw,¯ wh) = (∇wh,∇¯ wh) o all ¯ wh∈Wh. The es ic ion o his ope a o −∆h o Wh⊂H1 0(Ω) gi es a sel -adjoin , posi i e-de ini e ope a o . The e o e, we a e allowed o de ine he ac ional powe o −∆h, say (−∆h)s, o all s∈R, by he Hilbe -Schmid heo em. The domain o de ini ion o (−∆h)sis D((−∆h)s)≡Whsince dim Wh< ∞. Hence, Ws hmakes e e ence o Whequipped wi h he Hilbe no m kwhkWs h= ((−∆h)s 2wh,(−∆h)s 2wh)1 2. The amily {Ws h}s∈Ris a scale o Hilbe spaces wi h espec o he eal me hod o in e pola ion. Analogously, conside −∆⋆:W⋆→W⋆ o be he s abilized disc e e Laplacian ope a o de ined as −(∆⋆w⋆,¯ w⋆) = (∇πWhw⋆,∇πWh¯ w⋆) + h−2(π⊥ hw⋆, π⊥ h¯ w⋆) o all ¯ w⋆∈W⋆. I is easy o see ha −∆⋆w⋆=−πWh∆⋆w⋆−π⊥ Wh∆⋆w⋆=−∆hπWhw⋆−h−2π⊥ Whw⋆. We ha e ha −∆⋆is sel -adjoin and posi i e-de ini e. The e o e, we a e also allowed o de ine he ac ional powe o −∆⋆, say (−∆⋆)s, o all s∈R, by he Hilbe -Schmid heo em. Thus, Ws ⋆is W⋆equipped wi h he Hilbe no m kw⋆kWs ⋆= ((−∆⋆)s 2w⋆,(−∆⋆)s 2w⋆). Secondly, we will conside a non-con o ming app oxima ion o he S okes ope a o A:= P(−∆) : V∩H2(Ω) →Hwhe e Pis he Le ay-Helmhol z p ojec o ope a o . Le A⋆:V⋆→V⋆be de ined as (A⋆ ⋆,¯ ⋆) = (∇πWh ⋆,∇πWh¯ ⋆) + h−2(π⊥ Wh ⋆, π⊥ Wh¯ ⋆) o all ¯ ⋆∈V⋆. Equi alen ly, one can w i e A⋆=πWhA⋆+π⊥ WhA⋆:= Ah+˜ Ahsa is ying    (Ah ⋆,wh) + (∇ h,wh) = (∇πWh ⋆,∇wh) o all wh∈Wh, (Ah ⋆,∇qh) + ( ˜ Ah ⋆,∇qh) = 0 o all qh∈Qh, (˜ Ah ⋆,˜ wh) + (∇ h,˜ wh) = h−2(π⊥ Wh ⋆,˜ wh) o all ˜ wh∈˜ Wh. (19) Again, A⋆is a sel -adjoin , posi i e-de ini e ope a o . The e o e, he ac ional powe o A⋆, say As ⋆, is well-de ined o all s∈R. Mo eo e , Vs ⋆deno es V⋆equipped wi h he Hilbe no m k ⋆kVs ⋆= (As 2 ⋆ ⋆, As 2 ⋆ ⋆)1 2. The amily {Vs ⋆}s∈Ris a scale o Hilbe space wi h espec o he eal me hod o in e pola ion. Nex we will conside a non-con o ming app oxima ion o he Le ay-Helmhol z p ojec ion ope a o . Le P⋆:L2(Ω) →V⋆be de ined as (P⋆ ,¯ ⋆) = ( ,¯ ⋆) o all ¯ ⋆∈V⋆. Equi alen ly, one can w i e P⋆=πWhP⋆+π⊥ WhP⋆:= Ph+˜ Phsa is ying    (Ph ,wh) + (∇ h,wh) = (πWh ,wh) o all wh∈Wh (Ph ,∇qh) + ( ˜ Ph ,∇qh) = 0 o all qh∈Qh, (˜ Ph ,˜ wh) + (∇ h,˜ wh) = (π⊥ Wh ,˜ wh) o all ˜ wh∈˜ Wh. (20) CONVERGENCE TO SUITABLE WEAK SOLUTIONS FOR A SUBGRID FEM MODEL 9 Finally, we de ine he s abilized Ri z p ojec ion ope a o on o V⋆. Le R⋆:H1 0(Ω) = πWhH1 0⊕ π⊥ WhH1 0(Ω) →V⋆be de ined as (∇πWhR⋆ ,∇πWh ⋆) + h−2(π⊥ WhR⋆ , π⊥ Wh¯ ⋆) = (∇πWh ,∇πWh ⋆) + h−2(π⊥ Wh ,π⊥ Wh¯ ⋆), o all ⋆∈V⋆. Equi alen ly, one can w i e R⋆=πWhR⋆+π⊥ WhR⋆:= Rh+˜ Rhsa is ying    (∇Rh ,∇wh) + (∇ h,wh) = (∇πWh ,∇wh) o all wh∈Wh (Rh ,∇qh) + ( ˜ Rh ,∇qh) = 0 o all qh∈Qh, h−2(˜ Rh ,˜ wh) + (∇ h,˜ wh) = h−2(π⊥ Wh ,˜ wh) o all ˜ wh∈˜ Wh. (21) 4. Technical p elimina y esul s This sec ion is mainly de o e o some echnical esul s conce ning equi alence be ween no ms and in -sup condi ions in ac ional-o de Sobole spaces. Lemma 4.1. Suppose ha condi ions (H1)-(H3) hold. Then he e exis wo posi i e cons an s c, C such ha , o all s∈R, c(kwhkWs h+h−sk˜ whk)≤ kw⋆kWs ⋆≤C(kwhkWs h+h−sk˜ whk),(22) o all w⋆=wh+˜ wh∈W⋆. P oo . The p oo ollows by obse ing ha (−∆⋆w⋆)s= (−∆πWhw⋆)s+h−sπ⊥ Wh˜ w⋆ o all w⋆.  Co olla y 4.2. Suppose ha condi ions (H1)-(H3) hold. Then he e exis wo posi i e cons an s c, C such ha , o all s∈(−3 2,3 2), c(kwhk˜ Hs 0(Ω) +h−sk˜ whk)≤ kw⋆kWs ⋆≤C(kwhk˜ Hs 0(Ω) +h−sk˜ whk),(23) o all w⋆=wh+˜ wh∈W⋆. P oo . The p oo is based on he esul o [26, Lemma 2.2]: ckwhk˜ Hs 0≤ kwhkWs h≤Ckwhk˜ Hs 0 o all wh∈Wh. wi h s∈(−3 2,3 2).  In he nex lemma, we p o e he s abili y o he s abilized disc e e Le ay-Helmhol z ope a o P⋆= Ph+˜ Ph. Lemma 4.3. Assume ha condi ions (H1)-(H3) a e sa is ied. Then he e exis s a posi i e cons an C, independen o h, such ha , o all s∈[0,1 2), kPh k˜ Hs 0(Ω) +h−sk˜ Ph k ≤ Ck k˜ Hs 0(Ω) o all ∈˜ Hs 0(Ω),(24) whe e P⋆=Ph+˜ Phis he L2(Ω)-o hogonal p ojec ion ope a o on o V⋆. P oo . Le ∈˜ Hs 0(Ω). Then, by he Helmhol z-Hodge decomposi ion, he e exis s ∈H1 R=0(Ω) such ha =P +∇ , whose a ia ional o mula ion eads as: (P ,¯ ) + (∇ , ¯ ) = ( ,¯ ) o all ¯ ∈L2(Ω), ( ,∇q) = 0 o all H1 R=0(Ω),(25) No e ha p oblem (20) is he s abilized disc e e coun e pa o (25). F om [17, Chap e II, Theo em 1.1], we ge kP⋆ −P k+k∇ h− ∇ k ≤ C( in w⋆∈W⋆ kP −w⋆k+ in qh∈Qh k∇ − ∇qhk).(26) 16 S. BADIA AND J. V. GUTI´ ERREZ-SANTACREU whe e µ=2 2−γ. In eg a ing o e Rand using H¨olde ’s inequali y and Planche el’s equali y gi es ZR |ξ|2 2−γ−µkF˜ u⋆k2 V−α ⋆dξ ≤Ck˜ gk 2 2−γ H− (R;W−s ⋆)k˜ u⋆k 2(1−γ) 2−γ L2(R;V⋆), which implies ha ZR |ξ|2βkF˜ u⋆k2 V−α ⋆≤Ck˜ gk 2 2−γ H− (0,T ;W−s ⋆)k˜ u⋆k 2(1−γ) 2−γ L2(0,T ;V⋆)),(50) o β < ¯ βwi h ¯ β:= 1+α 1+s(1 −¯ ) coming om he de ini ion o γ,µ, ¯ , and α≤s≤1 + 2α. Nex obse e ha we ha e, om (15) and (46) o s∈[0,3 2), k˜ gkH− (R;W−s ⋆)≤C. (51) Inse ing (43) and (51) in o (50), we a i e a ZR |ξ|2βkF˜ u⋆k2 V−α ⋆≤C. Fo β≥0, we w i e ZR (1 + |ξ|)2βkF˜ u⋆k2 V−α ⋆dξ =Z|ξ|≤1 (1 + |ξ|)2βkF˜ u⋆k2 V−α ⋆dξ +Z|ξ|>1 (1 + |ξ|)2βkFu⋆k2 V−α ⋆dξ ≤CZ|ξ|≤1 kF ˜ u⋆k2 V−α ⋆dξ +CZ|ξ|>1 |ξ|2βkF˜ u⋆k2 V−α ⋆dξ. ≤CZR k˜ u⋆k2 V⋆dξ +CZR |ξ|2βkF˜ u⋆k2 V−α ⋆dξ, whe e Planche el’s equali y and he con inuous embedding be ween V⋆and V−α ⋆we e used in he las line. The abo e es ima e also holds i ially o β < 0. Thus we ge k∂ ˜ u⋆kHβ−1(R;V−α ⋆)+k˜ u⋆kHβ(R;V−α ⋆)≤C. As a esul o (33) o s∈[0,1 2), we ob ain k∂ ˜ u⋆kHβ−1(0,T ;W−α ⋆)+k˜ u⋆kHβ(0,T ;W−α ⋆)≤C, o all αsa is ying 0 ≤α≤s≤1 + 2α < 2, and o all βsa is ying β < ¯ β:= 1+α 1+s(s 2+1 4). This la e inequali y leads o (47). Nex , mul iply (49) by A1−s ⋆F˜ u⋆and ake he eal pa o ge νkA⋆F˜ u⋆k2 V−s ⋆≤CkF ˜ gkW−s ⋆kA1−s ⋆F˜ u⋆kVs ⋆≤CkF ˜ gkW−s ⋆kA⋆F˜ u⋆kVs ⋆, whe e we ha e applied (33) o s∈[0,2). Thus, (1 + |ξ|)−2 kA⋆F˜ u⋆k2 V−s ⋆≤C(1 + |ξ|)−2 kF˜ gk2 W−s ⋆, and hence kA⋆˜ u⋆kH− (R;V−s ⋆)≤Ck˜ gkH− (R;W−s ⋆). I is no ha d o see om (33) ha k∆⋆ ⋆kW−s ⋆≤CkA⋆ ⋆k o all ⋆∈V⋆and all s∈[0,3 2). Then, using (51) yields k∆⋆˜ u⋆kH− (0,T ;W−s ⋆)≤C, o all > ¯ , which implies (48).  Co olla y 5.7. Suppose ha assump ions (H1)-(H4) hold. Fo α∈[1 4,1 2)and β < ¯ β=2 5(1 + α), i ollows ha k∂ u⋆kHβ−1(0,T ;W−α ⋆)+ku⋆kHβ(0,T ;W−α ⋆)≤C, (52) whe e C > 0is a cons an e independen o h. CONVERGENCE TO SUITABLE WEAK SOLUTIONS FOR A SUBGRID FEM MODEL 17 P oo . F om s∈[1 2,3 2) and 0 ≤α≤s≤1 + 2α < 2, we ob ain α∈[1 4,1 2). Nex no e ha 1 1+s(s 2+1 4) eaches i s maximum 2 5a s=3 2. The e o e we can simpli y he exp ession ¯ βin e m o αonly as ¯ β=2 5(1 + α) in (47).  Using (23), one can also p o e he ollowing. Co olla y 5.8. Assume ha assump ions (H1)-(H4) hold. Then, o α∈[1 4,1 2)and β < ¯ β=2 5(1+α), i ollows ha he e exis s a cons an C > 0, independen o h, such ha k∂ uhkHβ−1(0,T ;˜ H−α 0(Ω)) +kuhkHβ(0,T ;˜ H−α 0(Ω)) ≤C. (53) Fu he mo e, o s∈[1 2,3 2]and such ha > ¯ =3 4−s 2=1 p−1 2, i ollows ha k∆huhkH− (0,T ;˜ H−s 0(Ω)) ≤C. (54) We now p oceed o ob ain an es ima e o ph. Lemma 5.9. Suppose ha condi ions (H1)-(H4) hold. The e exis s a cons an C > 0, independen o h, such ha , o s∈[1 2,7 10 ]and > ¯ =3 4−s 2, kphkH− (0,T ;H1−s(Ω)) ≤C, (55) whe e C > 0is a cons an independen o h. P oo . Fi s we w i e (52) as k∂ u⋆kH− (0,T ;W−α ⋆)≤C, whe e α∈[1 4,1 2) and > ˜ := 1 −¯ β=3 5−2 5α. As a esul , we ha e ha k∂ u⋆kH− (0,T ;W−s ⋆)≤C(56) holds o α≤sand ˜ ≤¯ p o ided ha s∈[1 2,7 10 ] and > ¯ =3 4−s 2. F om (45), we bound kphkH1−s(Ω) ≤sup ⋆∈W⋆ {0} (∇ph, ⋆) k ⋆kWs ⋆ ≤C(k∂ u⋆kW−s ⋆+k∆⋆u⋆kW−s ⋆+kN⋆(u⋆,u⋆)kW−s ⋆+k hkW−s ⋆) ≤C(k∂ u⋆kW−α ⋆+k∆⋆u⋆kW−s ⋆+kN⋆(u⋆,u⋆)kW−s ⋆+k hk˜ H−s 0(Ω)). The p oo is comple ed ia (56), (48), (46) and (16).  6. Con e gence owa ds weak and sui able weak solu ions In his sec ion we will p o e ha he sequence o he app oxima e solu ions p o ided by scheme (17) con e ges owa ds a weak solu ion in he sense o De ini ion 2.3 and owa ds a sui able weak solu ion in he sense o De ini ion 2.4. In o de o hese con e gence esul s o hold, we will need o use he ollowing compac ness esul s `a la Aubin-Lions. The ollowing compac ness esul is due o Lions [33]. Lemma 6.1. Le H0֒→H ֒→H1be h ee Hilbe spaces wi h dense and con inuous embedding. Assume ha he embedding H0֒→His compac . Then L2(0, T ;H0)∩Hγ(0, T ;H1)embeds compac ly in L2(0, T ;H) o γ > 0. The p oo o he wo ollowing compac ness esul can be ound in [23, Ap. A.1, A.2]. Lemma 6.2. Le X ֒→Ybe wo Hilbe spaces wi h compac embedding. Then Hβ(0, T ;X)embeds con inuously and compac ly in C0([0, T ]; Y) o β > 1 2. Lemma 6.3. Le H0֒→H1be wo Hilbe spaces wi h compac embedding. Le γ > 0and γ > µ, hen he injec ion Hγ(0, T ;H0)֒→Hµ(0, T H1)is compac . 18 S. BADIA AND J. V. GUTI´ ERREZ-SANTACREU Theo em 6.4. Assume ha hypo heses (H1)-(H4) a e sa is ied. Then he e exis s a subsequence (deno ed in he same way) o app oxima e solu ions (uh, ph)con e ging owa d a weak solu ion gi en in De ini ion (2.3) in he ollowing sense as h→0: uh→uin L2(0, T ;H1 0(Ω)) −weak and in L2(0, T ;Hβ(Ω)) −s ong o all β < 1 (57) and ph→pin H− (0, T ;Hδ(Ω)) −weak o all δ∈[3 10,1 2]and > 1 4+δ 2.(58) P oo . Le ∈H (0, T ;˜ Hs 0(Ω)), o s∈(1 2,7 10 ] and > 3 4−s 2, and q∈L2(0, T ;H1 R=0(Ω)). F om (11) and (12), we a e allowed o cons uc h ee sequences { h}h>0⊂H (0, T ;Wh), {˜ h}h>0⊂ H (0, T ;˜ Wh) and {qh}h>0⊂L2(0, T ;Qh) such ha h→ in H (0, T ;˜ Hs 0(Ω))-s ong, ˜ h→0in L2(0, T ;L2(Ω))-s ong and qh→qin L2(0, T ;H1 R=0(Ω))-s ong as h→0. By i ue o (43), (53), (54) and (55), we know ha he e exis a subsequence o { h}h>0and {ph}h>0, s ill deno ed by i sel , and a pai (u, p) such ha uh→uin L∞(0, T ;L2(Ω)) −weak-⋆, uh→uin L2(0, T ;H1 0(Ω)) −weak, ∂ uh→∂ uin H− (0, T ;˜ H−s 0(Ω)) −weak, ∆huh→∆uin H− (0, T ;˜ H−s 0(Ω)) −weak, and ∇ph→ ∇pin H− (0, T ;˜ H−s 0(Ω)) −weak, o all s∈(1 2,7 10 ] and > 3 4−s 2. Obse e ha we ha e used ha he ac ha ˜ H−s(Ω) coincides wi h H−s 0(Ω) o s∈(1 4,7 10 ] o he p essu e. We also ha e, om (43), ha ˜ uh→0 in L2(0, T ;L2(Ω)) −s ong,(59) since ν1 2 hk˜ uhkL2(0,T ;L2(Ω)) ≤ kτ−1 2˜ uhkL2(0,T ;L2(Ω)) ≤C. We can pass o he limi in (17b). Thus we ind ha ∇ · u= 0 in (0, T )×Ω, whence u∈ L∞(0, T ;H)∩L2(0, T ;V). Fo he ilinea e ms, we p oceed as ollows. By Lemma 6.1, we ha e ha uh→uin L2(0, T ;Hβ(Ω)) −s ong o all β < 1, since {uh}h>0is bounded in L2(0, T ;H1 0(Ω))∩Hβ((0, T ); ˜ H−α 0(Ω)) o α∈[1 4,1 2) and 0 < β < 2 5(1+α) om (43) and (53). The e o e, N(uh,uh)→ N(u,u) in D′((0, T )×Ω). As a consequence o (46), we ob ain N(uh,uh)→ N(u,u) in H− (0, T ;˜ H−s 0(Ω)). On passing o he limi in (17b), we ha e had ha ∇ · u= 0 in (0, T )×Ω, he eby N(uh,uh)→(u· ∇)uin H− (0, T ;˜ H−s 0(Ω)). By an analogous a gumen , we ind ha ˜ N(uh,˜ uh)→0in H− (0, T ;˜ H−s 0(Ω)), whe e h˜ N(uh,˜ wh), hi=b(uh, h,˜ wh) o all uh,wh∈Whand ˜ h∈˜ Wh. F om he abo e con e gences, i is easy o see ha ∂ uh+N(uh,uh)−ν∆huh+∇ph−˜ N(uh,˜ uh)− h→∂ u+ (uh· ∇)uh−ν∆u+∇p− . in H− (0, T ;˜ H−s 0(Ω)) as h→0. CONVERGENCE TO SUITABLE WEAK SOLUTIONS FOR A SUBGRID FEM MODEL 19 Fo he ini ial condi ion, we ha e ha uh→uin C0([0, T ]; ˜ H−α 0(Ω))-s ong o α∈(1 4,1 2) by Lemma 6.2; he e o e, uh(0) →u(0) in ˜ H−α 0(Ω). Fu he mo e, i ollows om (18) and (27) ha u0h→u0in ˜ H−α 0(Ω). We ha e hus shown ha u(0) = u0. The ene gy inequali y can be e i ied by he lowe semicon inui y o he no m o he weak opology; o comple e de ails, see [4].  Theo em 6.5. Unde hypo heses (H1)-(H4), he sequence o app oxima e solu ions (uh, ph)con e ges, up o a subsequence, o a sui able weak solu ion gi en in De ini ion 2.4 as h→0. P oo . Le φ∈ D((0, T )×Ω; R+) and subs i u e h=πWh(uhφ) in o (17a) o ge ZT 0 {(∂ uh, πWh(uhφ)) + b(uh,uh, πWh(uhφ)) + ν(∇uh,∇πWh(uhφ)) −(ph,∇ · πWh(uhφ)) −b(uh, πWh(uhφ),˜ uh)−( h, πWh(uhφ))}d = 0. (60) We a e eady o ake he limi in (60) as h→0 so as o p o e ha he weak solu ion (u, p) ound in Theo em 6.4 is sui able. We will only ocus on passing o he limi in he e ms o (60) in ol ing he subscale eloci y ˜ uhand he p essu e e m. The emaining e ms appea in a udimen a y ini e elemen o mula ion so ha a p oo can be ound in [25]. In pa icula , om (13) and (9) and in i ue o Lemma 6.3, i ollows ha lim h→0ZT 0 (∂ uh,uhφ) d =−1 2ZT 0 (|u|2, ∂ φ) d , lim h→0ZT 0 b(uh,uh, πWh(uhφ)) d =−1 2ZT 0 (|u|2u,∇φ)d , lim in h→0ZT 0 ν(∇uh,∇πWh(uhφ)) d ≥ZT 0 (|∇u|2, φ) d −ZT 0 (1 2|u|2,∆φ) d , and lim h→0−ZT 0 h h,uhφid =−ZT 0 h ,uφid . To begin wi h, we i s u n ou a en ion o passing o he limi in he con ec i e e m. b(uh, πWh(uhφ),˜ uh) d =b(uh,uhφ, ˜ uh) d +b(uh, πWh(uhφ)−uhφ, ˜ uh) d =b(uh,uh,˜ uhφ) + (uh· ∇φuh,˜ uh) + b(uh, π⊥ Wh(uhφ),˜ uh) = (π⊥ Wh(N(uh,uh)φ),˜ uh) + (π⊥ Wh(uh· ∇φuh),˜ uh) +b(uh, π⊥ Wh(uhφ),˜ uh). (61) F om (13) and (6), we ha e: ZT 0 (π⊥ Wh(N(uh,uh)φ),˜ uh) d ≤ZT 0 kπ⊥ Wh(N(uh,uh)φ),˜ uh)kk˜ uhkd ≤CZT 0 hkuhkL∞(Ω)k∇uhkk˜ uhkd ≤C ZT 0 τh2kuhk2 L∞(Ω)k∇uhk2d !1 2 ZT 0 τ−1k˜ uhk2d !1 2 ≤Ch3 4kuhk1 2 L∞(0,T ;L2(Ω))kuhkL2(0,T ;H1 0(Ω))k˜ uhkτ−1 2L2(0,T ;L2(Ω)) and hence lim h→0ZT 0 (π⊥ Wh(N(uh,uh)φ),˜ uh) d = 0. Analogously, we bound ZT 0 (uh· ∇φ, uh·˜ uh) d ≤CT h3 4kuhkL∞(0,T ;L2(Ω))k˜ uhkτ−1 2L(0,T ;L2(Ω)) 20 S. BADIA AND J. V. GUTI´ ERREZ-SANTACREU and ZT 0 b(uh, πWh(uhφ)−uhφ, ˜ uh) d ≤Ch3 4kuhk1 2 L∞(0,T ;L2(Ω))kuhkL2(0,T ;H1 0(Ω))k˜ uhkτ−1 2L2(0,T ;L2(Ω)). Thus lim h→0ZT 0 (uh· ∇φ, uh·˜ uh) d = 0, and lim h→0ZT 0 b(uh, πWh(uhφ)−uhφ, ˜ uh)d = 0. Fo he “ iscous” e m, i is no ha d o see ha lim in h→0ZT 0 τ−1(|˜ uh|2, φ) d ≥0. Fo he p essu e e ms, we w i e ZT 0 (ph,∇·πWh(uhφ)) d =ZT 0 (phuh,∇φ)d +ZT 0 (ph,∇·(πWh(uhφ)−(uhφ))) d +ZT 0 (φph,∇·uh) d . I was p o ed in [23] ha lim h→0ZT 0 (phuh,∇φ) d =ZT 0 (pu, ∇φ) d and lim h→0=ZT 0 (ph,∇ · (πWh(uhφ)−(uhφ))) d = 0. Fo he emaining p essu e e ms, we use (17b) wi h qh=πQh(φph) o ob ain ZT 0 (φph,∇ · uh) d +ZT 0 (∇ph,˜ uhφ) = ZT 0 (phφ−πQh(phφ),∇ · uh) d +ZT 0 (∇(phφ)− ∇πQh(phφ),˜ uh) d −ZT 0 (ph∇φ, ˜ uh) d . We know om [23] ha lim h→0ZT 0 (phφ−πQh(phφ),∇ · uh) d = 0. Le ε > 0 and se s=1 2+16 9εand ¯ =3 4−s 2=1 4−4 9ε. Now choose =1 2−4 9ε. Mo eo e , se α=1 4−5 9εand ¯ β=2 5(1 + α) = 1 5−2 9ε. Thus we ha e 1 −s > α and ¯ β > since 1−s=1 2−16 9ε > 1 2(1 2−16 9ε) = 1 4−4 9ε > 1 4−5 9ε=α and =1 2−4 9ε < 1 2−2 9ε=2 5(5 4−5 9ε) = 2 5(1 + 1 4−5 9ε) = 2 5(1 + α) = ¯ β. F om he a p io i ene gy es ima es (52) and (55) and he commu a o p ope y (14), ou choice o pa ame e s yields ZT 0 (∇(pφ)− ∇πQh(phφ),˜ uh) d ≤ k∇(pφ)− ∇πQh(phφ)kH− (0,T ;L2(Ω))k˜ uhkH (0,T ;L2(Ω)) ≤Ch1−s−αkphkH− (0,T ;H1−s(Ω))k˜ uhkhαH (0,T ;L2(Ω)) and hence lim h→0ZT 0 (∇(φ)− ∇πQh(phφ),˜ uh) d = 0. CONVERGENCE TO SUITABLE WEAK SOLUTIONS FOR A SUBGRID FEM MODEL 21 Finally, i is easy o see in a simila ashion ha lim h→0ZT 0 (φph,∇ · uh) d = lim h→ZT 0 (φph−πWh(φph),∇ · uh) d = 0.  Appendix A. P oo o he in e se inequali ies (8) To p o e inequali ies (8), we ollow e y closely he a gumen s de eloped in [8, Thm. 4.5.11]. We i s need o in oduce an equi alen no m o ac ional o de Hilbe spaces as ollows. Le s∈(0,1). Then kuk2 Hs(Ω) =kuk2+|u|2 Hs(Ω), whe e |u|2 Hs(Ω) =ZΩZΩ |u(x)−u(y)|2 |x−y|3+2sdxdy. Gi en (K, P,Σ), we de ine ( ˜ K, ˜ P, ˜ Σ) whe e ˆ K={(1/hK)x:x∈K}. Thus, i uhis a unc ion de ined on K, hen ˆuhis de ined on ˜ Kby ˆu(ˆ x) = u(h−1 Kx) o all ˆ x∈ˆ K. Thus we can w i e k∇uhkL2(K)=h1 2 Kkˆ ∇ˆuhkL2(K). As ˆ ∇ˆuhbelongs o a space o ini e and ixed dimension on ˆ K, on which all no ms a e equi alen , i is no ha d o see ha he e is a cons an Cˆ T>0 such ha kˆ ∇ˆuhkL2(K)≤Cˆ T|ˆuh|Hs(ˆ K). Re e ing o K, his leads o kˆ ∇ˆukL2≤Cˆ Th−3 2+s K|uh|Hs(K) and hence k∇uhkL2(K)≤Cˆ Th−1+s KkuhkHs(Ω). An a gumen in he p oo o [8, P op. 4.4.11 ] shows ha i ( ˜ K, ˜ P,˜ Σ) is a e e en elemen , we ha e ha he e exis s a cons an C˜ T>0 such ha Cˆ T≤C˜ T. Summing o e all elemen s Kand using he quasi-uni o mi y o he mesh leads o k∇uhk ≤ Ch−1+s X K∈Th kuhk2 Hs(Ω)!. Then (8) ollows because he sum o he ac ional no ms o e all elemen s is smalle han he ac ional no m o e he union o he elemen s. 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