ESAIM: M2AN 49 (2015) 1219–1238 ESAIM: Ma hema ical Modelling and Nume ical Analysis
DOI: 10.1051/m2an/2015008 www.esaim-m2an.o g
A REDUCED DISCRETE INF-SUP CONDITION IN Lp
FOR INCOMPRESSIBLE FLOWS AND APPLICATION
Tom´
as Chac´
on Rebollo1, Vi e e Gi aul 2,
Maca ena G´
omez M´
a mol3and Isabel S´
anchez Mu˜
noz4
Abs ac . In his wo k, we in oduce a disc e e speci ic in -sup condi ion o es ima e he Lpno m,
1<p<+∞, o he p essu e in a numbe o luid lows. I applies o p ojec ion-based s abilized ini e
elemen disc e iza ions o incomp essible lows, ypically when he eloci y ield has a low egula i y.
We de i e wo e sions o his in -sup condi ion: The i s one holds on shape- egula meshes and he
second one on quasi-uni o m meshes. As an applica ion, we de i e educed in -sup condi ions o he
linea ized P imi i e equa ions o he Ocean ha apply o he su ace p essu e in weigh ed Lpno m.
This allows o p o e he s abili y and con e gence o qui e gene al s abilized disc e iza ions o hese
equa ions: SUPG, Leas Squa es, Adjoin -s abilized and OSS disc e iza ions.
Ma hema ics Subjec Classi ica ion. 35Q35, 65N12, 76D05.
Recei ed June 12, 2014. Re ised Oc obe 15, 2014.
Published online July 6, 2015.
1. In oduc ion
S abilized me hods a e designed o p o ide s able disc e iza ions wi h educed compu a ional complexi y
o se e al sou ces o spu ious ins abili ies ha may a ise in he disc e iza ion o incomp essible flows ( ha
may be due ei he o incomp essibili y o o la ge con ec ion, Co iolis o eac ion e ms, among o he s). In
his pape , we ocus on he ea men o he incomp essibili y cons ain . Conc e ely, we deal wi h p ojec ion-
s abilized me hods, in oduced by Blasco and Codina in [3] by means o a local L2p ojec ion, ha p oduce
disc e iza ions wi h high-o de accu acy. This me hod was ex ended o he local p ojec ion-s abiliza ion me hods
ha use elemen -wise L2p ojec ions ins ead o he global L2p ojec ion, while sa is ying some o hogonali y
p ope ies. Among he many e e ences on diffe en e sions o local p ojec ion-s abiliza ion, le us quo e B aack
andBu man[4], B aack e al. [5], Ganesan e al. [17], Knobloch [19], Ma hies e al. [21], Roos e al. [23].
Keywo ds and ph ases. In -sup condi ion, Fini e elemen me hod, S abilized me hod, Incomp essible flows, P imi i e equa ions
o he Ocean.
1Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico and Ins i u o de Ma em´a icas de la Uni e sidad de Se illa
(IMUS), Apdo. de co eos 1160, Uni e sidad de Se illa, 41080 Se illa, Spain.
2Labo a oi e Jacques-Louis Lions, Uni e si ´e Pie e e Ma ie Cu ie & C.N.R.S, UMR 7598, Pa is 6, 4 Place Jussieu, 75252
Pa is cedex 05, F ance.
3Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico, Apdo. de co eos 1160, Uni e sidad de Se illa, 41080 Se illa,
Spain.
4Depa amen o de Ma em´a ica Aplicada I, Ca e e a de U e a Km 1, Uni e sidad de Se illa, 41013 Se illa, Spain.
[email p o ec ed]
A icle published by EDP Sciences c
EDP Sciences, SMAI 2015
1220 T. CHAC ´
ON REBOLLO ET AL.
A special class o p ojec ion-s abilized me hods is he in e io penal y me hod, in which s abiliza ion is achie ed
by in oducing in e -elemen jumps o he e ms o be s abilized. This me hod is equi alen o a p ojec ion-
s abilized me hod, whe e he L2p ojec ion ope a o is eplaced by he Oswald (c . [22]) quasi-in e polan
ope a o on he disc e e eloci y space (c . [7–9]).
A u he simplifica ion is in oduced in [15] whe e he local p ojec ion ope a o is eplaced by a quasi-local
app oxima ion ope a o ha does no need o sa is y any o hogonali y p ope y. This me hod has a mo e
compac s encil while e aining he same op imal accu acy as all p ojec ion-s abiliza ion me hods, in he sense
ha i s con e gence o de is op imal wi h espec o he deg ee o he fini e elemen spaces.
In he p esen wo k, we ex end he me hod in oduced in [15] o s abilize he disc e iza ion o he p essu e in
flows whe e he eloci y has a low accu acy, ypically in some space W1,s(Ω)wi h1<s<2. Then he p essu e
has only Lp(Ω) egula i y, whe e pis he conjuga e exponen o s. This is he case, o ins ance, o he weak
solu ions o he P imi i e equa ions o he Ocean, ha we conside as an applica ion o ou gene al se ing.
We in oduce a specific in -sup condi ion in Lpno ms, which is he main echnical con ibu ion o his pape .
As in [15], he de i a ion o his condi ion aces he difficul y o he educed numbe o deg ees o eedom o he
buffe space. This difficul y is sol ed by a fini e-dimensional a gumen o equi alence o no ms (c . Lem. 2.4).
We de i e wo e sions o his in -sup condi ion: he fi s one holds on shape- egula meshes and he second one
on quasi-uni o m meshes. As an applica ion, we de i e educed in -sup condi ions o he linea ized P imi i e
equa ions o he Ocean ha apply o he su ace p essu e in weigh ed Lpno ms. This allows o p o e he s abili y
and con e gence o qui e gene al s abilized disc e iza ions o hese equa ions: SUPG, Leas Squa es, Adjoin -
s abilized and OSS disc e iza ions. This condi ion gene alizes a simila one o L2weigh ed no ms in oduced
in [14].
The pape is s uc u ed as ollows: in Sec ion 2we in oduce he in -sup condi ion in Lpno ms o shape-
egula meshes, as well as a simplified condi ion o quasi-uni o m meshes. These condi ions a e applied o
s abilized disc e iza ions o a linea ized e sion o he P imi i e equa ions in Sec ion 3. In his sec ion, educed
in -sup condi ions o he su ace p essu e a e deduced, o bo h shape- egula and quasi-uni o m meshes. Also,
he s abili y and con e gence o he s abilized disc e iza ions a e p o ed by means o hese condi ions.
2. In -sup condi ion o shape- egula meshes
Le Ω⊂Rd(d= 2 o 3) be a bounded domain. We assume ha Ωis a polygon i d= 2 o a polyhed on
i d=3.Le {Th}h>0be a amily o con o ming iangula ions o ¯
Ω o med by simplicial elemen s, whe e he
pa ame e hdeno es he la ges diame e o he elemen s o Th.Weassume he ollowing.
Hypo hesis 2.1. The amily {Th}h>0is shape- egula in he sense o Cia le [16]and no elemen o Thhas
3nodes(when d=2)o 4 nodes (when d=3)on he bounda y o Ω.
We decompose Ωin o a fini e union o mac o-elemen s:
Ω=
R

i=1 Oi,
such ha each Oiis he suppo o he piecewise affine basis unc ion associa ed o he node i.No e ha any
elemen K∈T
hbelongs o a mos Mmac o-elemen s, whe e Mis independen o h.
Fo all i,1≤i≤R,wese
hi=max
K⊂Oi{hK}and ρi=min
K⊂Oi{ρK},
whe e hKdeno es he diame e o he elemen Ko Thand ρK he diame e o he ball insc ibed in K.
As he mesh is egula , hen i is locally uni o mly egula (o locally quasi-uni o m) and his implies ha
he e exis posi i e cons an s C1and C2independen o h, such ha o all K∈T
hand o all i o which
K⊂O
i,
C1ρi≤hK≤hi,hi
ρi≤C2.(2.1)
A REDUCED DISCRETE INF-SUP CONDITION IN LPFOR INCOMPRESSIBLE FLOWS AND APPLICATION 1221
This implies immedia ely ha
C1
C2
hi≤hK≤hi.(2.2)
Fo a domain O⊂Rd,wedeno eby·
k,p,Oand |·|
k,p,O he no m and, espec i ely, he semino m in
Wk,p(O). In Lp(Ω)/R,wealsodeno eby·0,p,Ω he quo ien no m in o de o simpli y he no a ion.
Fo all p∈[1,+∞)and o all ∈Lp(Ω)d, we define
 h,p =R

i=1
hp
i p
0,p,Oi
1
p
.(2.3)
Gi en an in ege l≥1, we deno e by Pl(K) he space o polynomials o deg ee smalle han, o equal o, l
defined on an elemen K∈T
hand define he ollowing fini e elemen spaces:
Vl
h={ h∈C0(¯
Ω) such ha h|K∈Pl(K),∀K∈T
h},
Xh=(Vl
h∩H1
0(Ω))d,
Mh=Vm
h/R,
Vl
h(Oi)={ h∈C0(Oi) such ha h|K∈Pl(K),∀K∈T
hsuch ha K⊂O
i},
Xh(Oi)=(Vl
h(Oi)∩H1
0(Oi))d.
We ecall now he in e se inequali ies in fini e elemen spaces, ha we use equen ly in he sequel.
Lemma 2.2. Le p1and p2be numbe s in [1,∞],andl1and l2 wo non-nega i e in ege s such ha l1≥l2and
l1−d
p1≥l2−d
p2.
[Local in e se inequali ies] ([2], P op. 4.2) Fo all K∈T
h,
∀ ∈Pl(K),| |Wl1,p1(K)≤Cρ
l2−l1−d
p2
Kh
d
p1
K| |Wl2,p2(K),(2.4)
whe e Cis a cons an independen o K.
[Global in e se inequali ies] ([6],Thm4.5.11) I p1≥p2and Zhis a ini e elemen space o polynomials in
each K, hen
∀wh∈Zh,
K∈Th
|wh|p1
Wl1,p1(K)1
p1
≤Cρ
l2−l1−d
p2+d
p1
min 
K∈Th
|wh|p2
Wl2,p2(K)1
p2
,(2.5)
whe e Cis a cons an independen o hand ρmin =in
K∈Th{ρK}.
Th oughou his wo k, C ep esen s a cons an ha is always independen o hbu may a y om an
inequali y o ano he .
Le us now conside an in e pola ion o p ojec ion ope a o
Jh:L2(Ω)d→(Vl−1
h)d,(2.6)
and se J∗
h=Id−Jh. Ou main esul is he ollowing.
1222 T. CHAC ´
ON REBOLLO ET AL.
Theo em 2.3. Assume ha Hypo hesis 2.1 holds. Then o any p∈(1,+∞) he e exis s a cons an γp>0
independen o hsuch ha o all qh∈Mh,
γpqh0,p,Ω ≤sup
h∈Xh
(∇· h,q
h)
| h|1,s,Ω
+R

i=1 sup
h∈Xh(Oi)
|(∇· h,q
h)Oi|p
| h|p
1,s,Oi
1
p
+J∗
h(∇qh)h,p (2.7)
whe e sis he conjuga e exponen o p(1
p+1
s=1).
To p o e his heo em we need he ollowing auxilia y esul .
Lemma 2.4. Unde Hypo hesis 2.1, he e exis s a posi i e cons an Cindependen o h, such ha o all i,
1≤i≤R,
∀gh∈Vl−1
h(Oi),gh0,p,Oi≤Csup
h∈Vl
h(Oi)∩H1
0(Oi)
(gh,
h)Oi
 h0,s,Oi
,(2.8)
whe e p∈(1,+∞)and sis he conjuga e exponen o p.
P oo . Fo p=s= 2 his esul was p o ed in [15]. We ex end i he e o any conjuga e pai o exponen s p
and s.
We deno e by {aj} he nodes o Th ha belong o Oiand we define he unc ion wh∈V1
h(Oi)∩H1
0(Oi)such
ha :
wh|K∈P1(K), o all K⊂O
i;
wh(aj)=1i ajis a in e io node o Oi;
wh(aj)=0i ajis a node belonging o ∂Oi.
Because each mac o-elemen Oiis he suppo o one piecewise affine basis unc ion and no elemen o Thhas
3nodes(whend= 2) o 4 nodes (when d= 3) on he bounda y o Ω, he e exis s a leas one in e io node
in Oi. So, his unc ion whis well-defined and i is posi i e in he in e io o Oi.
Le gh∈Vl−1
h(Oi)and h=ghwh. Then, h∈Vl
h(Oi)∩H1
0(Oi) and i sa isfies
|(gh,
h)Oi|≥Cgh2
0,Oi,(2.9)
wi h Ca posi i e cons an independen o h(c . [15], Lem. 3.2).
Gi en K⊂O
i,le ˆgK=gh|K◦FK,whe eFKis he affine mapping ha ans o ms he e e ence elemen
ˆ
Kon o K. The shape egula i y o he mesh implies ha
gh2
0,Oi=
K⊂OiK|gh|2=C
K⊂Oi|K|ˆ
K|ˆgK|2≥C|Oi|
K⊂Oiˆ
K|ˆgK|2.(2.10)
Deno e by Ni he numbe o elemen s o Th|Oiand conside he no m in RNi:
(x1,...,x
Ni)p,Ni=⎛
⎝
Ni

j=1 |xj|p⎞
⎠
1
p
.
As he g ids a e egula , Ni≤N(independen o h)and
(x1,...,x
Ni)p,Ni=(x1,...,x
Ni,0,...,0)p,N .
By he equi alence o no ms in RN, he e exis s a cons an Cp(independen o h) such ha
(x1,x
2,...,x
Ni,0,...,0)2,N ≥Cp(x1,x
2,...,x
Ni,0,...,0)p,N .(2.11)
A REDUCED DISCRETE INF-SUP CONDITION IN LPFOR INCOMPRESSIBLE FLOWS AND APPLICATION 1223
Then, applying (2.11)wi h
xj=ˆ
K|ˆgKj|p1
p
∀j=1,...,N
i,
we ha e

K⊂Oiˆ
K|ˆgK|21
2
≥Cp
K⊂Oiˆ
K|ˆgK|p1
p
.
Thus, om (2.10) and he shape egula i y o he mesh
gh2
0,Oi≥C|Oi|CpCs
K⊂Oiˆ
K|ˆgK|p
1
p
K⊂Oiˆ
K|ˆgK|s
1
s
≥C
K⊂Oi|K|ˆ
K|ˆgK|p1
p
K⊂Oi|K|ˆ
K|ˆgK|s1
s
.
Tha is,
gh2
0,Oi≥Cgh0,p,Oigh0,s,Oi.(2.12)
Mo eo e ,
 h0,s,Oi≤gh0,s,Oiwh0,∞,Oi≤gh0,s,Oi.(2.13)
Then, om (2.9) and aking in o accoun (2.12)and(2.13)weob ain
Cgh0,p,Oi≤(gh,
h)Oi
 h0,s,Oi
,
whence we deduce (2.8). 
P oo o Theo em2.3.We adap Ve ¨u h’s echnique (c . [24]). Gi en qh∈Mh, by Am ouche and Gi aul
(c . [1]), he e exis s a cons an C>0 independen o hsuch ha
Cqh0,p,Ω ≤sup
∈[W1,s
0(Ω)]d−{0}
(∇· ,q
h)
| |1,s,Ω ·
So he e exis s ∈[W1,s
0(Ω)]dsuch ha
1
2Cqh0,p,Ω ≤(∇· ,q
h)
| |1,s,Ω ·(2.14)
Since he amily o g ids is egula , ollowing he s anda d fini e elemen in e pola ion heo y (c . [2]o [6],
Sec . 4.8), he e exis s an in e pola e o , h∈Xh, such ha
| h|1,s,Ω ≤C| |1,s,Ω,(2.15)
 − h0,s,K ≤Ch
K| |1,s,ωK,(2.16)
whe e ωKdeno es he union o all elemen s o Th ha in e sec K.
Le us ew i e he .h.s. o (2.14)as
(∇· ,q
h)
| |1,s,Ω
=(∇· h,q
h)
| |1,s,Ω
+(∇·( − h),q
h)
| |1,s,Ω ·(2.17)

1224 T. CHAC ´
ON REBOLLO ET AL.
Using (2.15),
(∇· h,q
h)
| |1,s,Ω ≤C(∇· h,q
h)
| h|1,s,Ω ·(2.18)
Also, because qhbelongs o H1(Ω)and( − h)·n=0on∂Ω,
(∇·( − h),q
h)=−( − h,∇qh)
≤
K∈Th − h0,s,K ∇qh0,p,K
≤
K∈Th
Ch
K| |1,s,ωK∇qh0,p,K (using (2.16))
≤C
K∈Th| |s
1,s,ωK1
s
K∈Th
hp
K∇qhp
0,p,K1
p
≤C| |1,s,Ω 
K∈Th
hp
K∇qhp
0,p,K1
p
.
(2.19)
Then om (2.14), combining (2.17)−(2.19), we ha e
Cqh0,p,Ω ≤sup
h∈Xh
(∇· h,q
h)
| h|1,s,Ω
+
K∈Th
hp
K∇qhp
0,p,K1
p
.(2.20)
As each elemen K∈T
hbelongs o some mac o-elemen Oi,

K∈Th
hp
K∇qhp
0,p,K ≤C
R

i=1
hp
i∇qhp
0,p,Oi.
So, om (2.20)
Cqh0,p,Ω ≤sup
h∈Xh
(∇· h,q
h)
| h|1,s,Ω
+∇qhh,p.(2.21)
To es ima e he las e m in (2.21), we use he ela ion Jh+J∗
h=Id and w i e
∇qh0,p,Oi≤Jh(∇qh)0,p,Oi+J∗
h(∇qh)0,p,Oi.(2.22)
Since Jh(∇qh)|Oi∈(Vl−1
h(Oi))dwe can apply he in -sup condi ion (2.8) o each o i s componen s,
Jh(∇qh)0,p,Oi≤Csup
h∈Xh(Oi)
(Jh(∇qh), h)Oi
 h0,s,Oi·
Using again Jh+J∗
h=Id,
|(Jh(∇qh), h)Oi|≤|(∇qh, h)Oi|+|(J∗
h(∇qh), h)Oi|
≤|(∇· h,q
h)Oi|+J∗
h(∇qh)0,p,Oi h0,s,Oi,
as (∇qh, h)Oi=−(∇· h,q
h)Oibecause h=0on ∂Oi.So,
Jh(∇qh)0,p,Oi≤Csup
h∈Xh(Oi)
|(∇· h,q
h)Oi|
 h0,s,Oi
+J∗
h(∇qh)0,p,Oi.(2.23)
A REDUCED DISCRETE INF-SUP CONDITION IN LPFOR INCOMPRESSIBLE FLOWS AND APPLICATION 1225
Thus, om (2.22)and(2.23)
∇qhp
h,p ≤CR

i=1 sup
h∈Xh(Oi)
hp
i|(∇· h,q
h)Oi|p
 hp
0,s,Oi+J∗
h(∇qh)p
h,p.(2.24)
The local in e se inequali y (2.4) be ween W1,s(K)andLs(K)oneachK⊂O
iyields
| h|1,s,Oi≤Cρ
−1
i h0,s,Oi.
This inequali y and (2.1)imply ha
hp
i
 hp
0,s,Oi≤C1
| h|p
1,s,Oi
,
and we ob ain
∇qhp
h,p ≤CR

i=1 sup
h∈Xh(Oi)
|(∇· h,q
h)Oi|p
| h|p
1,s,Oi+J∗
h(∇qh)p
h,p.(2.25)
Finally, (2.7) ollows om (2.21)and(2.25). 
2.1. The case o uni o mly egula meshes
The in -sup condi ion (2.7) may be simplified i he g ids a e uni o mly egula . We assume he ollowing.
Hypo hesis 2.5. The amily {Th}h>0is uni o mly egula (also called quasi-uni o m): he e exis posi i e con-
s an s αand βindependen o hsuch ha
βh≤hK≤αρ
K,∀K∈T
h.(2.26)
We define he space
Y= ∈H1(Ω)dsuch ha ·n= 0 a.e. on ∂Ω,
and conside an in e pola ion o p ojec ion ope a o
Ih:L2(Ω)d→Yh,whe e Xh⊆Yh⊆(Vl
h)d∩Y. (2.27)
We shall deno e I∗
h=Id−Ih.
In his case he in -sup condi ion (2.7) educes o a simple one. This is s a ed as ollows.
Theo em 2.6. Assume ha Hypo hesis 2.5 holds. Then o any p∈(1,+∞) he e exis s a cons an λp>0
independen o hsuch ha o all qh∈Mh,
λpqh0,p,Ω ≤sup
h∈Yh
(∇· h,q
h)
| h|1,s,Ω
+hI∗
h(∇qh)0,p,Ω,(2.28)
whe e sis he conjuga e exponen o p.
P oo . As in Theo em 2.3 we ob ain (2.20) because his es ima e only equi es ha he g ids be egula .
Mo eo e , as Xh⊆Yh,
Cqh0,p,Ω ≤sup
h∈Yh
(∇· h,q
h)
| h|1,s,Ω
+h∇qh0,p,Ω.(2.29)
In o de o bound he las e m in (2.29), we a gue as in he p oo o Theo em 2.3 bu now we do no need o
ollow a local a gumen . Using he ela ion Ih+I∗
h=Id,
∇qh0,p,Ω ≤Ih(∇qh)0,p,Ω +I∗
h(∇qh)0,p,Ω.(2.30)
1226 T. CHAC ´
ON REBOLLO ET AL.
As Ih(∇qh)∈Lp(Ω)d, he eexis s ∈Ls(Ω)dsuch ha
(Ih(∇qh), )=Ih(∇qh)0,p,Ω  0,s,Ω.
Now, le hbe he L2o hogonal p ojec ion o on Yh. As he mesh is uni o mly egula , he L2p ojec ion is
s able in he Lsno m (c . [25]):
 h0,s,Ω ≤C 0,s,Ω.
Then, as Ih(∇qh)∈Yh,
(Ih(∇qh), h)=(Ih(∇qh), )≥CIh(∇qh)0,p,Ω  h0,s,Ω.
Thus,
Ih(∇qh)0,p,Ω ≤Csup
h∈Yh
(Ih(∇qh), h)
 h0,s,Ω ·
Using again Ih+I∗
h=Id and (∇qh, h)=−(∇· h,q
h) because h∈Y,weha e
Ih(∇qh)0,p,Ω ≤Csup
h∈Yh
|(∇· h,q
h)|
 h0,s,Ω
+I∗
h(∇qh)0,p,Ω.(2.31)
The e o e, om (2.30)and(2.31),
h∇qh0,p,Ω ≤Csup
h∈Yh
h|(∇· h,q
h)|
 h0,s,Ω
+hI∗
h(∇qh)0,p,Ω.(2.32)
The global in e se inequali y (2.5) be ween W1,s(Ω)andLs(Ω) and he quasi-uni o mi y o he mesh implies
| h|1,s,Ω ≤Ch
−1 h0,s,Ω.
Using his es ima e in (2.32), we ob ain
h∇qh0,p,Ω ≤Csup
h∈Yh
|(∇· h,q
h)|
| h|1,s,Ω
+hI∗
h(∇qh)0,p,Ω.(2.33)
Finally, (2.28) ollows om (2.29)and(2.33). 
Rema k 2.7. A he angula co ne poin s o he bounda y, he condi ion ·n= 0 implies ha =0.
So, when he domain Ωis app oxima ing a cu ed domain he space Yhin (2.27)coincideswi hXh.In he
applica ion o he P imi i e equa ions o he Ocean s udied in he nex sec ion, he bounda y o he domain
has a fla componen ( he su ace) and he space Xhis s ic ly con ained in o Yh.
3. Applica ion o he P imi i e equa ions o he Ocean
We s udy he fluid in a domain
Ω={(x,z)∈Rdsuch ha x∈ω, −D(x)≤z≤0},
whe e ωis a bounded domain in Rd−1and D:ω→Ris a Lipschi z-con inuous non-nega i e unc ion ha
ep esen s he dep h. The bounda y is spli as ∂Ω =Γs∪Γb,whe eΓs={(x,0) ∈Rdsuch ha x∈ω}
ep esen s he ocean su ace, and Γb ep esen s he ocean bo om and, e en ually, sidewalls.
A REDUCED DISCRETE INF-SUP CONDITION IN LPFOR INCOMPRESSIBLE FLOWS AND APPLICATION 1227
We conside a linea ized e sion o he s eady educed P imi i e equa ions model (c . [20]). The p oblem
consis s in finding a ho izon al eloci y field u:Ω→ Rd−1and a su ace p essu e p:ω→ Rsuch ha
⎧
⎪
⎨
⎪
⎩
W·∇u−μΔu+∇xp+ϕu⊥= in Ω,
∇x·u=0 inω,
u|Γb=0,μ
z∂zu|Γs=g.
(3.1)
He e W:Ω→ Rdis a gi en con ec ion eloci y and u:ω→ Rd−1is defined by
u(x)=0
−D(x)
u(x,s)ds. (3.2)
Also μis he iscosi y coefficien , ha we assume o be iso opic o simplici y. The e m ϕu⊥is due he Co iolis
accele a ion and i appea s only when d=3.In hiscase,u=(u1,u
2)andu⊥=(−u2,u
1). The unc ion ϕ
is defined by ϕ=2θsin φ,whe eθis he angula o a ion a e o he ea h and φis he la i ude. The sou ce
e m akes in o accoun a iable densi y effec s, due o a ia ions o empe a u e and salini y, and g ep esen s
he wind ension a he su ace.
This model includes he igid-lid assump ion, s a ing ha he ee su ace is fla (z= 0 in ou case), and
ha he e ical eloci y anishes a he su ace (c . [13]). This las condi ion and he incomp essibili y o he
eloci y U=(u,u
z) a e used o define he e ical eloci y uz:Ω→ R om he ho izon al eloci y:
uz(x,z)=0
z∇x·u(x,s)ds. (3.3)
To define weak solu ions o p oblem (3.1), we conside he spaces
W1,s
b(Ω)={ ∈W1,s(Ω) such ha |Γb=0}, o an in ege s≥1.
In pa icula H1
b(Ω)=W1,2
b(Ω). We deno e H−1
b(Ω) he dual space o H1
b(Ω)d−1and H−1
2(Γs) he dual space
o H1
2(Γs)d−1. We also conside he spaces
Lp
D(ω)=q:ω→ Rsuch ha ω
D(x)|q(x)|pdx<∞
and
Lp
D,0(ω)=Lp
D(ω)/R,
o all p∈(1,+∞).
P oblem (3.1) is a linea ized e sion o he non-linea P imi i e equa ions ha in ac may be used as an
in e media e s ep o p o e i s well posedness. We hus assume ha he eloci y field is W=(w,w
z)wi h
w∈H1
b(Ω)d−1and wz(x,z)=0
z∇x·w(x,s)ds. (3.4)
The weak o mula ion o p oblem (3.1) ha weconside is:Gi en ∈H−1
b(Ω)andg∈H−1
2(Γs),
⎧
⎨
⎩
Find (u,p)∈H1
b(Ω)d−1×L
3
2
D,0(ω) such ha
B((u,p),( ,q)) = L( ),∀( ,q)∈W1,3
b(Ω)d−1×L2
D,0(ω),
(3.5)
1234 T. CHAC ´
ON REBOLLO ET AL.
and simila ly
sup
Vh∈Xh(Oi)
(∇·Vh,˜qh)Oi
|Vh|1,s,Oi≤sup
h∈Uh(Oi)
(∇x· h,q
h)Oi
| h|1,s,Oi·(3.42)
Also, conside ing ha ∇˜qh=(∇xqh,0), we ha e
J∗
h(∇˜qh)0,p,Oi=Π∗
h(∇xqh)0,p,Oi.(3.43)
Then, (3.39) ollows om (3.40)by akingin oaccoun (3.41)–(3.43). 
We assume he ollowing local s abili y p ope ies o he ope a o Πh:
Hypo hesis 3.6. The ope a o Πhsa is ies o all ∈L2(Ω)d−1,
Πh( )0,K ≤C 0,wK,(3.44)
Πh( )0,3
2,K ≤C 0,3
2,wK,(3.45)
whe e wKis he union o all elemen s o Th ha in e sec K.
This hypo hesis is s onge han Hypo hesis 3.3, bu is also e ified by local L2p ojec ion o Lag ange
in e pola ion-based ope a o s. Howe e i is no e ified by he global L2(Ω) p ojec ion ope a o .
Obse e ha as a consequence o (3.44), he ope a o Πhis also s able wi h espec o he no m ·τ:
Πh( )τ≤C τ,∀ ∈L2(Ω)d−1.(3.46)
Theo em 3.7. Assume ha Hypo heses 2.1,3.1 and 3.6 hold. Then, he disc e e p oblem (3.13)and (3.14)
wi h l=1in (3.7)admi s a unique solu ion (uh,p
h)∈U
h×P
hwhich is bounded in H1
b(Ω)d−1×L
3
2
D,0(ω),
sa is ying he es ima es
|uh|1,Ω ≤C
μ H−1
b(Ω)+gH−1
2(Γs),(3.47)
phL
3
2
D,0(ω)≤C(1 + 1
μ+1
√μ)(1+|w|1,Ω) H−1
b(Ω)+gH−1
2(Γs),(3.48)
Π∗
h(Wh·∇uh+∇xph+ϕu⊥
h)τ≤C
√μ H−1
b(Ω)+gH−1
2(Γs),(3.49)
whe e Cis a cons an independen o h.
Mo eo e , he sequence {(uh,p
h)}h>0con ains a subsequence which is weakly con e gen in H1
b(Ω)d−1×
L
3
2
D,0(ω) o a solu ion o he con inuous p oblem (3.5). I his solu ion belongs o W1,3
b(Ω)d−1, hen he con e -
gence is s ong.
P oo . The es ima ions (3.47)and(3.49) a e ob ained in he same way as in Theo em 3.4 bu he es ima e o
he p essu e is now based on he in -sup condi ion (3.39)wi hp=s=2,whend=2,o p=3
2and s=3,when
d= 3. We indica e he es ima es in his las case. To es ima e he fi s summand in (3.39), we s a om (3.33).
We ea all e ms in he same way as be o e bu o bound he e m Π∗
h(Wh·∇ h+ϕ ⊥
h)τwe ha e o a gue
locally because in his case we only can use local in e se inequali ies:
F om (3.46),
Π∗
h(Wh·∇ h+ϕ ⊥
h)τ≤CWh·∇ h+ϕ ⊥
hτ.
Mo eo e ,
Wh·∇ h2
τ=
K∈Th
τKWh·∇ h2
0,K ≤α2
K∈Th
h2
KWh2
0,K ∇ h2
0,∞,K.

A REDUCED DISCRETE INF-SUP CONDITION IN LPFOR INCOMPRESSIBLE FLOWS AND APPLICATION 1235
Then, using (3.11), he local in e se inequali y (2.4) be ween W1,∞(K)andW1,3(K), and (2.1), we de i e
Wh·∇ hτ≤C|w|1,Ω | h|1,3,Ω.
Likewise,
ϕ ⊥
h2
τ=
K∈Th
τKϕ ⊥
h2
0,K ≤α2
K∈Th
h2
Kϕ2
0,∞,Ω  h2
0,K ≤α2ϕ2
0,∞,Ω h h2
0,Ω.
So,
ϕ ⊥
hτ≤Cϕ0,∞,Ω | h|1,3,Ω.
Also om (3.46),
Π∗
Uh( h)τ≤C hτ.
In his way, we ob ain (3.35).
To es ima e he second summand in (3.39), we conside he a ia ional o mula ion (3.13)and(3.14)wi h
h∈U
h(Oi)andqh= 0. Then,
(∇x· h,p
h)Oi=(Wh·∇uh, h)Oi+μ(∇uh,∇ h)Oi+(ϕu⊥
h, h)Oi
+(Π∗
h(Wh·∇ h+ϕ ⊥
h),Π∗
Uh(ch))τ,Oi−(Π∗
h(Wh·∇ h+ϕ ⊥
h),Π∗
Uh( h))τ,Oi−L( h).(3.50)
All e ms in he igh -hand side a e es ima ed as be o e bu he e again we a gue locally. Fo he s abilizing
e m, we ha e:
(Π∗
h(Wh·∇ h+ϕ ⊥
h),Π∗
Uh(ch))τ,Oi≤Π∗
h(Wh·∇ h+ϕ ⊥
h)τ,OiΠ∗
Uh(ch)τ,Oi.
To es ima e he fi s ac o , we fi s obse e ha we ha e he local e sion o (3.46):
Π∗
h(Wh·∇ h+ϕ ⊥
h)τ,Oi≤CWh·∇ h+ϕ ⊥
hτ,Oi,
because h∈U
h(Oi). Now, we expand he e ms Wh·∇ hτ,Oiand ϕ ⊥
hτ,Oi:
Wh·∇ h2
τ,Oi=
K∈Oi
τKWh·∇ h2
0,K
≤α2
K∈Oi
h2
KWh2
0,K ∇ h2
0,∞,K ≤C|w|2
1,Ω 
K∈Oi| h|2
1,3,K ,
applying (3.15), (3.11), he local in e se inequali y (2.4) be ween W1,∞(K)andW1,3(K)and(2.1). Nex ,
ϕ ⊥
h2
τ,Oi=
K∈Oi
τKϕ ⊥
h2
0,K
≤α2
K∈Oi
h2
Kϕ2
0,∞,Ω  h2
0,K ≤Cϕ2
0,∞,Ω h2
K∈Oi| h|2
1,3,K .
Bu

K∈Oi| h|2
1,3,K 1
2
≤C| h|1,3,Oi,
because he numbe o elemen s in Oiis bounded by a cons an independen o hand i.
1236 T. CHAC ´
ON REBOLLO ET AL.
Then,
(Π∗
hWh·∇ h+ϕ ⊥
h,Π∗
Uh(ch))τ,Oi≤C(1 + |w|1,Ω)| h|1,3,OiΠ∗
Uh(ch)τ,Oi.
F om he e, applying Jensen’s inequali y,
⎡
⎣
R

i=1
sup
h∈Uh(Oi)|(Π∗
h(Wh·∇ h+ϕ ⊥
h),Π∗
Uh(ch))τ,Oi|
| h|1,3,Oi3
2⎤
⎦
2
3
≤C(1 + |w|1,Ω)R

i=1 Π∗
Uh(ch)2
τ,Oi
1
2
≤C(1 + |w|1,Ω)Π∗
Uh(ch)τ,
because he numbe o epe i ions o a gi en elemen Kin all mac o-elemen s is bounded by a fixed cons an
independen o h. Hence, by (3.49),
⎛
⎝
R

i=1
sup
h∈Uh(Oi)|(Π∗
h(Wh·∇ h+ϕ ⊥
h),Π∗
Uh(ch))τ,Oi|
| h|1,3,Oi3
2⎞
⎠
2
3
≤C1
√μ(1 + |w|1,Ω) H−1
b(Ω)+gH−1
2(Γs).
We p oceed in a simila way o he emaining e ms and ob ain
R

i=1
sup
h∈Uh(Oi)|(∇x· h,p
h)Oi|
| h|1,3,Oi3
2
2
3
≤C(1 + 1
μ+1
√μ)(1+|w|1,Ω) H−1
b(Ω)+gH−1
2(Γs).
(3.51)
I emains o es ima e he hi d summand in (3.39). Fi s , we w i e:
Π∗
h(∇xph)h, 3
2≤Π∗
h(ch)h, 3
2+Π∗
h(Wh·∇uh)h, 3
2+Π∗
h(ϕu⊥
h)h, 3
2·(3.52)
Again we a gue locally o bound he fi s e m in he .h.s. o (3.52). By defini ion,
Π∗
h(ch)h, 3
2=R

i=1
h
3
2
iΠ∗
Uh(ch))
3
2
0,3
2,Oi
2
3
.
To bound his e m, i is con enien o use he unc ion
H(x)= 
K∈Th
hKχK(x),
whe e χKis he cha ac e i ic unc ion o K.Then(2.2) implies ha
Π∗
h(ch)h, 3
2≤CR

i=1 HΠ∗
h(ch)
3
2
0,3
2,Oi
2
3
.
Since any elemen K∈T
hbelongs o a mos Mmac o-elemen s, his yields
Π∗
h(ch)h, 3
2≤C
K∈ThHΠ∗
h(ch)
3
2
0,3
2,K 2
3
.
A REDUCED DISCRETE INF-SUP CONDITION IN LPFOR INCOMPRESSIBLE FLOWS AND APPLICATION 1237
Finally, we associa e wi h τk he analogue o H:
T(x)= 
K∈Th
τKχK(x).
Then we in e om (3.15) and Cauchy−Schwa z’s inequali y:
Π∗
h(ch)h, 3
2≤C
K∈Th
√TΠ∗
h(ch)
3
2
0,3
2,K 2
3
=C√TΠ∗
h(ch)0,3
2,Ω
≤C√TΠ∗
h(ch)0,2,Ω =CΠ∗
h(ch)τ,(3.53)
which is bounded by (3.49).
To es ima e he second e m in he .h.s. o (3.52), we use fi s he local s abili y o Πh:
Π∗
h(Wh·∇uh)
3
2
h, 3
2≤C
R

i=1
h
3
2
i
K∈OiWh·∇uh
3
2
0,3
2,wK.
Then we use he local quasi-uni o mi y o he mesh and he local in e se inequali y (2.4) be ween W1,6(K)and
W1,2(K),
∇uh0,6,K ≤Cρ
−1
K∇uh0,K.
The e o e
Π∗
h(Wh·∇uh)
3
2
h, 3
2≤C
K∈Th
h
3
2
KWh·∇uh
3
2
0,3
2,wK≤C
K∈Th
h
3
2
K1
ρKWh0,wk∇uh0,wK3
2
≤C
K∈ThWh
3
2
0,wk∇uh
3
2
0,wK.
Hence,
Π∗
h(Wh·∇uh)h, 3
2≤C
K∈Th
Wh
3
2
0,K∇uh
3
2
0,K2
3
≤C
K∈Th
Wh6
0,K1
6
|uh|1,Ω
≤CWh0,Ω|uh|1,Ω ≤C|w|1,Ω |uh|1,Ω,(3.54)
by ano he applica ion o Jensen’s inequali y and (3.11). Simila ly, we bound he las e m in he .h.s. o (3.52)
using again (3.45)andob ain
Π∗
h(ϕu⊥
h)h, 3
2≤Chϕ0,∞,Ω |uh|1,Ω.(3.55)
Then, om (3.52) and aking in o accoun (3.53)–(3.55):
Π∗
h(∇xph)h, 3
2≤1
μ(1 + |w|1,Ω)+ 1
√μ H−1
b(Ω)+gH−1
2(Γs).(3.56)
Finally, we de i e (3.48) om(3.39) by combining (3.35), (3.51)and(3.56). The es o he p oo ollows as in
Theo em 3.4.
Rema k 3.8. When l>1in(3.7), Theo ems 3.4 and 3.7 also hold i he cons an α2in (3.15) is small enough.
In hiscase,weha ealso obound he e mΔuhτ.(See[14] o de ails).
Acknowledgemen s. This esea ch was pa ially suppo ed by he Spanish Minis e io de Educaci´on y Ciencia Resea ch
and EU Fede Fund P ojec MTM2012 36124-C02-1.
1238 T. CHAC ´
ON REBOLLO ET AL.
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