Ma hema ical Modelling and Nume ical Analysis ESAIM: M2AN
Mod´elisa ion Ma h´ema ique e Analyse Num´e ique Vol. 35, No1, 2001, pp. 57–89
AN ANALYSIS TECHNIQUE FOR STABILIZED FINITE ELEMENT SOLUTION
OF INCOMPRESSIBLE FLOWS ∗
Tom´
as Chac´
on Rebollo1
Abs ac . This pape p esen s an ex ension o s abilized me hods o he s anda d echnique o he
nume ical analysis o mixed me hods. We p o e ha he s abili y o s abilized me hods ollows om an
unde lying disc e e in -sup condi ion, plus a uni o m sepa a ion p ope y be ween bubble and eloci y
ini e elemen spaces. We apply he echnique in oduced o p o e he s abili y o s abilized spec al
elemen me hods so as s abilized solu ion o he p imi i e equa ions o he ocean.
Ma hema ics Subjec Classi ica ion. 65N30, 76M10.
Recei ed: No embe 24, 1999. Re ised: Oc obe 19, 2000.
1. In oduc ion and mo i a ion
This pape deals wi h he nume ical analysis o he solu ion o incomp essible low p oblems by s abilized
ini e elemen s. We shall be in e es ed in he Oseen equa ions (S okes equa ions plus a linea anspo e m),
also called in some wo ks “linea ized Na ie -S okes equa ions”.
S abilized me hods p o ide e icien and compu a ionally cheap echniques o sol e incomp essible luids.
His o ically, hese me hods ha e been he objec o a speci ic analysis, di e en om ha o mixed me hods.
Indeed, he p oo o s abili y is no based upon he exis ence o a disc e e eloci y – p essu e in -sup condi ion,
bu a he upon speci ic a gumen s ha s ongly ely on he elemen wise egula i y o ini e elemen unc ions.
Based upon such kind o a gumen s, he pape s o Hugues, F anca and Bales a [21] and Hughes and F anca [20]
con ained an e o analysis ha was imp o ed in B ezzi and Douglas [8] and in Pie e [25]. In F anca and
S enbe g [15] a gene al s abili y and e o analysis echnique was in oduced, which was summa ized in F anca,
Hugues and S enbe g [16]. Also, he pape o Tobiska and Ve ¨u h [27] de elops an analysis o s abili y and
con e gence o he solu ion o Na ie -S okes equa ions by s abilized me hods.
Ano he way o analysis is sugges ed by he ela ionship be ween s abilized and mixed me hods. In F anca
and F ey [14] i is p o ed ha he S eamline Upwind/Pe o -Gale kin (SUPG) me hod is equi alen o he
s anda d mixed me hod cons uc ed wi h he mini-elemen . This equi alence is unde s ood in he sense ha
bo h me hods yield he same o mula ion i he deg ees o eedom associa ed o he bubbles a e elimina ed
by s a ic condensa ion. This equi alence yields he s abili y o SUPG me hod om ha o he mixed me hod
Keywo ds and ph ases. Oseen equa ions, ini e elemen s, mixed me hods, s abilized me hods, disc e e in -sup condi ion, spec al
me hods, p imi i e equa ions.
∗This esea ch was pa ially suppo ed by Spanish M. E. C. P ojec MAR97-1055-C02-02 and by EU HCM ERBCHB ICT
94.1823 G an .
1Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa. C/ Ta ia, s/n. 41080 Se illa, Spain.
e-mail: [email p o ec ed]
c
EDP Sciences, SMAI 2001
58 T. CHAC ´
ON REBOLLO
cons uc ed wi h he mini-elemen . I is a di ec consequence o he ac ha his elemen sa is ies he disc e e
in -sup condi ion. Such analysis is essen ially pe o med in Chac´on Rebollo [10].
We add ess in his pape he ques ion o whe he his way o analysis may be applied o s abilized me hods
o he han SUPG. We de elop a echnique o he nume ical analysis o s abilized me hods ha gi es a posi i e
answe o ha ques ion. Conc e ely, we p o e he exis ence o an unde lying disc e e in -sup condi ion om
which we deduce he s abili y o s abilized me hods. Once his poin has been se up, ou echnique allows o
analyze s abilized me hods as i hey whe e mixed me hods (Th. 1). They appea as in e nal app oxima ions o
a weak o mula ion, whose s abili y elies on an in -sup condi ion. Then, ou analysis may be applied o mo e
complex si ua ions, whe e we use he ools p o ided by unc ional analysis o ob ain gains wi h espec o he
s anda d analysis. We include in his pape wo o such applica ions:
•To p o e he s abili y o a spec al elemen app oxima ion o he gene alized S okes equa ions, in oduced
in Ge asio and Sale i [19]. He e, we ob ain L2es ima es o he p essu e, while he s anda d analysis,
used in ha pape , allows only o es ima e a semino m o he p essu e g adien .
•To sol e a linea model o p imi i e equa ions o he ocean by s abilized ini e elemen s. Fo such equa ions,
he e is some lack o egula i y o he con ec ion e m, so ha he p essu e has only Lp egula i y, o
some p∈(1,2). In his case, we ob ain Lpes ima es o he disc e e p essu e, and p o e con e gence in
H1×Lpno m o he con inuous solu ion. The s anda d analysis in his con ex would be qui e di icul
o be ca ied on.
Ou analysis may also be applied o nonlinea lows. Fo ins ance, in Chac´on Rebollo and Dom´ınguez Del-
gado [11], i is applied o he analysis o he app oxima ion o Na ie -S okes equa ions by s abilized me hods,
in pa allel o he analysis o hei app oxima ions by mixed me hods. S abili y and e o es ima es a e de i ed.
This analysis also applies o nonlinea s abilized me hods, such as he op imal one in oduced in Russo [26].
Up o ou knowledge, he s anda d analysis is unable o handle nonlinea s abiliza ion, which u ns ou o be
a he simple o manage wi h ou echnique.
Ano he possible applica ion is he analysis o he solu ion o Oseen equa ions by he educed Q1/Q1s abilized
me hods in oduced in Knobloch and Tobiska [22]. This is a new amily o compu a ionally cheap me hods ha
may be di ec ly analyzed wi h ou analysis. In ac , all hypo hesis o Theo em 1 a e eadily p o ed o be
sa is ied, using he analysis de elopped in ha pape .
We would like o poin ou ha he analysis echnique ha we in oduce is a he complex om a echnical
poin o iew. Howe e , we hink ha i is wo h o be used, as i essen ially educes he di icul ies o he
analysis o s abilized me hods o ha o mixed me hod. Mo eo e , we ha e ied o p esen he echnique in a
sys ema ic way, so ha i may be applied o si ua ions o he han he conside ed he e, wi h ela i e ease.
The pape is o ganized as ollows. In Sec ion 2 we in oduce an abs ac disc e iza ion o Oseen equa ions,
whose s abili y is analyzed in Sec ion 3. In Sec ion 4, we apply he abs ac heo y o s abilized me hods.
Sec ion 5 is de o ed o he analysis o spec al elemen s abilized me hods. Finally, in Sec ion 6 we sol e a
linea model o p imi i e equa ions o he ocean by s abilized ini e elemen s.
2. Abs ac disc e iza ion
In his Sec ion we in oduce an abs ac disc e iza ion o Oseen equa ions which is he base o ou analysis.
Le us conside a connec ed bounded domain Ω ⊂Rd(d= 2 o 3), wi h Lipschi z-con inuous bounda y Γ.
We a e gi en a “d i ing” eloci y ield u:Ω−→ Rd, ha we assume o be di e gence- ee. Ou pu pose is o
sol e nume ically he ollowing bounda y alue p oblem:



Find y:Ω−→ Rd,p:Ω−→ Rsuch ha
u·∇y−ν∆y+∇p= ,∇·y=0 inΩ,
y=0 onΓ.
(1)
UNIFIED MIXED AND STABILIZED SOLUTIONS 59
He e, ν>0 is he iscosi y coe icien , and ∈H−1(Ω)dis a gi en sou ce e m. Only homogeneous Di ichle
bounda y condi ions a e conside ed, in o de o no in oduce nonessen ial di icul ies in ou de i a ion.
Le us de ine he bilinea o m on H1
0(Ω)d×H1
0(Ω)d,
a(w, )=(u·∇w, )+ν(∇w,∇ ),∀w, ∈H1
0(Ω)d,(2)
whe e we deno e by (·,·) heL2scala p oduc , ei he o scala , ec o o enso unc ions. I we assume ha
u∈[Lp(Ω)]d o some p>d,and∇·u=0, hena(·,·) is well de ined and is con inuous and H1
0(Ω)d-ellip ic;
i.e., i e i ies
a(w, )≤M(u)|w|1| |1,a( , )≥ν| |2
1∀ ,w∈H1
0(Ω)d.(3)
He e, we ha e deno ed by |·|
1 he H1(Ω)dsemino m. Also, M(u)=C(kuk0,p +ν) o some cons an C
appea ing om Sobole injec ions, whe e k·k0,p deno es he Lpno m.
The o m a(·,·) de ines a linea bounded ope a o A om H1
0(Ω)din o H−1(Ω)d,gi enby
hAw, i=a(w, ),∀w, ∈H1
0(Ω)d.
Thus, Aw=u·∇w−ν∆w.
The s anda d mixed o mula ion o p oblem (1) eads as ollows:
(Ob ain (y,p)∈H1
0(Ω)d×L2
0(Ω) such ha
B(y,p; ,q)=h , i,∀( ,q)∈H1
0(Ω)d×L2
0(Ω); (4)
whe e
B(y,p; ,q)=a(y, )−(p, ∇· )−(∇·y,q).
Also, h·,·i s ands o he H−1(Ω)d−H1
0(Ω)dduali y, and L2
0(Ω) is he subspace o L2(Ω) gi en by
L2
0(Ω) = {q∈L2(Ω) such ha ZΩ
qdx=0}·
The pai o spaces (H1
0(Ω)d,L
2
0(Ω)) e i ies he con inuous in -sup condi ion (c . Gi aul and Ra ia [17]).
Then, due o p ope ies (3), p oblem (4) has a unique solu ion ha depends con inuously on he da a .
In o de o desc ibe ou abs ac disc e iza ion o p oblem (4) we shall conside wo amilies o subspaces
{Yh}h>0and {Zh}h>0o H1
0(Ω)dand ano he amily o subspaces {Mh}h>0o L2
0(Ω), all o hem o ini e
dimension. These spaces may be, o ins ance, s anda d ini e elemen spaces. We shall also conside a amily
o bilinea con inuous o ms on H1
0(Ω)d×H1
0(Ω)d,{Sh(·,·)}h>0. These o ms a e assumed o be coe ci e in
H1no m on Zh.
We shall deno e by Rh he “s a ic condensa ion” ope a o
Rh:H−1(Ω)d→Zh,
de ined as ollows. Gi en ϕ∈H−1(Ω)d,Rh(ϕ) is he only elemen o Zh ha sa is ies
Sh(Rh(ϕ),zh)=hϕ, zhi,∀zh∈Zh.(5)
60 T. CHAC ´
ON REBOLLO
We disc e ize p oblem (4) by
Ob ain (yh,p
h)∈Yh×Mhsuch ha
Bh(yh,p
h; h,q
h)=Fh( h,q
h),∀( h,q
h)∈Yh×Mh;(6)
whe e
Bh(w, ; ,q)=B(w, ; ,q)−Sh(Rh(B +∇q),Rh(Aw+∇ )) ;
Fh( ,q)=h , i−Sh(Rh(B +∇q),Rh( )) ;
whe e Bdeno es he ope a o
Bw=−u·∇w+εν∆w,∀w∈H1
0(Ω)d,
o agi enε∈R.
We shall use me hod (6) as an abs ac amewo k o analyze a ious s anda d s abilized me hods. To
desc ibe hese me hods, we shall conside a ine-equi alen ini e elemen spaces, as desc ibed in Hughes, F anca
and Bales a [21]. Assume ha he domain Ω is polyhed ic. Le us conside a iangula ion Tho Ω o med by
ei he simplicial o pa allelepipedic elemen s. We assume ha he elemen s o Tha e a ine- ans o med o a
e e ence elemen K∗(ei he he uni simplex o pa allelepiped), in he sense o Cia le [13]. Gi en an in ege
numbe k≥0, and an elemen K∈T
h, deno e by Pk(K) he space o polynomials o deg ee smalle han, o
equal o, k, de ined on K. Also, deno e by Qk(K) he space o polynomials o deg ee smalle han, o equal o,
k, in each a iable, de ined on K. Deno e by Rk(K)ei he Pk(K), i Kis a iangle o e ahed on, o Qk(K)
i Kis a quad ila e al o hexaed on. Gi en wo in ege numbe s m≥1, l≥0, conside he ollowing ini e
elemen spaces.
Y(m)
h=n ∈H1
0(Ω)d| |K∈[Rm(K)]d,∀K∈T
ho;(7)
M(l)
h=q∈L2
0(Ω) |q|K∈Rl(K),∀K∈T
h,(8)
o
M(l)
h=q∈L2
0(Ω) ∩C0(Ω) |q|K∈Rl(K),∀K∈T
h.(9)
We conside he ollowing s abilized me hods.
(Find (yh,p
h)∈Y(m)
h×M(l)
hsuch ha
BS(yh,p
h; h,q
h)=FS( h,q
h),∀( h,q
h)∈Y(m)
h×M(l)
h;(10)
whe e
BS(w, ; ,q)=B(w, ; ,q)−X
K∈Th
τK(B +∇q;Aw+∇ )K;
FS( ,q)=h , i− X
K∈Th
τK(B +∇q, )K,
whe e he τKa e gi en s abilizing coe icien s, and (·,·)Kdeno es he inne p oduc in L2(K)d.When
l=m= 1, me hod (10) is independen o he ac ual alue o he coe icien ε, and i is known as S eamline
UNIFIED MIXED AND STABILIZED SOLUTIONS 61
Upwind/Pe o -Gale kin (SUPG) me hod. Fo o he alues o m≥1andl≥0, when ε=−1,0and1,
me hod (10) is espec i ely known as Adjoin s abilized (AdS), gene alized SUPG and Gale kin-Leas Squa es
(GaLS) me hod.
Typically, he coe icien s τKa e con inuous unc ions o he local P´ecle numbe on elemen K,
PeK=UKhK
νwi h UK=ZK|u|p1/p
;
τK(PeK)=AhK
UK
min(PeK,P)=






Ah2
K
νi PeK≤P,
AP hK
UK
i PeK>P;
(11)
whe e Ais a nume ical cons an and Pis a p ese h eshold o he P´ecle numbe . This allows on one hand
o in oduce some sui able s abiliza ion o high equence componen s o he anspo ope a o (o o de hK),
due o con ec ion dominance (La ge PeK). Also, his in oduces low le els o nume ical di usion (o o de h2
K)
in egions whe e di usion is dominan (Low PeK). On he o he hand, his s abilizes he spu ious modes o he
p essu e g adien .
Also, o easons o compu abili y, in p ac ice he con ec ion eloci y uis eplaced in he s abilizing e ms
by some s able in e pola e uh∈Y(m)
h. We shall assume i so in ou analysis.
The s anda d analysis o s abilized me hods, summa ized in F anca, Hughes and S enbe g [16], s a es ha
SUPG and GaLS me hods a e s able o any posi i e coe icien s τK, and ha AdS and gene alized SUPG
me hods a e s able i he τKa e small enough. The ob en ion o op imal bounds o hese coe icien s o ensu e
s abili y equi es he compu a ion o he bes cons an CIin he in e se inequali y
CIX
K∈Th
h2
Kk∆ hk2
K≤k∇ hk2
0,∀ h∈Y(m)
h.(12)
Tha analysis applies o ei he con inuous p essu es combined wi h eloci ies o a bi a y in e pola ion deg ee,
o o discon inuous p essu es combined wi h high-deg ee in e pola ion eloci ies. Conc e ely, i holds unde he
ollowing condi ion:
Ei he M(l)
h⊂C0(¯
Ω),o m≥n, (13)
whe e
n=di This o med by iangles o e ahed a, and
2i This o med by quad ila e als o hexaed a.
In Tobiska and Ve ¨u h [27] his es ic ion is emo ed by in oducing in he s uc u e o he me hod some
addi ional e ms ha ake in o accoun in e elemen p essu e jump e ms. Howe e , i seems ha me hod (10),
wi hou hese jump e ms, is no able o s abilize he disc e iza ion o discon inuous p essu es combined wi h
low-deg ee eloci ies.
In his pape we shall analyze me hods sa is ying condi ion (13). Ou analysis also applies o gene al dis-
c e iza ions ha do no necessa ily sa is y his condi ion. Howe e , i s p oo equi es a a he leng hy de i a ion
ha shall appea in a o hcoming pape .
No ice ha me hod (6) applies o gene al in e nal app oxima ions o H1
0(Ω)dand L2
0(Ω), while s abilized
me hods only apply o app oxima ions by piecewise smoo h unc ions. We a e, hus, conside ing a genuine
gene aliza ion o s abilized me hods.
In he nex wo Sec ions we i s de elop a s abili y and con e gence analysis o he abs ac me hod (6)
which ex ends he s anda d analysis o mixed me hods, and nex apply i o analyze he s abilized me hods (10).

62 T. CHAC ´
ON REBOLLO
3. Analysis o abs ac me hod
In his sec ion we p o e ha he s abili y o he abs ac me hod (6) ollows om a disc e e in -sup B ezzi-
Babuˇska condi ion, simila ly o mixed me hods.
The s abili y o abs ac me hod (6), in addi ion o he in -sup condi ion, equi es he ollowing hypo heses
on he new elemen s appea ing in me hod (6):
Hypo hesis 1. The e exis s a cons an C0>0 independen o hsuch ha
|yh|1+|zh|1≤C0|yh+zh|1,∀yh∈Yh,zh∈Zh,∀h>0.(14)
Hypo hesis 2. The e exis wo cons an s νs>0,M
s>0 such ha
|Sh(wh, h)|≤Ms|wh|1| h|1,Sh( h, h)≥νs| h|2
1,∀wh, h∈Zh.
Bo h hypo heses play a c ucial ole in he ob en ion o es ima es o bo h eloci y and p essu e, and hus in he
p oo o s abili y o me hod (6). Hypo hesis 1 is a gene aliza ion o he well known H1
0-o hogonali y be ween
piecewise a ine and bubble ini e elemen s. Hypo hesis 2 is a gene aliza ion o he ac ha he s abilizing
coe icien s in (11) a e o o de h2
K.
Le us ecall he de ini ion o s abili y o me hod (6) (c . Babuˇska [3], B ezzi [7]):
De ini ion 1. Me hod (6) is said o be s able on Yh×Mhi he e is a cons an γ>0 independen o hsuch
ha o any (w, )∈Yh×Mh,
sup
( ,q)∈Yh×Mh
( ,q)6=(0,0)
Bh(w, ; ,q)
| |1+kqk0≥γ(|w|1+k k0);
sup
( ,q)∈Yh×Mh
( ,q)6=(0,0)
Bh( ,q;w, )
| |1+kqk0≥γ(|w|1+k k0).

We now s a e ou basic s abili y esul .
Theo em 1. Assume ha he pai s o spaces {(Yh+Zh,M
h)}h>0sa is y a uni o m disc e e B ezzi-Babuˇska
condi ion, and ha Hypo heses 1 and 2 hold. Assume ha a leas one o he wo ollowing sen ences hold:
i) Zhand Yha e o hogonal wi h espec o he H1
0(Ω)dinne p oduc and νs>0,o
ii) νs≥1−ε
22
ν,whenε6=1,o νs>0when ε=1.
Then, he abs ac me hod (6) is s able.
F om his heo em we deduce he main esul o his pape :
Theo em 2. Assume ha he amily o iangula ions {Th}h>0is egula . Assume ha condi ion (13) holds.
Then, he s abilized me hod (10) coincides wi h an abs ac me hod (6) cons uc ed wi h a ini e elemen space
Zho bubble unc ions and a bilinea o m Sh, e i ying
1. The pai s o spaces {Yh+Zh,M
h}h>0sa is y a uni o m disc e e in -sup condi ion.
2. The pai s o spaces {Yh,Z
h}h>0sa is y Hypo hesis 1.
3. The o ms {Sh}h>0sa is y Hypo hesis 2.
UNIFIED MIXED AND STABILIZED SOLUTIONS 63
As a consequence,
•GaLS and SUPG me hods a e s able o any A>0in (11)
•The gene al s abilized me hod (10) is s able i A≤A02
ε−12
,whe eA0is a compu able posi i e
cons an . In pa icula , AdS me hod is s able i A≤A0, and gene alized SUPG me hod is s able i
A≤4A0.
Thus, unde ou analysis, he s abili y o s abilized me hods ollows om a disc e e in -sup condi ion, simila ly
o mixed me hods. We shall p o e his esul in Sec ion 4. In addi ion, we shall p o e ha he cons an A0
depends on he aspec a io o he g id and on he e e ence elemen s o spaces Yhand Mh,andshallgi e
compu able ine es ima es o his cons an .
P oo o Theo em 1.
Veloci y es ima e. We shall ea sepa a ely cases i)andii).
i) Assume ha spaces Zhand Yha e o hogonal wi h espec o he H1
0(Ω)dinne p oduc . In his case, all
me hods (10) coincide, independen ly o he ac ual alue o ε, as such o hogonali y implies Rh(∆w)=0,∀w∈
Yh.
Conside a pai (wh,
h)∈Yh×Mh. De ine ch=Rh(Awh+∇ h).As Rh(∆ h)=0, hen
ch=Rh(−Bwh+∇ h). Consequen ly,
Bh(wh,
h;wh,− h)=a(wh,wh)+Sh(ch,ch)≥ν|wh|2
1+νs|ch|2
1.
ii) Conside a pai (wh,
h)∈Yh×Mh. De ine ch=Rh(Awh+∇ h). Then,
B(wh,
h;wh,− h)=a(wh,wh)+Sh(ch,ch)+(1−ε)νSh(Rh(∆wh),ch) (15)
=a(wh,wh)+Sh(ch,ch)−(1 −ε)ν(∇wh,∇ch)
Due o Hypo hesis 1,
|(∇yh,∇zh)|≤(1 −δ0)|yh|1|zh|1,∀yh∈Yh,zh∈Zh,whe e δ0=2
C2
0·(16)
Then, using Young’s inequali y, (15) implies
B(wh,
h;wh,− h)≥˜ν|wh|2
1+˜νs|ch|2
1,(17)
whe e
˜ν=ν[1 −(1 −δ0)|1−ε|
2µ],˜νs=νs−ν(1 −δ0)|1−ε|
2µ−1
o any µ>0. When ε=1,˜ν=νand ˜νs=νs>0. When ε6=1,wemaychoose
ν
νs
|1−ε|
2(1 −δ0)<µ< 2
|1−ε|(1 −δ0)−1,
and hen ˜ν>0, ˜νs>0.
Deno e
S=sup
( ,q)∈Yh×Mh
( ,q)6=(0,0)
Bh(wh,
h; ,q)
| |1+kqk0·
Then, in all cases
˜ν|wh|2
1+˜νs|ch|2
1≤(|wh|1+k hk0)S, (18)
64 T. CHAC ´
ON REBOLLO
whe e o case i) we de ine ˜ν=νand ˜νs=ν.
P essu e es ima e. Conside a nonze o elemen h∈Yh.Weha e
( h,∇· h)=−Bh(wh,
h; h,0) + a(wh, h)−Sh(Rh(B h),ch).(19)
Rema k ha Bh(wh,
h;− h,0) ≤S| h|1. Obse e also ha
Sh(Rh(B h),ch)=hB h,chi=−(u·∇ h,ch)−εν(∇ h,∇ch)
≤[M(u)+|ε−1|ν]| h|1|ch|1.
Consequen ly,
( h,∇· h)≤{S+M(u)|wh|1+[M(u)+|ε−1|ν]|ch|1}| h|1≤
≤C1(S+|wh|1+|ch|1)| h|1,
whe e C1=max{1,M(u)+|ε−1|ν}.
Also, gi en a nonze o elemen zh∈Zh,
( h,∇·zh)=−h∇ h,zhi=−Sh(Rh(∇ h),zh)
=Sh(Rh(Awh),zh)−Sh(ch,zh) (20)
≤M
s[|Rh(Awh)|1+|ch|1]|zh|1
≤M
sν−1
s|Awh|−1+|ch|1|zh|1
≤C2(|wh|1+|ch|1)|zh|1,
whe e C2=Msmax{ν−1
sM(u),1}. Then, using Hypo hesis 1,
( h,∇·(zh+ h)) ≤C3(S+|wh|1+|ch|1)(| h|1+|zh|1)
≤C0C3(S+|wh|1+|ch|1)| h+zh|1,(21)
whe e C3=max{C1,C
2}. Now, we use he disc e e in -sup condi ion: The e exis s a cons an α>0 such ha
αkqhk0≤sup
xh∈Yh+Zh
(qh,∇·xh)
|xh|1
,∀qh∈Mh.
The e o e,
k hk0≤C4(S+|wh|1+|ch|1),(22)
whe e C4=α−1C0C3.
Conclusion. Combining (18) and (22) and applying Young’s inequali y yields
˜ν|wh|2
1+˜νs|ch|2
1≤C4S2+[(1+C4)|wh|1+C4|ch|1]S
≤1
2[(1 + C4)ε1|wh|2
1+C4ε2|ch|2
1]
+[C4+1
2(1 + C4)ε−1
1+1
2C4ε−1
2]S2,
o any ε1>0, ε2>0. Le us ake ε1=˜ν
1+C4
,ε2=˜νs
C4
. Then,
˜ν|wh|2
1+˜νs|ch|2
1≤C2
5S2,(23)
UNIFIED MIXED AND STABILIZED SOLUTIONS 65
whe e C5=2C4+(1 + C4)2
˜ν+C2
4
˜νs1/2
.Thus,
|wh|1≤C5
√˜νS, |ch|1≤C5
√˜νs
S. (24)
Combining now (23) wi h (22), we ob ain
k hk0≤C6S, whe e C6=C4+C4C51
√˜ν+1
√˜νs.(25)
F om (24) and (25) we inally deduce
S≥γ(|wh|1+k hk0+|ch|1),whe e γ=C6+C5
√˜ν+C5
√˜νs−1
·(26)
The p oo o he second inequali y in De ini ion 1 ollows om simila a gumen s. 
The ollowing esul closes he equi alence be ween disc e e in -sup condi ion and s abili y o me hod (6).
Thus, he s abili y analysis o mixed me hod and me hod (6) a e ully pa allel.
Theo em 3. Assume ha abs ac me hod (6) is s able o some νs>0. Assume ha Hypo hesis 1 and 2 hold.
Then, he pai s o spaces {Yh+Zh,M
h}h>0sa is y he disc e e B ezzi-Babuˇska condi ion.
We omi he p oo o his esul as i again ollows om a gumen s simila o hose used in he p oo o
Theo em 1.
The s abili y o o m Bhyields he well-possedness o ou me hod, and allows o de i e e o es ima es,
simila ly o he s anda d analysis o mixed me hods:
Co olla y 1. Unde he hypo heses o Theo em 1, p oblem (6) admi s a unique solu ion (yh,p
h)∈Yh×Mh,
ha e i ies, o some cons an C>0,
|yh|1+kphk0+|zh|1≤Ck k−1,(27)
and
|y−yh|1+kp−phk0+|zh|1≤Cin
h∈Yh|y− h|1+in
qh∈Mhkp−qhk0,(28)
whe e zh=Rh(Ayh+∇ph− ).
Rema k 1. F om his esul , he “bubble” space Zhappea s as a con ol space o high- equency componen s
o he esidual Ayh+∇ph− . In ac , (28) shows ha he high equency componen s o he esidual which
a e ep esen able on Zh, ia he condensa ion ope a o Rh, a e bounded.
4. Applica ion o s abilized me hods
In his sec ion we p o e ha s abilized me hods (10) may be o mula ed as pa icula cases o abs ac
me hod (6), and hen apply he gene al s abili y analysis o Sec ion 3.
Ou de i a ion s a s om he cons uc ion o i ual bubbles de elopped in Baiocchi e al. [4]. Le us ecall
he main esul o ha pape , ha we adap o ou con ex . Conside a Hilbe space ( H,(·,·)H). Gi en a
subse Bo Ho ini e dimension, we de ine he abs ac s a ic condensa ion ope a o R:H0→Bby:
Gi en ϕ∈H0,R(ϕ) is he only elemen o B ha sa is ies
(R(ϕ),ζ)H=hϕ, ζi,∀ζ∈B.
72 T. CHAC ´
ON REBOLLO
whe e
BH(wH,
H; H,q
H)=B(wH,
H; H,q
H)−X
K∈Th
τK(B H+∇qH;AwH+∇ H)N,K;
FH( H,q
H)=h , Hi− X
K∈Th
τK(B H+∇qH, )N,K.
The essen ial di e ence be ween SSE me hod and s abilized me hod (10) is ha he L2inne p oduc s (·,·)K
ha appea in (10) in he s abilizing e ms a e he e eplaced by he disc e e inne p oduc s (·,·)N,K. In Ge asio
and Sale i [19], he disc e e inne p oduc (·,·)His also used o app oxima e he in eg al e ms appea ing in o m
B. He e, o simplici y we p e e o conside he abo e disc e iza ion. Howe e , we may ex end ou analysis o
he ac ual disc e iza ion conside ed in ha pape i he p essu es a e app oxima ed by piecewise polynomials
o deg ee a mos N−1(seeRem.5).
In Ge asio and Sale i [19], he s abilizing coe icien s τKa e s ill gi en by (11), wi h
P=2N2
m,A=m
4N4, o some m>0.
The pa ame e mis de e mined in ha pape in o de o ob ain uni o m-in- ime s abili y o he linea p oblems
ha a ise a e ime disc e iza ion. We shall simply assume ha he s abilizing coe icien s τKa e gi en by (11).
Ou analysis allows o s a e he ollowing esul :
Theo em 5. Assume he iangula ions {Th}h>0a e egula . Then, he SSE me hod (43) is s able o any
A>0i ε=1, and o 0<A<2
1−ε2
ˆ
A0i ε6=1,whe e ˆ
A0is a compu able posi i e cons an .
As a consequence, i ∈C0(Ω)d, p oblem (43) admi s a unique solu ion ha sa is ies
|yH|1+kpHk0≤Ck kC0,(44)
o some cons an C>0independen o H.
P oo . We p oceed as in he p oo o Theo em 2.
S ep 1: Embedding o SSE me hod in abs ac me hod.
Le us de ine he local in e pola ion ope a o IK
N:C0(K)→QN(K)by
(IK
Nw)(P(K)
ijk )=w(P(K)
ijk ),i,j,k=1,···,N +1.
Conside he space o piecewise con inuous unc ions on Th,
Cp,h(Ω) = { ∈L2(Ω) | |K∈C0(K),∀K∈T
h};
and de ine he global in e pola ion ope a o IH:Cp,h(Ω) →WHby
(IHw)|K=IK
N(w|K),∀K∈T
h.
Obse e ha C0(¯
Ω) ⊂Cp,h(Ω) and ha IHw∈VHi w∈C0(¯
Ω).

UNIFIED MIXED AND STABILIZED SOLUTIONS 73
The ollowing ep esen a ion o mula holds:
Lemma 4. The e exis s a ini e-dimensional bubble ini e elemen space
ZH⊂H1
0(Ω)dsuch ha ,
Sh(RH(IH 1),RH(IH 2)) = X
K∈Th
τK( 1, 2)N,K,∀ 1, 2∈[Cp,h(Ω)]d; (45)
whe e Shis he bilinea o m de ined by (32).
This lemma is p o ed in he Appendix.
As a consequence, o all wH, H∈YH; H,qH∈MH,
BH(wH,
H; H,q
H)=B(wH,
H; H,q
H) (46)
−Sh(RH(IH(B H+∇qH)),RH(IH(AwH+∇ H)));
FH( H,q
H)=h , Hi−Sh(RH(IH(B H+∇qH)),RH(IH )) .
This occu s because ∈C0(Ω)dand B H+∇qH,AwH+∇ H∈[Cp,h(Ω)]d.
S eps 2 and 3: P oo o Hypo heses 1 and 2.
Hypo heses 1 and 2 ha e espec i ely been p o ed in he S eps 3 and 4 o he p oo o Theo em 2.
S ep 4: Disc e e in -sup condi ion.
Obse e ha i H∈MH, henIH(∇ H)=∇ H, because IK
N(qN)=qN,∀qN∈QN(K). Then, by (45),
Sh(RH(∇ H),RH(∇ H)) = X
K∈Th
τK(∇ H,∇ H)N,K,∀ H∈MH.
By Be na di and Maday [5],
kqNk0,K ≤kqNkN,K ≤3kqNk0,K,∀qN∈QN(K).(47)
These es ima es a e ob ained by a ine anspo a ion o simila es ima es ob ained in he e e ence elemen .
As he coe icien s τKa e o o de h2
K, hen he e exis s a cons an C>0 such ha
X
K∈Th
h2
Kk∇ Hk2
0,K ≤CSh(RH(∇ H),RH(∇ H)) .
Then, by Lemma 3, he pai s o spaces {YH+ZH,M
H}H>0sa is y he disc e e in -sup condi ion.
S ep 5: Conclusion.
Following he p oo o Theo em 1, we p o e ha i νs≥(1 −ε)2
4νwhen ε6=1,o νs>0whenε=1, hen
he o m BHis s able. Then, p oblem (43) admi s a unique solu ion ha sa is ies, o some cons an C>0,
|yH|1+kpHk0+|zH|1≤C(k k−1+|RH(IH )|1),
whe e zH=RH(IH(AyH)+∇pH). As ∈C0(Ω)d, using (47),
|RH(IH )|1≤ν−1
skIH k0≤ν−1
skIH kH=ν−1
sk kH≤Ck kC0.
Thus, es ima e (44) ollows.
The emaining o he p oo is simila o he conclusion o he p oo o Theo em 2. 
74 T. CHAC ´
ON REBOLLO
Rema k 5. A sligh modi ica ion o he abo e a gumen allows o p o e an unde lying in -sup condi ion and
hus he s abili y o a s abilized ull spec al elemen disc e iza ion o Oseen equa ions.
Indeed, le us eplace he p essu e space MHby MH0, wi h H0=(h, (N−1)−1), o N≥2; i.e.,weconside
p essu es o deg ee a mos N−1 elemen wise. We conside he ollowing disc e e p oblem:
Ob ain (yH,p
H0)∈YH×MH0such ha
B0
H(yH,p
H0; H,q
H0)=F0
H( H,q
H0),∀( H,q
H0)∈YH×MH0;(48)
whe e
B0
H(wH,
H0; H,q
H0)=1
2[(u·∇wH, H)H−(u·∇ H,wH)H]+ν(∇wH,∇ H)H
−(∇·wH,q
H0)H−( H0,∇· H)H
−X
K∈Th
τK(B H+∇qH0;AwH+∇ H0)N,K;
F0
H( H,q
H0)=( , H)H−X
K∈Th
τK(B H+∇qH0, )N,K.
Then, ou analysis allows o p o e ha he o m B0
His s able. This holds because he quad a u e o mula
ZK∗
gdx∗≃
N
X
i,j,k=0
ωiωjωkg(ξi,ξ
j,ξ
k)
is exac o g∈Q2N−1(K∗).
6. Solu ion o linea p imi i e equa ions
In his sec ion we apply ou analysis o he solu ion o a linea model o he p imi i e equa ions o he ocean
by a penal y s abilized echnique. This model includes he main di icul y o hese equa ions: The e ical
con ec ion is degene a ed. This makes he p essu e o be only in some space Lp o 1 <p<2. We p o e
a disc e e in -sup condi ion in his no m, and p o e he con e gence o he app oxima ed solu ions o a weak
solu ion o he con inuous p oblem.
To desc ibe ou model equa ions, le us conside a connec ed 2D bounded domain ω⊂R2, and a piecewise
con inuous unc ion D:¯ω→Rsuch ha D(x)>0, ∀x=(x1,x
2)∈ω. This unc ion ep esen s he sea dep h.
We conside he domain
Ω={(x,z)∈R3|x∈ω, −D(x)<z<0},
which is in ended o ep esen a piece o he ocean wi h la su ace. To a oid some echnical complexi ies, we
shall assume ha ωis polygonal and Dis piecewise a ine on some iangula ion o ¯ω, so ha Ω is polyhed ic.
Ou analysis can be ex ended o piecewise C1dep h unc ions, simila ly o he analysis o he app oxima ion
o p imi i e equa ions by mixed me hods (c . Chac´on Rebollo and Guill´en Gonz´alez [12]).
We assume he domain Ω o be Lipschi z con inuous. This occu s, o ins ance, i he no mal de i a i e o
Dsa is ies ∂D
∂n ≤α o some α<0 a. e. on he pa o ∂ω whe e D=0. No ice ha Dmay be ze o pa ially
o o ally on ∂ω. Also, ha we allow he sea bo om o ha e e ical walls when Dhas a jump, and sidewalls
i D>0 on a pa o ∂ω.
We also conside he ollowing subse s o ∂Ω:
Γs={(x,0) ∈R3|x∈¯ω},(sea su ace),
Γb=∂Ω−Γs(sea bo om and, e en ually, sidewalls).
UNIFIED MIXED AND STABILIZED SOLUTIONS 75
We assume known a con ec ion eloci y W=(w1,w
2,w
3)onΩ,such ha
(w=(w1,w
2)∈H1(Ω)2,w
3∈L2(Ω),
∇·W=0 inΩ,w
3|Γs=0,w
3·n3|Γb=0,w|Γb=0,(49)
whe e n3deno es he hi d componen o he ou wa d no mal o ∂Ω, n=(n1,n
2,n
3). We a e hus o cing
he incomp essibili y o he sea wa e (Boussinesq’s hypo hesis). The i s bounda y condi ion means ha we
assume he sea su ace o no mo e in he e ical di ec ion ( igid lid hypo hesis), while he second and hi d
ones a e a he echnical bounda y condi ions, meaning ha we ea he whole Γbas a solid wall.
We also assume known a dis ibu ed sou ce e m , ep esen ing he e ec s o empe a u e, salini y and
Co iolis o ce (assumed o be cons an on he whole domain o simplici y), and a “su ace wind ension” g.We
se he ollowing p oblem:















Ob ain y:¯
Ω→R2,y=(y1,y
2),(ho izon al eloci y)
and p:ω→Rsuch ha (su ace p essu e)
W·∇y−ν∆y+∇Hp= in Ω,
∇H·hyi=0 inω,
y|Γb=0,ν
∂y
∂n|Γs
=g.
(50)
He e, ∇H=(∂1,∂
2) s ands o he ho izon al g adien , and he symbols h·i deno e e ical mean,
hyi(x)=Z0
−D(x)
y(x,z)dz, o x∈ω.
In his p oblem he su ace p essu e pac s as a Lag ange mul iplie associa ed o he condi ion ∇H·hyi=0.
P oblem (50) is a educed e sion o a linea model o he p imi i e equa ions o he ocean (in oduced in
Lions, Temam and Wang [24]), ha eads as ollows:























Ob ain a eloci y ield (y,y
3):¯
Ω→R3,
and a p essu e P:Ω→Rsuch ha
W·∇y−ν∆y+∇HP= in Ω,
∇·(y,y
3)=0 in Ω,
∂3P=−ρg in Ω,
y|Γb=0,ν
∂y
∂n|Γs
=g,
y3·n3|Γb=0,y
3|Γs=0.
(51)
He e, ρ ep esen s he sea wa e densi y, assumed o be cons an , and g he accele a ion o he g a i y.
This model is o mally ob ained om he Na ie -S okes equa ions by neglec ing in he e ical momen um
equa ion all o ces (con ec ion, di usion and Co iolis) bu he g a i y. This leads o he hyd os a ic p essu e
app oxima ion. A igo ous de i a ion o his app oxima ion is ound in Besson and Laydi [2], as an asymp o ic
limi as he a io be ween e ical and ho izon al dimensions ends o ze o. The physically meaning ul –
nonlinea – p oblem would be o ind a “ ixed poin ” o equa ions (50), in he sense ha y=w. This jus i ies
he choice o egula i y and bounda y condi ions sa is ied by w(see (49)).
Equa ions (50) may be iewed as a model p oblem o he nonlinea p imi i e equa ions, much as he Oseen
equa ions a e a linea model o he Na ie -S okes equa ions.
76 T. CHAC ´
ON REBOLLO
Rema k 6. P oblems (50) and (51) a e equi alen . The key poin o his equi alence is he ollowing: I a
ho izon al eloci y y=(y1,y
2)∈H1(Ω)2sa is ies y|Γb=0, hen
h∇H·yi=∇H·hyi.
As a consequence, he e exis s a e ical eloci y y3∈L2(Ω) such ha
∇·(y,y
3)=0in Ω,y
3|Γs=0and y3·n3|Γb=0
i and only i
y3(x,x
3)=Z0
x3∇H·y(x,s)dsin Ω,(52)
and
∇H·hyi=0 in ω.
This allows o elimina e he e ical eloci y y3 om p oblem (51). Also, he condi ion ∂3P=−ρg allows o
eco e he p essu e P om he su ace p essu e p,by
P(x,z)=ρgz+p(x).(53)
A igou ous p oo o his equi alence may be ound in Lewandowski [23].
To gi e a a ia ional o mula ion o p oblem (51), le us de ine he spaces
Vk={ =( 1,
2)∈W1,k(Ω)2| |Γb=0} o k≥1,in ege ;
Lα
D(ω)={q:ω→Rmeasu able such ha Zω
D(x)|q(x)|αdx<+∞} o α≥1;
Lα
D,0(ω)=Lα
D(ω)/R(quo ien space).
Spaces Lα
D(ω)andLα
D,0(ω) a e Banach spaces – e lexi e i 1 <α<+∞–, espec i ely endowed wi h he no ms
kqkLα
D(ω)=Zω
D(x)|q(x)|αdx1/α
,
kqkLα
D,0(ω)=in
c∈Rkq+ckLα
D(ω).
Space Lα
D(ω) is isomo phic, and, mo e speci ically, isome ic, o he space
Lα(∂3,Ω) = {q∈Lα(Ω) such ha ∂3q=0}.
Indeed, we iden i y each q∈Lα
D(ω) wi h i s ex ension o Ω as a cons an unc ion wi h espec o he x3 a iable.
Then, we ha e kqkLα
D(ω)=kqkLα(Ω).
Mo eo e , i we conside he space Lα
0(∂3,Ω) = Lα(∂3,Ω)/R, henLα
D,0(ω)andLα
0(∂3,Ω) also a e isomo phic,
and kqkLα
D,0(ω)=kqkLα
0(Ω),∀q∈Lα
D,0(ω).
We u he assume ∈V0
2and g∈H−1/2(Γs)d, he dual space o [H1/2(Γs)]d. This space is well de ined
as Γsis C∞.
UNIFIED MIXED AND STABILIZED SOLUTIONS 77
We conside he ollowing weak o mula ion o p oblem (51):
(Ob ain (y,p)∈V2×L3/2
D,0(ω) such ha
B(PE)(y,p; ,q)=F( ); ∀( ,q)∈V4×L2
D,0(ω); (54)
whe e
B(PE)(y,p; ,q)=hW·∇y, iV0
4−V4+ν(∇y,∇ )Ω−(p, ∇H·h i)ω
−(∇H·hyi,q)ω,
F( )=h , iV0
2−V2+hg, iH−1/2(Γs)−H1/2(Γs).
This o m is well de ined, due o he ollowing:
Lemma 5. The ollowing s a emen s hold.
i) Conside a unc ion W=(w,w
3)∈V2×L2(Ω) such ha ∇·W=0,w3|Γs=0.Then,∀u∈V2,
W·∇u∈V0
k o k≥3,and
kW·∇ukV0
k≤ˆ
Ck|w|1,Ω|u|1,Ω,(55)
o some cons an ˆ
Ck>0.
ii) I w∈Vk o some k≥1, henhwi∈[W1,k(ω)]2and ∂ihwi=h∂iwi,i=1,2.
P oo . i) Obse e ha , gi en w∈V2,andw3∈L2(Ω) such ha ∂3w3=−∇H·w,andw3|Γs=0weha e
w3(x,x
3)=Z0
x3∇H·w(x,s)ds.Thus,
kw3k0,Ω+k∂3w3k0,Ω≤C1|w|1,Ω,(56)
o some cons an C1>0.
Now, i Wis smoo h, we see by in eg a ions by pa s ha o u∈V2and ∈Vk,
ZΩ
(W·∇u)· dxdx3=ZΩ
[(w·∇Hu)· −∂3w3u· −w3u·∂3 ]dxdx3.
Then, we may de ine he duali y hW·∇u, iby
hW·∇u, i=ZΩ
[(w·∇Hu)· −∂3w3u· −w3u·∂3 ]dxdx3.(57)
Using (56),
|hW·∇u, i| ≤ C2(kwk0,4;Ω|u|1,Ωk k0,4;Ω +|w|1,Ωkuk0,4;Ωk k0,4;Ω (58)
+kwk0,Ωkuk0,6;Ωk∂3 k0,3;Ω )≤ˆ
Ck|w|1,Ω|u|1,Ω| |1,k;Ω.
This p o es ha W·∇u∈V0
k. Nex , conside a ield W=(w,w
3)∈V2×L2(Ω) wi h ∇·W=0,w3|Γs=0.
Then, he e exis s a sequence {wn}n≥1⊂[D(¯
Ω)]2such ha wn=0onΓ
b, which con e ges o win V2.
This is p o ed by a s anda d a gumen ( o ins ance, by symme iza ion wi h espec o Γs)using ha ∂Ωis
Lipschi z-con inuous. Le Wn=(wn,w
3n), wi h w3n(x,x
3)=Z0
x3∇H·wn(x,s)ds.

78 T. CHAC ´
ON REBOLLO
Following Dau ay and Lions [18], Chap e XXI, we may ensu e ha i a unc ion z∈L2(Ω) is such ha
∂3z∈L2(Ω), hen he ace o zon Γsbelongs o H1/2(Γs). Mo eo e , a Poinca ´e inequali y holds i z|Γs=0:
kzk0,Ω≤C3k∂3zk0,Ω,
o some cons an C3>0. The e o e,
kw3−w3nk0,Ω≤C3k∇H·(w−wn)k0,Ω,
and w3ncon e ges o w3in L2(Ω). Thus, we may pass o he limi in he .h.s. o (57), and de ine W·∇uas
a linea o m on Vk. Now, passing o he limi in (58) we deduce W·∇u∈V0
kand es ima e (55).
ii) Conside w∈Vk.Asw∈[Lk(Ω)]2, one eadily p o es hwi∈[Lk(ω)]2. Also, le ϕ∈D(ω). Then, i wis
smoo h, o i=1,2,
Zωh∂iwi(x)ϕ(x)dx=ZΩ
∂iw(x,x
3)ϕ(x)dxdx3= (59)
=Z∂Ω
niwϕd(∂Ω) −ZΩ
w(x,x
3)∂iϕ(x)dxdx3= (60)
=−Zωhwi(x)∂iϕ(x)dx,=−Zω
∂ihwi(x)ϕ(x)dx,(61)
as ni=0onΓ
sand w=0onΓ
b.Thus,∂ihwi=h∂iwiand w∈[W1,k(ω)]2.
I wis any elemen o Vk, he same esul s ollows om a densi y a gumen simila o ha o he p oo o
s a emen i) abo e. 
Rema k 7. Any solu ion (y,p) o p oblem (54) is a weak solu ion o p oblem (50) in he dis ibu ion sense.
Fu he mo e, i we eco e he e ical eloci y y3by (52), and he physical p essu e Pby (53), hen he couple
((y,y
3),P) is a solu ion o p oblem (51) in he dis ibu ion sense.
We shall disc e ize p oblem (54) by a penal y s abilized me hod, o B ezzi and Pi k¨a an a’s kind (c . [9]).
Conside a iangula ion Cho ¯ωsuch ha Dis a ine on each iangle T∈C
h. Conside also a pa i ion Pho
¯
Ω by se s o he o m
PT={(x,x
3)∈R3,such ha x∈T, −D(x)≤x3≤0} o some iangle T∈C
h.
No ice ha i a iangle T∈C
his no adjacen o ∂ω, o i i is adjacen o ∂ω and D>0on ¯
T, heni s
associa ed se PTis a iangula p ism wi h uppe base T×{0}and possibly non-ho izon al lowe base. Howe e ,
i Tis adjacen o ∂ω and D= 0 on a pa o ∂T, henPTis a non-p isma ic polyhed on.
We shall conside a iangula ion Tho Ω cons uc ed by subdi iding each elemen Phin o e ahed a. Le
us de ine he ini e elemen spaces,
Vh={ h∈C0(¯
Ω) | h|K∈P1(K),∀K∈T
h}; (62)
Yh={ h∈V2
h| h|Γb=0};
˜
Nh={qh∈C0(¯ω)|qh|T∈P1(T),∀T∈C
h};Nh=˜
Nh/R.
We in oduce he ollowing disc e iza ion o (54):
Ob ain (yh,p
h)∈Yh×Nhsuch ha
B(PE)
h(yh,p
h; h,q
h)=F( h); ∀( h,q
h)∈Yh×Nh;(63)
UNIFIED MIXED AND STABILIZED SOLUTIONS 79
whe e
B(PE)
h(uh,
h; h,q
h)=B(PE)(uh,
h; h,q
h)+ X
K∈Th
τ(c)
K(Wh·∇uh,Wh·∇ h)K
−X
T∈Ch
τ(p)
T(∇H h,∇Hqh)T.
The s abilizing coe icien s o con ec ion τ(c)
Ka e assumed o be s ill gi en by (11). This will p o ide some
s abiliza ion o he con ec i e de i a i e. Also, o ensu e he s abili y o he p essu e disc e iza ion we shall
assume ha he s abilizing coe icien s o p essu e τ(p)
Tsa is y he ollowing condi ion: The e exis wo cons an s
β1>0,β
2>0 such ha
β1h2
TZT
Ddx
|T|≤τ(p)
T≤β2h2
TZT
Ddx
|T|,∀T∈C
h.(64)
Obse e ha hese inequali ies make sense as we assume D>0onω. In he s abilizing e ms o (63), we
eplace he con ec ion eloci y W=(w,w
3)bysomein e pola eWh=(wh,w
3h)∈Yh×Vh, sa is ying o
some cons an C>0,
|Wh|1≤C|w|1.(65)
We now s a e he main esul o his sec ion.
Theo em 6. Assume he con ec ion eloci y W=(w,w
3)lies in he space V2×L2(Ω) and e i ies ∇·W=0,
w3|Γs=0. Assume he iangula ions {Th}h>0a e egula . Then, he ollowing s a emen s hold.
i) P oblem (63) admi s a unique solu ion (yh,p
h)∈Yh×Nhwhich is bounded in
V2×L3/2
D,0(ω).
ii) The sequence {(yh,p
h)}h>0con ains a subsequence which is weakly con e gen in
V2×L3/2
D,0(ω) o a solu ion o (54) sa is ying he es ima e
|y|1+kpkL3/2
D,0(ω)≤Ck kV0
2+kgk−1/2,Γs(1 + |w|1,Ω),(66)
o some cons an C>0independen o h.
P oo . We p oceed by s eps.
S ep 1: Embedding o me hod (63) in abs ac me hod.
Gi en an elemen T∈C
h, le us de ine τ(p)
K=|T|
ZT
Ddx
τ(p)
T, o any elemen K∈T
h ha be in he p ism PT
ha lies on T. We assume ha he p essu es o Nha e de ined on he whole Ω, as cons an unc ions in he x3
80 T. CHAC ´
ON REBOLLO
a iable. Then,
X
K∈Th
τ(p)
K(∇ h,∇qh)K=X
T∈Ch
|T|
ZT
Ddx
τ(p)
TZPT∇H h·∇Hqhdxdx3
=X
T∈Ch
|T|
ZT
Ddx
τ(p)
T(∇H h)|T·(∇Hqh)|TZPT
dxdx3
=X
T∈Ch
τ(p)
T(∇H h,∇Hqh)T,∀ h,q
h∈Nh.(67)
Le us de ine Mh=Vh/R,whe eVhis gi en by (62). We now apply Lemma 1: The e exis s a bubble ini e
elemen space B1h, gene a ed on Thby a e e ence elemen B∗
1⊂H1
0(K∗)3, and a bilinea coe ci e o m S1h
on H1
0(Ω)3, such ha
X
K∈Th
τ(p)
K(∇ h,∇qh)K=S1h(R1h(∇ h),R1h(∇qh)),∀ h,q
h∈Mh; (68)
whe e R1his he s a ic condensa ion ope a o on B1hwi h espec o o m S1h.Wemayiden i yNhwi h he
subspace o Mhde ined by {qh∈Vh|∂3qh=0}.Then, om (67) and (68) we deduce
X
T∈Ch
τ(p)
T(∇H h,∇Hqh)T=S1h(R1h(∇ h),R1h(∇qh)),∀ h,q
h∈Nh.(69)
Also, again by Lemma 1, he e exis s a bubble ini e elemen space B2h, gene a ed on Thby a e e ence elemen
B∗
2⊂H1
0(K∗)2, and a bilinea coe ci e o m S2hon H1
0(Ω)2, such ha ∀uh, h∈Yh,
X
K∈Th
τ(c)
K(Wh·∇uh,Wh·∇ h)K=S2h(R2h(Wh·∇uh),R2h(Wh·∇ h) ); (70)
whe e R2his he s a ic condensa ion ope a o on B2hwi h espec o o m S2h. Then,
B(PE)
h(uh,
h; h,q
h)=B(PE)(uh,
h; h,q
h)
+S2h(R2h(Wh·∇uh),R2h(Wh·∇ h))
−S1h(R1h(∇ h),R1h(∇qh)),∀uh, h∈Yh,∀ h,q
h∈Nh.
We ecall ha by Theo em (2) (S ep 3), he o ms {S2h}h>0a e uni o mly con inuous and coe ci e in H1no m.
Also, due o (64) and he egula i y o iangula ions Th, he coe icien s τ(p)
Ka e o o de h2
K. Then, he o ms
{S1h}h>0also a e uni o mly con inuous and coe ci e.
S ep 2: Disc e e in -sup condi ion.
We s a e he ollowing:
Lemma 6. Gi en α∈(1,2], he e exis s a cons an Cα>0such ha ∀qh∈Nh,
CαkqhkLα
D,0(ω)≤sup
h∈Yh−{0}
(∇H·h hi,q
h)ω
| h|1,α0,Ω
+[S1h(R1h(∇qh),R1h(∇qh))]
1/2,(71)
whe e α0is he conjuga e exponen o α.
UNIFIED MIXED AND STABILIZED SOLUTIONS 81
P oo . De ine he space Wh={( h,
3h)∈V3
h|( h,
3h)|∂Ω=0}. I is enough o p o e ha
CαkqhkLα
0(Ω) ≤sup
( h, 3h)∈Wh−{0}
(∇·( h,
3h),q
h)Ω
|( h,
3h)|1,α0,Ω
(72)
+[S1h(R1h(∇qh),R1h(∇qh))]
1/2,∀qh∈Vh.
Indeed, i qh∈Nh,( h,
3h)∈Wh,
(∇·( h,
3h),q
h)Ω=(∇H· h,q
h)Ω−( 3h,∂
3qh)Ω=(∇H·h hi,q
h)ω.
Then,
sup
( h, 3h)∈Wh−{0}
(∇·( h,
3h),q
h)Ω
|( h,
3h)|1,α0,Ω
=sup
( h, 3h)∈Wh−{0}
(∇H·h hi,q
h)ω
|( h,
3h)|1,α0,Ω
≤sup
h∈Yh−{0}
(∇H·h hi,q
h)ω
| h|1,α0,Ω·
Also, kqhkLα
0(Ω) =kqhkLα
D,0(ω)i qh∈Nh. Thus, (71) ollows om (72).
To p o e (72), conside qh∈Vh. As Ω is polyhed ic, hen ∂Ω is Lipschi z, and he con inuous in -sup
condi ion in Lα(Ω) no m is sa is ied (c . Am ouche and Gi aul [1]): The e exis s a cons an Dα>0 such ha
DαkqhkLα
0(Ω) ≤sup
∈
h
W1,α0
0(Ω)
i
3−{0}
(∇· ,q
h)Ω
| |1,α0,Ω
,∀q∈Lα
0(Ω).
As [D(Ω)]3is dense in hW1,α0
0(Ω)i3, he e exis s a 0∈[D(Ω)]3such ha
1
2DαkqhkLα
0(Ω) ≤(∇· 0,q
h)Ω,| 0|1,α0,Ω=1.
Following he s anda d ini e elemen s in e pola ion heo y (c . Cia le [13]), he e exis s an in e pola e 0h∈
Whsuch ha
| 0h|1,α0,Ω≤C1| 0|1,α0,Ω; (73)
k 0h− 0k0,K ≤C1hK| 0|1,K,∀K∈T
h; (74)
o some cons an C1>0 independen o h. Then, as qhis con inuous,
1
2DαkqhkLα
0(Ω) ≤(∇· 0h,q
h)Ω+( 0h− 0,∇qh)Ω
≤C1|(∇· 0h,q
h)Ω|
| 0h|1,α0,Ω
+"X
K∈Th
h−2
Kk 0h− 0k2
0,K#1/2"X
K∈Th
h2
Kk∇qhk2
0,K#1/2
.
As α0≥2, hen (74) yields
"X
K∈Th
h−2
Kk 0h− 0k2
0,K#1/2
≤C2| 0|1,α0,Ω.
88 T. CHAC ´
ON REBOLLO
whe e kAKkdeno es he spec al ma ix no m. Then,
kzhk2
0≤C∗γ2X
K∈ThkAKk2|zh|2
1,K ≤C1h2|zh|2
1,
o some cons an C1>0.
Le us now conside he in e pola ion es ima e (c . B ´ezis [6]),
kwk0,q ≤C2kwkβ
0kwk1−β
0,6,∀w∈L6(Ω)d,
i 2 ≤q≤6, wi h βgi en in (78). As H1
0(Ω)dis con inuously embedded in L6(Ω)di d=2o d=3,
hen (78) ollows.
ii) Conside a sequence {zh}h>0, wi h zh∈Zh. This sequence con ains a subsequence, ha we s ill deno e
in he same way, weakly con e gen o some elemen zin H1
0(Ω)d.As
H1
0(Ω)dis compac ly embedded in
L2(Ω)d, we may assume ha his sequence con e ges s ongly in L2(Ω)d.
Recall ha space Y(0)
his de ined by
Y(0)
h=n ∈L2(Ω)d| |Kis cons an ,∀K∈T
ho.(84)
Due o s anda d ini e elemen in e pola ion analysis, he e exis s a sequence {yh}h>0, wi h yh∈Y(0)
h, s ongly
con e gen o zin L2(Ω)d(e en i he amily o iangula ions is no egula ).
Deno e by Y∗ he e e ence space ha gene a es space Y(0)
h. By hypo hesis, Y∗∩Z∗={0}. Then, he e
exis s ρ>0 such ha
|(z∗,y∗)K∗|≤(1 −ρ)kz∗k0,K∗ky∗k0,K∗,∀z∗∈Z∗,y∗∈Y∗.
This is p o ed simila ly o es ima e (83) in Lemma 2. Thus,
|(zh,yh)|=X
K∈Th|de AK||(zK
h,yK
h)K∗|≤(1 −ρ)X
K∈Th|de AK|kzK
hk0,K∗kyK
hk0,K∗
≤(1 −ρ)kzhk0kyhk0.
Consequen ly, z=0askzk2
0= lim
h→0|(zh,yh)|≤(1 −ρ)kzk2
0.
As he limi o any weakly con e gen subsequence is necessa ily ze o, he he whole sequence {zh}h>0
con e ges weakly o ze o. 
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