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.
Acknowledgmen
The au ho s a e e y g a e ul o P o esso Vi e e Gi aul who p o ided a p oo o a pa icula case
o inequali y (8).

22 S. BADIA AND J. V. GUTI´
ERREZ-SANTACREU
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