SIAM J. NUMER. ANAL.c
2005 Socie y o Indus ial and Applied Ma hema ics
Vol. 43, No. 3, pp. 1091–1111
THE POSTPROCESSED MIXED FINITE-ELEMENT METHOD
FOR THE NAVIER–STOKES EQUATIONS∗
BLANCA AYUSO†, BOSCO GARC´
IA-ARCHILLA‡,AND JULIA NOVO†
Abs ac . A pos p ocessing echnique o mixed fini e-elemen me hods o he incomp essible
Na ie –S okes equa ions is s udied. The echnique was ea lie de eloped o spec al and s anda d
fini e-elemen me hods o dissipa i e pa ial diffe en ial equa ions. The pos p ocessing amoun s o
sol ing a S okes p oblem on a fine g id (o highe -o de space) once he ime in eg a ion on he
coa se mesh is comple ed. The analysis p esen ed he e shows ha his echnique inc eases he
con e gence a e o bo h he eloci y and he p essu e app oxima ions. Nume ical expe imen s a e
p esen ed ha confi m bo h his inc ease in he con e gence a e and he co esponding imp o emen
in compu a ional efficiency.
Key wo ds. Na ie –S okes equa ions, mixed fini e-elemen me hods
AMS subjec classifica ions. 65M60, 65M20, 65M15, 65M12
DOI. 10.1137/040602821
1. In oduc ion. This pape in a sense culmina es he de elopmen o a pos -
p ocessing echnique o inc ease he accu acy and compu a ional efficiency o Gale kin
me hods o dissipa i e pa ial diffe en ial equa ions in oduced in [18]. We u n o
he equa ions which ga e ise o his pos p ocessing echnique, he incomp essible
Na ie –S okes equa ions, and we add ess hose Gale kin me hods o hese equa ions
which, when complex-shaped bodies a e p esen , a e acknowledged o be o wide
applicabili y, mixed fini e-elemen (MFE) me hods.
The pos p ocessing echnique we s udy he e was o iginally de eloped o spec-
al me hods [18], [19]. A ha momen , ei he i s analysis and unde s anding o
i s de elopmen seemed o depend hea ily on he p ope ies o he Fou ie modes,
al hough his was no a sho coming o p o e i s use ulness in he s udy o nonlinea
shell ib a ions [27]. In la e wo ks [13], [14], he dependence on he Fou ie modes
was o e come. O pa icula impo ance o he p esen wo k, besides [14], has been
he de elopmen o he pos p ocessing echnique o fini e-elemen me hods in [20],
[15]. In [20], i was de ised how o ca y ou he pos p ocessing wi hou he help o
an app oxima e ine ial mani old [11], [12], a concep mo e sui ed o spec al me hods
and eigen unc ion expansion. In [15], i is shown wha gains can be expec ed when
pos p ocessing low-o de elemen s.
As is usually he case wi h MFE me hods, i is he expe ience and unde s anding
gained in p e ious wo ks (see [14], [15], [16], [17], [18], [19], [20], and he e e ences
ci ed he ein) wi h simple equa ions and me hods which has allowed he p esen one
o be w i en. Fu he mo e, al hough o simplici y we ocus on Hood–Taylo [26]
elemen s, he pos p ocessing echnique can be easily adap ed o o he kinds o mixed
∗Recei ed by he edi o s Janua y 2, 2004; accep ed o publica ion (in e ised o m) Ma ch 18,
2005; published elec onically Sep embe 23, 2005.
h p://www.siam.o g/jou nals/sinum/43-3/60282.h ml
†Depa amen o de Ma em´a icas, Uni e sidad Au ´onoma de Mad id, Mad id, Spain (blanca.
a[email p o ec ed], julia.no[email p o ec ed]). The esea ch o he fi s au ho was suppo ed by p ojec HPRN-
CT-2002-00284. The esea ch o he hi d au ho was suppo ed by DGI-MCYT unde p ojec
MTM2004-02847 (cofinanced by FEDER unds) and by JCYL unde p ojec VA044/03.
‡Depa amen o de Ma em´a ica Aplicada II, Uni e sidad de Se illa, Se illa, Spain (bosco@
ma ina.us.es). This au ho ’s esea ch was suppo ed by DGICYT p ojec BFM2003-00336.
1091
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1092 BLANCA AYUSO, BOSCO GARC´
IA-ARCHILLA, AND JULIA NOVO
elemen s. In ac , in [3] (see also [5]) he so-called mini-elemen is shown o ende
simila gains as Hood–Taylo elemen s when pos p ocessed i he p o isions in [15]
a e aken in o accoun .
Le us desc ibe wha his pos p ocessing echnique is. We conside he incom-
p essible Na ie –S okes equa ions, which, in app op ia e dimensionless a iables, can
be w i en as
u −νΔu+(u·∇)u+∇p= ,(1.1)
di (u)=0
in a bounded domain Ω ⊂Rd(d=2,3) wi h smoo h bounda y subjec o homoge-
neous Di ichle bounda y condi ions u=0on∂Ω. In (1.1), uis he eloci y field,
p he p essu e, and a gi en o ce field. Suppose ha o he solu ion uand p
co esponding o a gi en ini ial condi ion
u(·,0) = u0;(1.2)
we a e in e es ed in i s alue a a ce ain ime T>0. We fi s compu e MFE
app oxima ions uhand ph o he eloci y and p essu e, espec i ely, by in eg a ing in
ime he co esponding disc e iza ion o (1.1)–(1.2) om =0 o =T. Then, in he
pos p ocessing s ep, we ob ain an app oxima ion o he solu ion ˜u,˜po he S okes
p oblem
−νΔ˜u+∇˜p= −d
d uh(T)−(uh(T)·∇)uh(T)
di (˜u)=0
in Ω,
˜u=0 on∂Ω.
(1.3)
The MFE o his las s ep is ei he he same-o de Hood–Taylo elemen o e a fine
g id o a highe -o de Hood–Taylo elemen o e he same g id. The a e o con e -
gence o he disc e e eloci y and p essu e in he esul ing me hod is p o ed o be
he same as he a e o con e gence o he MFE used in he pos p ocessed s ep. The
o e cos o he pos p ocessed p ocedu e is nea ly negligible since he S okes p oblem
using he enhanced MFE is sol ed only once, when he ime in eg a ion has been
comple ed. In his espec , i adically diffe s om some o he esea ch [2], [32], wi h
low-o de MFEs o he Na ie –S okes equa ions ha also de eloped om he ideas
in [11] and [12], since in [2] and [32] compu a ions wi h he enhanced elemen o on
he fine g id a e ca ied ou all he way h ough he in e al (0,T].
Some supe con e gence esul s a e ob ained in he pape and a e used as a ool
o ge he a e o con e gence o he pos p ocessed me hod. In pa icula , we de i e a
supe con e gence esul o he e o be ween he MFE app oxima ion o he eloci y
and he disc e e S okes p ojec ion in oduced in [24]. Fo simplici y o analysis, we
de i e hese esul s unde he s ong egula i y hypo heses in (2.2), which, as poin ed
ou in [24], a e un ealis ic in p ac ical si ua ions. In a mo e p ac ical se ing, assump-
ions (2.2) should be assumed om some posi i e ime 0>0 onwa ds, and, as we
commen in sec ion 2, compu a ions (and hei analysis) up o his ime should ake
in o accoun he lowe egula i y a =0.
Finally, we ema k ha ecen esea ch [16], [17] has shown he use ulness o he
pos p ocessing echnique in ob aining efficien a pos e io i e o es ima o s in pa ial
diffe en ial equa ions o e olu ion, a field much less de eloped han in he case o
s eady p oblems. The applica ion o he pos p ocessing echnique o ge a pos e io i
e o es ima es o Na ie –S okes equa ions using he esul s ob ained in his pape
will be he subjec o u u e wo k.
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POSTPROCESSED MFE METHOD FOR NAVIER–STOKES EQUATIONS 1093
The es o he pape is as ollows. In sec ion 2 we ecall some p ope ies o MFE
me hods and collec some inequali ies o be used la e . In sec ion 3 we fi s speci y
he pos p ocessing echnique and hen ca y ou he con e gence analysis. Finally,
in sec ion 4 nume ical expe imen s a e p esen ed o assess he capabili ies o he new
echnique.
2. P elimina ies and no a ions. Le Ω be a bounded domain in Rd,d=2,3,
no necessa ily con ex, bu o class Cm,m≥3, and le Hand Vbe he Hilbe spaces
H={u∈L2(Ω))d,|di (u)=0,u·n|∂Ω=0},V={u∈H1
0(Ω))d,|di (u)=0},
endowed wi h he inne p oduc o L2(Ω)dand H1
0(Ω)d, espec i ely. Fo 1 ≤q≤∞
and l≥0, we conside he s anda d Sobole spaces, Wl,q(Ω)d, o unc ions wi h
de i a i es up o o de lin Lq(Ω), and Hl(Ω)d=Wl,2(Ω)d. The no m in Hl(Ω)dwill
be deno ed by ·lwhile ·−lwill ep esen he no m o i s dual space. We conside
also he quo ien spaces Hl(Ω)/Rwi h no m pHl/R= in {p+cl|c∈R}.
We shall equen ly use he ollowing Sobole ’s imbeddings [1]. The e exis s a
cons an C=C(Ω,q) such ha o q∈[1,∞),q
<∞, i holds ha
 Lq(Ω)d≤C Ws,q(Ω)d,1
q≥1
q≥1
q−s
d>0, ∈Ws,q(Ω)d.(2.1)
Fo q=∞, (2.1) holds wi h 1
q<s
d.
Le Π : L2(Ω)d−→ Hbe he Le ay p ojec o ha maps each unc ion in L2(Ω)d
on o i s di e gence- ee pa . We deno e by A he S okes ope a o in Ω:
A:D(A)⊂H−→ H, A=−ΠΔ,D(A)=H2(Ω)d∩V.
Applying Le ay’s p ojec o o (1.1), he equa ions can be w i en in he o m
u +νAu+B(u, u)=Π in Ω,
whe e B(u, u) = Π((u·∇)u).
In wha ollows we will assume ha he solu ion (u, p) o (1.1)–(1.2) sa isfies
max
0≤ ≤Tu( ) +p( )H −1/R<∞,max
0≤ ≤Tu ( ) +p ( )H −1/R<∞.(2.2)
We e e he eade o [30] o a s udy abou he egula i y o he solu ions o he
Na ie –S okes equa ions. No ice, howe e , ha , as poin ed ou in [24], i is un ealis ic
o assume such a s ong egula i y up o ime = 0. The assump ion in (2.2) is o
simplici y in he analysis. In a mo e ealis ic se ing, = 0 should be eplaced by
some posi i e ime 0, and e o bounds equi ing less egula i y such as hose in [24]
and [25] should be conside ed om =0 o = 0. In o de o main ain he accu acy
le els ha a highe egula i y would allow om 0onwa ds, compu a ions up o = 0
should be ca ied ou on an adequa e fine g id. No ice also ha among he condi ions
o ensu e (2.2) (see, e.g., Theo em 4 in [23]) is ha Ω is o class C .
Le Th=(τh
i,φ
h
i)i∈Ih,h>0, be a amily o pa i ions o sui able domains Ωh,
whe e he pa ame e his he maximum diame e o he elemen s τh
i∈T
hand φh
i
a e he mappings o he e e ence simplex τ0on o τh
i. We es ic ou sel es o quasi-
uni o m and egula meshes Th.
Le ≥2, we conside he fini e-elemen spaces

Sh, =χh∈C0(Ωh)|χh|τh
i◦φh
i∈P −1(τ0)⊂H1(Ωh),
◦
Sh, =χh∈C0(Ωh)|χh|τh
i◦φh
i∈P −1(τ0),χ
h(x)=0∀x∈∂Ωh⊂H1
0(Ωh),
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1094 BLANCA AYUSO, BOSCO GARC´
IA-ARCHILLA, AND JULIA NOVO
whe e P −1(τ0) deno es he space o polynomials o deg ee a mos −1onτ0.Asa
consequence o es ic ing ou s udy o quasi-uni o m pa i ions, he ollowing in e se
inequali y holds (see, e.g., [9, Theo em 3.2.6]) ∀τ=τh
i∈T
h, wi h diam(τ)=hτ≤h,
h∈(◦
Sh, )d:
(2.3)
 hWm,q(τ)d≤Chl−m−d(1
q−1
q) hWl,q(τ)d,0≤l≤m≤2,1≤q≤q≤∞.
In o de o gua an ee con e gence o he MFE app oxima ion, we choose a s able
combina ion o wo fini e-elemen spaces (see [7]). We in oduce he fini e-elemen
spaces in which ou MFE app oxima ion o (u, p) will be ca ied ou . We shall deno e
by (Xh, ,Q
h, −1) he so-called Hood–Taylo elemen , whe e
Xh, =◦
Sh, d,Q
h, −1=
Sh, −1∩L2(Ωh)/R, ≥3.
Fo his mixed elemen a uni o m in -sup condi ion is sa isfied (see [26], [6]), ha is,
he e exis s a cons an β>0 independen o he mesh g id size hsuch ha
in
qh∈Qh, −1
sup
h∈Xh,
(qh,∇· h)
 h1qhL2/R≥β.(2.4)
The app oxima e eloci y solu ion belongs o he disc e ely di e gence- ee space
Vh, =Xh, ∩χh∈H1
0(Ωh): Ωh
qhdi (χh)=0 ∀qh∈Qh, −1.
We obse e ha o he Hood–Taylo elemen , Vh, is no a subspace o V.
Fo any ∈C0(Ω)d, we conside he s anda d in e polan ope a o Ih:C0(Ω)d−→
Xh, . Le ∈H (Ω)d∩H1
0(Ω)d; i is well known ha Ihsa isfies
 −Ih( )L2(Ω∩Ωh)d+h −Ih( )H1(Ω∩Ωh)d≤Ch  H (Ω)d.(2.5)
We b iefly discuss nex unde wha ci cums ances (2.5) can be ex ended o a global
es ima e (i.e., o an es ima e in Ω and no jus in Ω∩Ωh). The in e pola ion ope a o
Ih( ) is ex ended by ze o in Ω Ωh, and defining δ(h) = maxx∈∂Ωhdis (x, ∂Ω), one
ob ains
 −Ih( )L2(Ω)d+h −Ih( )H1(Ω∩Ωh)d≤C(h +δ(h)) H (Ω)d.(2.6)
Fo x∈Ω∩Ωh, (2.5) (and so (2.6)) ollows om s anda d heo y o in e pola ion
and he B amble–Hilbe lemma (see, e.g., [9, p. 192]). Fo x∈Ω Ωh, (x) can be
bounded by means o he mean- alue heo em,
 −Ih( )L2(Ω Ωh)d= L2(Ω Ωh)d≤δ(h)∇ L2(Ω)d.
We obse e ha using isopa ame ic elemen s δ(h)≤Ch , and so in (2.6) he igh -
hand side is u he bounded by Ch  W ,q (Ω)d(see [9, sec ion 4.4]). As ega ds he
global es ima e o he g adien , isopa ame ic modifica ion is no enough o p ese e
he op imal app oximabili y p ope ies o he fini e-elemen space. Following [3], we
shall assume in wha ollows he use o supe pa ame ic elemen s a he bounda y. By
his ype o app oxima ion we mean ha δ(h)≤Ch2 −2so ha he ou side effec s will
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POSTPROCESSED MFE METHOD FOR NAVIER–STOKES EQUATIONS 1095
no pollu e he op imal es ima e. Unde hese assump ions [3], [4], he in e polan Ih
sa isfies
 −Ih( )L2(Ω)d+h −Ih( )H1(Ω)d≤Ch  H (Ω)d.(2.7)
No ice hen ha he condi ion δ(h)≤Ch2 −2allows us o o ge abou he disc ep-
ancies be ween Ω and Ωhin mos o he a gumen s ha ollow. Obse e, howe e ,
ha one mus hen assume ha Ω is piecewise o class C2 −2.
Fo each fixed ime ∈[0,T] he solu ion (u, p) o (1.1)–(1.2) is also he solu ion
o a S okes p oblem wi h igh -hand side −u −(u·∇)u. We will deno e by
(sh,q
h)∈(Xh, ,Q
h, −1),i s MFE app oxima ion sa is ying
ν(∇sh,∇φh)−(qh,∇·φh)=ν(∇u, ∇φh)−(p, ∇·φh)
=( −u −(u·∇u),φ
h)∀φh∈Xh, ,(2.8)
(∇·sh,ψ
h)=0 ∀ψh∈Qh, −1.
We obse e ha sh=Sh(u):V−→ Vh, is he so-called disc e e S okes p ojec ion o
he solu ion (u, p) o (1.1)–(1.2) (see [24]) and sa isfies
(∇Sh(u),∇χh)=(∇u, ∇χh)−(p, ∇·χh)=( −u −(u·∇)u, χh)∀χh∈Vh, .
The ollowing bound holds o 2 ≤l≤ :
u−sh0+hu−sh1≤Chlul+pHl−1/R.(2.9)
The p oo o (2.9) o Ω = Ωhcan be ound in [25]. Fo he gene al case supe -
pa ame ic app oxima ion a he bounda y is assumed; see [3], [4]. Unde he same
condi ions, he bound o he p essu e is [21]
p−qhL2/R≤Cβhl−1ul+pHl−1/R,(2.10)
whe e he cons an Cβdepends on he cons an βin he in -sup condi ion (2.4).
Since we a e assuming ha Ω is o class Cmwi h m≥3 (and ha δ(h)≤Ch2 −2)
using s anda d duali y a gumen s and (2.9), one ob ains [3], [4]
u−sh−s≤Ch +s(u +pH −1/R),0≤s≤min( −2,1).(2.11)
Le Πh, :L2(Ω)d−→ Vh, be he disc e e Le ay’s p ojec ion defined by demand-
ing ha (Πh, (u),χ
h)=(u, χh)∀χh∈Vh, . By defini ion, he p ojec ion is s able in
he L2no m. Fo di e gence- ee unc ions, by w i ing Πh, u=(Π
h, u−Sh(u))+Sh(u)
and using he quasi-uni o mi y o he meshes, one easily shows ha
Πh, u1≤Cu1∀u∈V.(2.12)
We will deno e by Ah he disc e e S okes ope a o defined by
(∇ h,∇φh)=(Ah h,φ
h)=A1/2
h h,A1/2
hφh∀ h,φ
h∈Vh, .
Since Ahis a disc e e sel -adjoin ope a o , i is easy o show ha , o each 0 ≤α<1,
he e exis s a posi i e cons an Cα, which is independen o h, such ha
Aα
he− Ah0≤Cα −α∀0≤α<1.(2.13)
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1096 BLANCA AYUSO, BOSCO GARC´
IA-ARCHILLA, AND JULIA NOVO
In ou analysis we shall equen ly use he ollowing ela ions o ∈L2(Ω)d:
A−s/2
hΠh, 0≤Chs 0+A−s/2Π 0,s=1,2,(2.14)
A−s/2Π 0≤Chs 0+A−s/2
hΠh, 0,s=1,2.(2.15)
These inequali ies a e eadily deduced om he es ima es A−s/2−A
−s/2
hΠh, 0≤
Chs o s=1,2 [29]. Simila ly, since ∀ h∈Vh, ,(A−1/2
hΠh, , h)=( ,A−1/2
h h), i
ollows ha
A−1/2
hΠh, 0≤C −1,(2.16)
and since ∀ ∈V, we ha e (A−1/2Π(Πh, ), )=(Π
h, ,A−1/2 )=( ,Πh, A−1/2 ),
om (2.12) i ollows ha
A−1/2Π(Πh, )0≤C −1, ∈L2(Ω)2.(2.17)
2.1. The sugges ed me hod. Le us suppose ha we wan o app oxima e he
solu ion o (1.1)–(1.2) a ime T.Fo d= 3, he final ime Tis assumed o sa is y
0<T<T
∗, whe e T∗is he c i ical ime un il which he exis ence and uniqueness
o a s ong solu ion o (1.1)–(1.2) has been p o en. The pos p ocessing echnique
can be seen as a wo-le el me hod. We fi s compu e he MFE app oxima ion o
(1.1)–(1.2) a ime T. Gi en uh(0) an ini ial app oxima ion o u(0), we find ha
uh:[0,T]−→ Xh, and ph:[0,T]−→ Qh, −1sa is y
(˙uh,φ
h)+ν(∇uh,∇φh)+bh(uh,u
h,φ
h)+(∇ph,φ
h)=( ,φh)∀φh∈Xh, ,(2.18)
(∇·uh,ψ
h)=0 ∀ψh∈Qh, −1,(2.19)
whe e bh(·,·,·) is a sui able disc e e app oxima ion o i s con inuous coun e pa . As
an ini ial condi ion we will ake uh(0) = Sh(u0), al hough o he choices a e possible.
In he second s ep, he disc e e eloci y and p essu e (uh(T),p
h(T)) a e pos p o-
cessed. Basically, we enhance his app oxima ion by sol ing a single disc e e S okes
p oblem, ia MFE. The MFE in his s ep, deno ed by ( 
X, 
Q), is ei he
• he same-o de Hood–Taylo elemen o e a fine g id ( 
X, 
Q)=(X˜
h, ,Q
˜
h, −1),
≥3,˜
h<h,o
•a highe -o de Hood–Taylo elemen o e he same g id ( 
X, 
Q)=(X˜
h, +1,Q
˜
h, ),
≥3, ˜
h=h.
Tha is, we shall sea ch o (˜uh,˜ph)∈(
X, 
Q) sa is ying
ν∇˜u˜
h,∇˜
φ+∇˜p˜
h,˜
φ=( , ˜
φ)−b˜
h(uh(T),u
h(T),˜
φ)−(˙uh(T),˜
φ)∀˜
φ∈
X,(2.20) ∇·˜u˜
h,˜
ψ=0 ∀˜
ψ∈
Q.(2.21)
We will deno e by 
V he co esponding disc e ely di e gence- ee space ha can be
ei he 
V=V˜
h, o 
V=Vh, +1 depending on he selec ion o he pos p ocessed space.
The disc e e Le ay’s p ojec ion in o 
Vwill be deno ed by 
Π˜
h, and we will ep esen
by 
A˜
h he disc e e S okes ope a o ac ing on unc ions in 
V.
The pos p ocessed Hood–Taylo app oxima ion o he eloci y, ˜u˜
h, is he solu ion
o he p essu e- ee o mula ion
ν∇˜u˜
h,∇˜χh= , ˜χh−b˜
huh(T),u
h(T),˜χh−˙uh(T),˜χh∀˜χh∈
V.(2.22)
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POSTPROCESSED MFE METHOD FOR NAVIER–STOKES EQUATIONS 1097
In he nex sec ion, we show ha he solu ion (˜uh,˜ph) o (2.20)–(2.21) is a mo e
accu a e app oxima ion o he solu ion o (1.1)–(1.2) han he Gale kin MFE app ox-
ima ion (uh,p
h) ha sol es (2.18)–(2.19).
Fo he disc e e app oxima ion o he nonlinea e m, ollowing [24], we define bh
in he ollowing way:
bh(uh,
h,φ
h)=((uh·∇) h,φ
h)+1
2(di (uh) h,φ
h)∀uh,
h,φ
h∈Xh, ⊂H1
0(Ω)d.
Fo all u, ∈H1
0(Ω)d, he co esponding con inuous ope a o will be deno ed by
F(u, )=(u·∇) +(1/2) di (u) . Ex ending he defini ion o bh o unc ions in
H1
0(Ω)d(no necessa ily in Xh, ), we obse e ha ∀u, , w ∈H1
0(Ω)d,b
h(u, , w)=
(F(u, ),w).I is s aigh o wa d o e i y ha bhenjoys he skew-symme y p ope y
bh(u, , w)=−bh(u, w, )∀u, , w ∈H1
0(Ω)d.(2.23)
Le us obse e ha B(u, )=ΠF(u, )i u∈V. Finally, we shall deno e by
Bh(u, )=Π
h, F(u, )∀u, ∈H1
0(Ω)d.
3. Analysis o he pos p ocessed me hod. This sec ion is de o ed o he
analysis o con e gence o he pos p ocessed MFE me hod. Ou fi s aim will be o
show a supe con e gence esul o he e o be ween he MFE app oxima ion o he
eloci y uhand he S okes p ojec ion o he eloci y field u,sh. This supe con e gence
beha io occu s o bo h he L2and H1no ms, as will be shown in Theo em 3.7 and
Co olla y 3.8, espec i ely. In he fi s pa o he sec ion, we shall concen a e ou
effo s in Theo em 3.7. I will be achie ed by a s abili y plus consis ency a gumen
(P oposi ions 3.2 and 3.6, espec i ely). Fo he pu pose o analysis, we shall mainly
be conce ned wi h he p essu e- ee o mula ion associa ed wi h (2.18)–(2.19). I
(uh,p
h) is he MFE app oxima ion o he solu ion (u, p) o (1.1)–(1.2), hen uh∈Vh,
is he solu ion o
(˙uh,χ
h)+ν(∇uh,∇χh)+bh(uh,u
h,χ
h)=( ,χh)∀χh∈Vh, ,(3.1)
which can also be exp essed in abs ac ope a o o m as
˙uh+νAhuh+Bh(uh,u
h)=Π
h, .(3.2)
The S okes p ojec ion shsa isfies he abs ac equa ion
˙sh+νAhsh+Bh(sh,s
h)=Π
h, +Th,(3.3)
whe e Th( ) is he unca ion e o , defined as
Th( )= ˙sh−Πh, (u )+Bh(sh,s
h)−Bh(u, u).(3.4)
Le us now conside mappings h:[0,T]−→ Vh, sa is ying he ollowing h eshold
condi ion:
sh( )− h( )0≤cτh2∀ ∈[0,
1],0<
1≤T.(3.5)
We define hei unca ion e o as

Th=˙ h+νAh h+Bh( h,
h)−Πh, .(3.6)
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1098 BLANCA AYUSO, BOSCO GARC´
IA-ARCHILLA, AND JULIA NOVO
P io o es ablishing he s abili y es ic ed o he h eshold (3.5) (P oposi ion 3.2),
we p o e a lemma which p o ides some es ima es o he con ec i e e m.
Lemma 3.1. Le (u, p)be he solu ion o he Na ie –S okes p oblem (1.1)–(1.2).
Le sh=Sh(u)be he disc e e S okes p ojec ion o he eloci y field uand le h:
[0,T]−→ Vh, sa is y he h eshold condi ion (3.5). Then, he e exis s a cons an
K>0, independen o 1in (3.5), such ha ∀ ∈[0,
1],
F(sh( ),s
h( )) −F( h( ),
h( ))0≤Ksh( )− h( )1,(3.7)
F(sh( ),s
h( )) −F( h( ),
h( ))]−1≤Ksh( )− h( )0,(3.8)
whe e he cons an K=Kcτ,max0≤ ≤T(u( )2+p( )H1/R).
P oo . In o de o simpli y he no a ion, we shall omi he dependence on in he
p oo . Deno e by eh= h−sh. We p oceed by s anda d duali y a gumen s, using he
spli ing
F( h,
h)−F(sh,s
h)=F( h,e
h)+F(eh,s
h).(3.9)
We s a by showing (3.7). We fi s obse e ha
F(eh,s
h)0= sup
φ0=1 (eh·∇sh,φ)+1
2((∇·eh)sh,φ)
≤CehL2d/(d−1)(Ω)d∇shL2d(Ω)d+Ceh1sh∞.
Le us show ha bo h sh∞,∇shL2d(Ω)da e bounded. Since, by i ue o Sobole ’s
imbeddings (2.1), we ha e sh∞≤C∇shL2d(Ω)d, we only need o bound he second
e m. Applica ion o he in e se inequali y (2.3) and he e o es ima es (2.9) and
(2.7) oge he wi h (2.1) gi e
∇shL2d(Ω)d≤Ch
−(1+d)
2(sh−u0+u−Ihu0)+∇IhuL2d(Ω)d
(3.10)
≤Ch(3−d)/2(u2+pH1/R)+CuW1,2d(Ω)d≤K.
Using again (2.1) we ob ain
ehL2d/(d−1)(Ω)d≤Ceh1/2≤Ceh1,
and so F(eh,s
h)0≤Keh1.As ega ds he o he e m in (3.9), he same a gumen s
lead o
F( h,e
h)0= sup
φ0=1 ( h·∇eh,φ)+1
2((∇· h)eh,φ)
≤C h∞eh1+C∇ hL2d(Ω)dehL2d/(d−1)(Ω)d.
As be o e, o conclude we mus show ha he abo e no ms o ha e bounded. We
only need o handle ∇ hL2d(Ω)d. Using he in e se inequali y (2.3) and he h eshold
condi ions (3.5) and (3.10), we find
∇ hL2d(Ω)d≤h
−(1+d)
2 h−sh0+∇shL2d(Ω)d≤cτh(3−d)/2+K≤K.
The e o e, (3.7) ollows. We now show (3.8). Applying (3.9), we find
F( h,
h)−F(sh,s
h)−1≤F( h,e
h)−1+F(eh,s
h)−1,(3.11)
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POSTPROCESSED MFE METHOD FOR NAVIER–STOKES EQUATIONS 1099
so ha he p oo is educed o es ima e each o he abo e nega i e no ms on he
igh -hand side. Using he skew-symme y p ope y (2.23), one ge s o he fi s
e m:
F( h,e
h)−1= sup
φ1=1 −(( h·∇)φ, eh)−1
2((∇· h)φ, eh)
≤sup
φ1=1 eh0 h∞φ1+eh0∇· hL2d/(d−1) φL2d(Ω)d)≤Keh0.
Rega ding he o he e m in (3.11), in eg a ing by pa s, we ob ain
F(eh,s
h)−1= sup
φ1=1 
1
2(eh·∇)sh,φ
−1
2(eh·∇)φ, sh
≤sup
φ1=1
(eh0∇shL2d/(d−1)(Ω)dφL2d(Ω)d+eh0φ1sh∞)≤Keh0.
This finishes he p oo o (3.8).
P oposi ion 3.2 (s abili y). Le T>0be fixed; le sh=Sh(u)be he disc e e
S okes p ojec ion o he eloci y field usolu ion o (1.1)–(1.2) and le h:[0,T]−→
Vh, sa is y he h eshold condi ion (3.5). Then, he e exis s a posi i e cons an Ks>0
such ha ∀ 1≤T, he ollowing es ima e holds:
max
0≤ ≤ 1sh( )− h( )0≤eKs 1sh(0) − h(0)0
(3.12)
+ max
0≤ ≤ 1



0
e−ν( −s)Ah[Th(s)−
Th(s)]ds


0,
whe e Th(s)and 
Th(s)a e he unca ion e o s gi en in (3.4) and (3.6), espec i ely.
P oo . We deno e by eh=sh− h. Sub ac ing (3.6) om (3.3), i ollows ha
ehsa isfies he e o equa ion
˙eh( )+νAheh( )=Bh( h( ),
h( )) −Bh(sh( ),s
h( )) + Th( )−
Th( ).
Then, by in eg a ing he abo e e o equa ion om ime 0 up o ime , we find ha
eh( )=e−ν AhΠh, eh(0) + 
0
e−ν( −s)AhΠh, [Bh( h,
h)−Bh(sh,s
h)]ds
+
0
e−ν( −s)AhΠh, [Th(s)−
Th(s)]ds.
Since {e−ν AhΠh, } >0is a con ac ion e−ν AhΠh, eh(0)0≤eh(0)0.As ega ds
he second e m, es ima es (2.13), (2.16), and (3.8) om Lemma 3.1 lead o




0
e−ν( −s)Ah[Bh(sh,s
h)−Bh( h,
h)]ds


0
≤C1/2
√ν
0
A−1/2
hΠh, F(sh,s
h)−Πh, F( h,
h)
0
√ −sds ≤KC1/2
√ν
0
eh(s)0
√ −sds.
Then,
eh( )0≤eh(0)0+KC1/2
√ν
0
eh(s)0
√ −sds +
0
e−ν( −s)Ah[Th(s)−
Th(s)]ds.
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1106 BLANCA AYUSO, BOSCO GARC´
IA-ARCHILLA, AND JULIA NOVO
Then, applying 
A−1
˜
h o bo h sides o he abo e equa ion, we ob ain
˜u˜
h−
S˜
h(u)0≤1
ν

A−1
˜
h
Π˜
h[F(u(T),u(T)) −F(uh(T),u
h(T))]
0
+

A−1
˜
h
Π˜
h[u (T)−˙uh(T)]
0.
Thus, ou aim is educed o es ima e each o he abo e no ms. As ega ds he nonlinea
e m, aking in o accoun (2.14), wi h s=2,wefind


A−1
˜
h
Π˜
h[F(u, u)−F(uh,u
h)]
0≤C˜
h2
F(u, u)−F(uh,u
h)
0
+
A−1Π[F(u, u)−F(uh,u
h)]
0.
Now, using es ima es (3.7) om Lemma 3.1 and (3.13) om Lemma 3.4, we ge

A−1
˜
h
Π˜
h[F(u, u)−F(uh,u
h)]
0≤C˜
h2u−uh1+C(u−uh−1+u−uh0u−uh1).
To conclude, we shall es ima e each e m in bo h sums. The equi ed es ima es in he
L2and H1no ms a e g an ed by Co olla y 3.9. As ega ds he es ima e in he H−1
no m, no e ha by means o (2.11) and (3.17), one eadily finds
u−uh−1≤u−sh−1+sh−uh−1≤u−sh−1+sh−uh0
≤Ch +1(u +pH −1/R)+Kh +1|log(h)|.
Then, we finally ge 

A−1
˜
h
Π˜
h[F(u, u)−F(uh,u
h)]
0≤Kh +1|log(h)|. We nex deal
wi h he es ima e o he ime de i a i e. Applying again (2.14) wi h s= 2 oge he
wi h es ima es (3.20) and (3.21) om Lemma 3.10, we each


A−1
˜
h
Π˜
h[u (T)−˙uh(T)]
0≤˜
h2u (T)−˙uh(T)0+A−1Π[u (T)−˙uh(T)]0
≤K˜
h2h −1|log(h)|+Kh +1|log(h)|≤Kh +1|log(h)|.
Hence he p oo o he L2no m is also finished.
Theo em 3.15. Le T>0be fixed. Le (uh,p
h)be he MFE app oxima ion o
he solu ion (u, p)o (1.1)–(1.2).Le (˜u˜
h,˜p˜
h)be he pos p ocessed MFE app oxima ion
a ime T. Then, he e exis s a cons an K(u, p, ν)such ha
(i) i he pos p ocessing elemen is (
X, 
Q)=(X˜
h, ,Q
˜
h, −1), hen
p(T)−˜p˜
hL2/R≤Cβ(˜
h) −1u(T) +p(T)H −1/R
(3.34)
+K(u, p, ν, β)h |log (h)|;
(ii) i a ime T he solu ion (u(T),p(T)) belongs o (H +1(Ω)d∩V)×H (Ω)/R,
and he pos p ocessing elemen is (
X, 
Q)=(Xh, +1,Q
h, ), hen
p(T)−˜p˜
hL2/R≤Cβh u(T) +1 +p(T)H /R+K(u, p, ν, β)h |log (h)|.(3.35)
P oo . Le us deno e by ˜q˜
h he MFE app oxima ion o he p essu e p(T) ob ained
by sol ing he S okes p oblem (2.8) a ime Tin he pos p ocessed space ( 
X, 
Q).
Adding and sub ac ing ˜q˜
h,wege
p(T)−˜p˜
hL2/R≤p(T)−˜q˜
hL2/R+˜q˜
h−˜p˜
hL2/R.
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POSTPROCESSED MFE METHOD FOR NAVIER–STOKES EQUATIONS 1107
The fi s e m can easily be es ima ed applying (2.10):
p(T)−˜q˜
hL2/R≤⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
Cβ(˜
h) −1u(T)
+p(T)H −1/R,
X, 
Q=(X˜
h, ,Q
˜
h, −1),
Cβh u(T) +1
+p(T)H /R,
X, 
Q=(Xh, +1,Q
˜
h, ).
Le us now bound he second e m. Using he equa ions ha sa is y ˜p˜
hand ˜q˜
h((2.20),
(2.8), espec i ely), we deduce
˜p˜
h−˜q˜
h,∇·˜
φ=ν∇˜u˜
h−
S˜
h(u),∇˜
φ+F(uh,u
h)
−F(u, u),˜
φ+˙uh−u ,˜
φ∀˜
φ∈
X.
Using he in -sup condi ion (2.4), we ob ain
β˜p˜
h−˜q˜
hL2/R≤ν˜u˜
h−
S˜
h1+F(uh,u
h)−F(u, u)−1+uh−u −1.
Taking in o accoun (3.33), (3.8) om Lemma 3.1, and (3.22) om Lemma 3.10, we
each
˜p˜
h−˜q˜
hL2/R≤1
β(Kh |log(h)|+u−uh0+Kh |log(h)|),
so ha , applying Co olla y 3.8, we ha e comple ed he p oo .
Rema k 3.1. Obse e ha o he eloci y we used piecewise polynomials o
deg ee a leas 2. In gene al, he pos p ocessed me hod does no inc ease he a e o
con e gence in he L2no m in he linea case al hough an imp o emen in he ene gy
no m is ob ained. The applica ion o he pos p ocessing echnique o he mini-elemen
app oxima ion o Na ie –S okes equa ions is s udied in [3], [5].
4. Nume ical expe imen s. In his sec ion, we p esen some nume ical expe -
imen s in o de o suppo he analysis de eloped in he pape and o assess he me i
o he pos p ocessed me hod when compa ed wi h he s anda d MFE me hod. We
conside he Na ie –S okes equa ions (1.1) o e he domain Ω = [0,1] ×[0,1] subjec
o homogeneous Di ichle bounda y condi ions. The alue o he iscosi y in he ex-
pe imen s is ν= 1, and he final ime is T=1.2. We se o ze o he ini ial eloci y
field u0(1.2) and choose he ex e nal o ce so ha he exac solu ion is
u1(x, y, )=−6·[1 −cos(π )]sin3(πx) sin2(πy) cos(πy),(x, y, )∈Ω×[0,T],
u2(x, y, )=6·1−cos(π )sin2(πx) sin3(πy) cos(πx),(x, y, )∈Ω×[0,T],
p(x, y, ) = (sin(2π )/2)sin4(πx) + sin3(πy)−p0,(x, y, )∈Ω×[0,T],
whe e p0deno es he mean o he p essu e. In spi e o he simplici y o his solu ion
and i s lack o physical meaning, we ema k ha ou main in e es has been o check
he imp o emen in he a e o con e gence achie ed wi h he pos p ocessing echnique
and whe he his also inc eases he efficiency o he s anda d MFE app oxima ion.
In ou calcula ions we ake he so-called egula pa e n iangula ions o Ω, which
a e induced by he se o nodes (i/N, j/N), 0 ≤i, j ≤N, whe e N=|Ω|/h is an
in ege . The MFE app oxima ion o (1.1)–(1.2) is ca ied ou using he Hood–Taylo
elemen (Xh,3,Q
h,2) ha we will deno e by P2P1. Tha is, we use Lag ange quad a ic
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1108 BLANCA AYUSO, BOSCO GARC´
IA-ARCHILLA, AND JULIA NOVO
10
1
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
N=|Ω|/h
L
2
−no m eloci y e o
slope=−3.7521
slope=−4.0632
slope=−3.0694
10
1
10
−5
10
−4
10
−3
10
−2
10
−1
N=|Ω|/h
H
1
−no m eloci y e o
slope=−2.9489
slope=−2.9671
slope=−1.9714
Fig. 4.1.Con e gence diag ams o he fi s componen o he eloci y wi h P2P1(con inuous
line), P3P2(dashed-do ed line), and he pos p ocessed me hod wi h P3P2(dashed line). On he le
he e o s a e measu ed in he L2no m (ci cles ◦) and on he igh in he H1no m (diamonds ♦).
elemen s o he app oxima ion o he eloci y and linea elemen s o app oxima e
he p essu e. Fo he pos p ocessing s ep, due o he smoo hness o he solu ion
(u, p), we pe o m he expe imen s no only wi h he same MFE o e a fine g id,
(Xh,3,Q
h,2), h<h, bu also wi h he highe -o de Hood–Taylo elemen o e he
same g id, (Xh,4,Q
h,3); i.e., Lag ange cubic o he eloci y and Lag ange quad a ic
o he p essu e. This elemen will be deno ed by P3P2.
Fo he ime in eg a ion we use he well-known semi-implici me hod whe e lin-
ea e ms a e app oxima ed by he implici midpoin ule (i.e., he C ank–Nicolson
me hod) and nonlinea e ms by he wo-s ep explici Adams o mula (see, e.g., [8,
p. 105]). The modified S okes p oblems ha a ise a each s ep a e sol ed by means
o a s anda d p ojec ion me hod [31, pp. 27–28] (see also [3, sec ion 4.6]).
Fo each hused in he iangula ions o Ω, e e y expe imen was ca ied ou wi h
diffe en alues o he ime s ep d . The e is always a poin , depending on h, a which
u he educ ion o he ime s ep d does no educe he e o s anymo e. This means
ha he e o a ising om he ime disc e iza ion is smalle han he e o a ising
om he MFE disc e iza ion. To a oid w ong conclusions om ou nume ical expe i-
men s, we ha e been ca e ul o ensu e ha he dominan e o in all he compu a ions
p esen ed he e is he spa ial disc e iza ion e o . Fo he compu a ional cos in he
efficiency diag ams shown he e, we use he la ges ime s ep among hose in which
he spa ial disc e iza ion e o is dominan .
In wha ollows, we use he same symbols in all he plo s o ep esen he ela i e
e o s. Fo he eloci y we plo he e o s in he fi s componen . Simila e o s a e
ob ained o he second. The diffe en me hods a e dis inguished by he line used o
join he symbols. Fo he MFE-P2P1 app oxima ion, we use con inuous line, and o
he MFE-P3P2 dashed-do ed line. The MFE-P2P1 has been pos p ocessed in wo
diffe en ways: using P3P2 (dashed line) and efining he mesh (do ed line).
In Figu e 4.1 we p esen wo con e gence diag ams showing he e o s commi ed
by he me hods when used wi h h=|Ω|/N, N =8,16,32,64, bo h in he L2no m
(le ) and he H1no m ( igh ). We ha e plo ed he e o s o he MFE-P2P1 and
P3P2 me hods and he pos p ocessed e o s wi h P3P2. One can obse e ha he
pos p ocessing echnique wi h P3P2 p o ides an app oxima e eloci y wi h abou he
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POSTPROCESSED MFE METHOD FOR NAVIER–STOKES EQUATIONS 1109
10
1
10
−4
10
−3
10
−2
10
−1
N=|Ω|/h
L
2
/ℜ−no m p essu e e o
slope=−2.9076
slope=−2.9864
slope=−1.9714
101
10−4
10−3
10−2
10−1
N=|Ω|/h
H
1
0
−no m eloci y e o
h´=h/2
h´ ≈ h
3/2
Fig. 4.2.Le : con e gence diag am o he p essu e app oxima ion wi h P2P1(con inuous
line), P3P2(dashed-do ed lines), and he pos p ocessed me hod wi h P3P2(dashed lines). Righ :
con e gence diag am o he fi s componen o he eloci y app oxima ion wi h P2P1(con inuous
line) and he pos p ocessed P2P1o e fine g ids.
same accu acy as ha co esponding o he MFE-P3P2 me hod. This is especially
ue o he H1no m, in which he wo me hods p oduce i ually he same e o s.
Measu es o he slopes o he plo s confi m he a es p edic ed by he heo y (i.e., he
e o s in he plo s dec ease like Nslope = cons .h−slope).
Simila conclusions can be eached om he e o s o he app oxima ions o he
p essu e in Figu e 4.2 (le ). Excep o he fi s poin , which co espond o h=1/8,
he pos p ocessed e o s lies on a line (almos ) pa allel o he one joining he MFE-
P3P2 e o s. The a e o con e gence o hese wo me hods is one uni la ge han
ha o he MFE-P2P1 in ag eemen wi h wha he heo y p edic s.
In Figu e 4.2 ( igh ), we plo he e o s ob ained pos p ocessing he MFE-P2P1
efining he g id. We ha e ep esen ed he e o s measu ed in he H1no m; simila
esul s ha e been ob ained o he L2no m. In iew o Theo em 3.14, in o de o
ge a gain o one o de o con e gence in he H1no m, we should use a mesh o size
h≈h3/2. The imp o emen in he a e o con e gence o he pos p ocessed me hod
can be obse ed in he figu e. We can also obse e in he plo ha using a efined mesh
o size h=h/2 (only one egula efinemen ), he e o s a e conside ably educed. In
ac , obse e ha he pos p ocessed e o wi h h=h/2 is almos he same as ha
o he s anda d MFE-P2P1 ca ied ou using a mesh o size h/2 o e he ull in e al
[0,T]. This ac can be o in e es when he cos o he pos p ocessing s ep wi h a
efined mesh o size a powe o his no affo dable o compu a ional easons.
The ele an ques ion now is whe he he imp o emen in he a e o con e gence
also implies imp o ed efficiency. In Figu e 4.3, we ha e ep esen ed he same e o s as
in Figu e 4.1 ( igh ) and Figu e 4.2 (le ) agains he smalles amoun o ime needed
o achie e hem. We ha e also plo ed he e o s o he pos p ocessed me hod efining
he mesh (Figu e 4.2 ( igh )). In he plo we obse e ha he efficiency o he wo
pos p ocessing p ocedu es is e y simila . We can conclude ha he pos p ocessed
me hod eally imp o es he efficiency o he s anda d MFE me hod o bo h app ox-
ima ions o he eloci y and o he p essu e. Fo any e o ha we may demand, he
pos p ocessed me hod achie es ha e o in less compu ing ime han he s anda d
P2P1 and P3P2-MFE me hods. The eason o his imp o emen is ha he e o o
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1110 BLANCA AYUSO, BOSCO GARC´
IA-ARCHILLA, AND JULIA NOVO
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
−4
10
−3
10
−2
10
−1
cpu ime
H
1
0
−no m eloci y e o
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
−3
10
−2
10
−1
cpu ime
L
2
/ℜ−no m p essu e e o
Fig. 4.3.Efficiency diag ams o he fi s componen o he eloci y in he H1no m (le ) and
he p essu e ( igh ) wi h P2P1(con inuous line), P3P2(dashed-do ed line) and he pos p ocessed
me hod wi h P3P2(dashed line) and efining he g id (do ed line).
he MFE-P2P1 me hod is educed when he pos p ocessing is done, bu his is done
a e y li le cos : ha o sol ing a single disc e e S okes p oblem a he final ime.
All nume ical expe imen s we e ca ied ou on a Pen ium IV, wi h 1 GB o Rimm
memo y, unde he Sola is8 (In el) ope a ing sys em, wi h SUN Wo kshop 5 compile s.
The p og ams we e w i en in Fo an 77.
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