edge stabilization for the incompressible navier-stokes … · 2020-07-04 · issn 0249-6399 isrn...

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HAL Id: inria-00070653 https://hal.inria.fr/inria-00070653 Submitted on 19 May 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Edge Stabilization for the Incompressible Navier-Stokes Equations: a Continuous Interior Penalty Finite Element Method Erik Burman, Miguel Angel Fernández, Peter Hansbo To cite this version: Erik Burman, Miguel Angel Fernández, Peter Hansbo. Edge Stabilization for the Incompressible Navier-Stokes Equations: a Continuous Interior Penalty Finite Element Method. [Research Report] RR-5349, INRIA. 2004, pp.34. inria-00070653

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Page 1: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

HAL Id: inria-00070653https://hal.inria.fr/inria-00070653

Submitted on 19 May 2006

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

Edge Stabilization for the Incompressible Navier-StokesEquations: a Continuous Interior Penalty Finite

Element MethodErik Burman, Miguel Angel Fernández, Peter Hansbo

To cite this version:Erik Burman, Miguel Angel Fernández, Peter Hansbo. Edge Stabilization for the IncompressibleNavier-Stokes Equations: a Continuous Interior Penalty Finite Element Method. [Research Report]RR-5349, INRIA. 2004, pp.34. inria-00070653

Page 2: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--53

49--

FR

+E

NG

ap por t de r ech er ch e

Thème BIO

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Edge Stabilization for the IncompressibleNavier-Stokes Equations: a Continuous Interior

Penalty Finite Element Method

Erik Burman — Miguel Ángel Fernández — Peter Hansbo

N° 5349

Octobre 2004

Page 3: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE
Page 4: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

Unité de recherche INRIA RocquencourtDomaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)

Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30

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Page 5: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

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− 〈β · nuh,vh〉∂Ωin+

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h

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γβ

h2K

‖β‖0,∞,K

∂K

(βh · J∇uhK) · (βh · J∇vhK) ds

+∑

K∈Th

γdivh2K‖β‖0,∞,K

∂K

J∇ · uhKJ∇ · vhK ds,ÆÇ[ªÊ

jp(ph, qh)def=

K∈Th

γpξ(ReK)h2

K

‖β‖0,∞,K

∂K

J∇phK · J∇qhK ds,ÆÇmÇTÊ

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) def= (f ,vh),

nuym¶m®Hº e P xPmH

∂ΩÃm wwº uy Fº

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ν, ξ(λ)

def= min1, λ,

〈x, y〉∂Ωdef=

e⊂∂Ω

e

xy ds, ∂Ωindef= x ∈ ∂Ω : (β · n)(x) < 0.

t3uwy¨y e(x2yxP*y[Ìy¼uyyHeyT'EÇmÂ*FºÐTm ¹Auwym Ìy[,ºÌyLyx2%ÆÙuwy#2wLwx ÓÆÇ[ªmÊÊG©Â3m̺ Pºm wm5m e"m¹uy wxwyTºwº×Öm w׺ ÝÆÔuy%y[m wx ÓÆÇ[ªmÊÊG¤Â3xP·º wCuyyÐ[m 54 09 mºyÖÆÔuyx º ÓÆÇÇTÊÊÂ

ÜÝy%¶*yy% uwyC H*uHLuy[yuy[ymy[,ºÌy[¼ymº yTº Áuymxy¹Úuwº m wuw¼yTy ePuyCmFmy e(x2ÁHy[mL[ ½wººÐ[y% º wmººwÌm·myTÖ2HmFy¨y[xÂ2 %um§HyP myTÁuw¹ÔuyÖ[myCm¹suwyPº wmx2y[yÀ¼n̺y[Íe·myTyTewº w2×xmue®"yÏÌ emmywCÒwºy½uy½y[xM emº ÝuyÖx2wÖ XuyºÌymy Hy2Ïuy®m[y[%ewy[mºÐ ½y[x

(γdivh∇ · uh,∇ · vh)º Ûy

·yy[϶* Áuy2w H¶*×uÏm¹uy2xPº¸6 DL¶yÌyL¹Ôxyxwºy¸ÁyxPwFu§m¨º¶3ÍOÏmFuw¶9uy3x2yx 3y¸"y[y[uHnÌmyx2yÌ emmyTÂA x2º wÓ@?Öy ePymwmº§m¹1uyy¸m2mm ÎuwymHÌyÖ¹ÔxCm Î2m º

m wy e[Â

å)(+*5åÙí

Page 10: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

!#"$%'&('*)+-,.0/ È

<ß Û²²%' ß ± 2±n° ãHã ² ' Ô° à 7 @&(Q-,.(u, p)

$, @ 4 ÆÇTÊ <7% -,. 97

[H3/2+ε(Ω)]d+1-,

ε > 0!/ %

(uh, ph) ∈ W kh

<7#-,. @+ 4 ÆÚÉÊ ,

A((u − uh, p− ph), (vh, qh)

)= 0, ∀(vh, qh) ∈ W k

h . 5C4 t3uAA Cºx2x2y[ym wy[ey[ wy<¹uy3m Hy w¹uym wm%Ñmºy[· x2yuwm wÏuy2¹Ú,uw wy[uyPy×½mx2m w

jp(p, qh) = 0 w

ju(u,vh) = 0Ã+ wy

J∇pKe = 0 w

J∇uKe = 0¹ÔmLmº6 ey[m*yTmyT

CA FHG %ÒA $MP

wº×Cº uyLy[myLwººÐTº Cx2yu2wmy[m uy3¹Ômº¶* yx2xP¶*uFu2¶3m`wÌy[¹Ôsy[y¶*y¼º yT<m e mHswn¸xPm Ͼ È[ËbÂD1yy1¶yLy¸y[ wPusy[º×m"m mx2symyyÂÀ1yPuwC¶yÖmÌmyÖ¶y"m HÕm¶ybÂt3uw%% % wyy[y[ݹÔm%uym w[Ãew1uw¶Luw3 ½x2yy Hy¨uwyFºÐ y[xP*mymx2mVÂ<ß Û²"!# ,.' Q%$K 8'9% 9'

π∗h : [H2(Th)]d −→ [V k

h ]d /&7.

γβ

γlow

/K89/' -,.J%7!& C,K7&(' / ,#93 & /=(= 7 <'@ # -,. 7 &( , %>0 )7, -,.

γlowjβ(vh,vh) ≤ ‖h12

(βh · ∇vh − π∗

h(βh · ∇vh))‖20,Ω ≤ jβ(vh,vh)

4 vh ∈ [V k

h ]d* ,

jβ(vh,vh) = γβ

K∈Th

∂K

h2K |βh · J∇vhK|2 ds.

5C4 m¨y[FuÁ yxiÃ6ºy

ni"yuy x%"y¹<yyx2y em emº

xim¨® wymÂt3uy[

¶yy2H y2Hm×Í º eyHm π∗

h

¹Aymyyk

π∗hv(xi)

def=

1

ni

K : xi∈K

v|K(xi), ∀v ∈ [H2(Th)]d.

wm*yTmFu®y[ºy[x2y eK ∈ Th

m Hyuy¹Ô wm

δKdef= h

12

K

(βh · ∇vh|K − π∗

h(βh · ∇vh)|K).

M<y[mmÃδK(xj) = 0

¹ÔmLyTmFuº eyº* yxj ∈

K¶*uyy[*m uwyyyx2y eLyTmyTÃb ymÂ<¹Ôm1mº

wy[xj ∈ ∂K

ö<yuwnÌy

δK(xj) = h12

K

1

nj

K′ : xj∈K′

βh ·(∇vh|K(xj) −∇vh|K′(xj)

)

= h12

K

1

nj

K′ : xj∈K′

e∈P (K,K′)

βh(xj) · J∇vhKe(xj),ÆÇT©mÊ

* * æRSTEH

Page 11: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

Å "@& ) /K >P '<

¶*uwyyP (K,K ′)

F wws¹Ôsuyym¹+y[my[<"y¶yy Km w

K ′ ÆÙuwy¨umyTwuHÊ,Â`ÜÝy ¶ ewy¨uyy¹Ôyy wy¼y[ºy[x2y eK w6ùÔm*y[Fu

K ∈ ThÃuy?2 wyx2m

FK(x) = BK x + bK , ∀x ∈ K,

wFuÏuwFK(K) = K

 wmºmÃ5yϕK

j

ÃB¹Ôj = 1, . . . , k

Ã6uy2wm1¹Ôw w,º wm KÂ2º Hy

δK(xj) = 0¹Ôm<y[Fu ey[m wy

xj ∈

KÃ‖δK FK‖2

0,∂K= 0

x2ºyT`uHδK FK = 0

t3uwyy¹ÔmymÃ6§y[eÌnmºy[ wy2m¹< xPm ÕyyPwmy[[Ã6wº Ám wFÝ[º mwxy[ eÆVy[y2¾ ©ÇÃ5¦ÉËÔÊ<m wÝÆÇT©Ê,ú*¹Ôm¶Lsuw

‖δK‖20,K = detBK‖δK FK‖2

0,K

≤ C detBK‖δK FK‖20,∂K

=

∂K

1

|B−TK n|

|δK FK |2 detBK |B−TK n| ds

︸ ︷︷ ︸

ds

≤ C|BTK |

∂K

|δK |2 ds

≤ ChK

∂K

|δK |2 ds

≤ ChK

∂K

k∑

j=1

|δK(xj)|2(ϕK

j )2 ds

≤ Ch2K

∂K

k∑

j=1

1

nj

K′ : xj∈K′

e∈P (K,K′)

|βh(xj) · J∇vhKe(xj)|2(ϕK

j )2 ds

≤ Ch2K

∂K

k∑

j=1

1

nj

e∈E(K)

|βh(xj) · J∇vhKe(xj)|2(ϕK

j )2 ds

≤ Ch2K

e∈E(K)

e

|βh · J∇vhKe|2 ds,

¶*uwyymÃw ½uy¶P¼º yTew׺yTÃE(K)

y[ y[*uyCy¨¹`y[my[1m eF º w®x2y ym¹K¼|¼ uyCuy¼uw w5Ãwuy%Hm+ymwmº®m¹

Thxwºy[Luw1uwy%xP¸x%x x%"y

m¹+y Hy[<¹(%yTmy¨ mº"uwyyFE(K)

H wyT®Ö 2¸y[m HF e wy[Hy[ wy[ *m¹hKÂt3uy +ÃewxxPm m

Köymy3uy"yL"mw w

‖h12

(βh · ∇vh − π∗

h(βh · ∇vh))‖20,Ω ≤ C

K∈Th

h2K

e∈E(K)

e

|βh · J∇vhKe|2 ds,

≤ C∑

K∈Th

∂K

h2K |βh · J∇vhK|2 ds,

å)(+*5åÙí

Page 12: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

!#"$%'&('*)+-,.0/ ¦

t3uwy¶yAH w%¹Ômº¶LA wy[ ¼uyL2 Íb wmx¡¹uyLm e mw+¹Ô wm δ Ìyuyy¹Ôy[y[ wyHFu

Gm wº Pm¹(uyCy¹Ôyy wyyyx2y e

K wº1 y[myTL y[ºuH[Â M<ºyTº

º¹‖δ‖G = 0

uy βh · ∇vh = π∗

hβh · ∇vh GÂ`t3u<xyT wsuH

βh · ∇vh e ewmw<º

Gm w2uy[ wy ∑

e∈E(K)

e hKJβh ·∇vhK2 ds = 0ÂDLy[ wy¼ wmx yTwºÌy HyL ®yy1wmy[

¶yuwnÌmy∑

e∈E(K)

e

Jβh · ∇vhK2 ds ≤ ‖δG‖20,G

t3uwymºx.uy ιÔmº¶LC Òuy½mxy¹Úmum ÒCuy 2wF2wm2¹Luyem¹1Û[º w§ Hy¸y Hm P5m¹

ThÂ

:¼ uy1x2y3yTFu wy*¶<y*x2xyTy[ºuwnÌy3uy3¹Ôº¶*º wmm¶*uyy3¹Ômsxwº,ͺPuyº¶y3"m HLy¨mx2×yT6 㠱 ã ' 'Ú²± à !# /=' ,# '& ( 7 @& 9 . & &( ! ,

‖h12

(∇ · vh − π∗

h(∇ · vh))‖20,Ω ≤ γdiv

K∈Th

∂K

h2KJ∇ · vhK2 ds,

‖h12

(∇qh − π∗

h(∇qh))‖20,Ω ≤ γp

K∈Th

∂K

h2K |J∇qhK|2 ds,

ÆÇ[¤Ê

4 (vh, qh) ∈ W k

h

/ -,γdiv, γp > 0

7. # /K89/=' 4h

À1¶ÃH Ïy[*mÌmy1xPmº Ïmººy[º[ÃB wmxy[ºuwy% ¹ÙÍ Ým H׺ ¹ÔmAÃ"¶<y

y2w yy["y[,ºÌy2uy¹Ôº¶* 2xyTuÍ y[Hy[ wy[ Lyx2ºÍb x w mx W k

h

E

|||(vh, qh)|||2 def

= ‖σ12 vh‖

20,Ω + ‖ν

12∇vh‖

20,Ω + ju(vh,vh) + jp(qh, qh) + ‖|β · n|

12 vh‖

20,∂Ωin

+ ‖γ12ν (ν/h)

12 vh‖

20,∂Ω + ‖γ

12n max

|β|,ν

h

12

vh · n‖20,∂Ω,

|||(vh, qh)|||2∗

def= |||(vh, qh)|||

2+ ‖qh‖

20,Ω,

¹ÔL(vh, qh) ∈ W k

h

Ât3uy§¹Ôm¶*º Òºy[xxPÒmÌmy[®uwy§y[ºÌºKm¹mm"yFmÖ¶*ºuyT"y[®Òuy§mHÌy

x2yTu͵y[Hy[ wy e1yx2ºÍb xÂ<ß Û²"!# ã5ß ±³ Ô° à , %$K J .

C > 0 /=9.' /

Ω/

γν

)7, -,.ah(vh,vh) + ju(vh,vh) + jp(qh, qh)︸ ︷︷ ︸

A((vh, qh), (vh, qh)

)

≥ C|||(vh, qh)|||2,

4 (vh, qh) ∈ W k

h

* * æRSTEH

Page 13: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

ÇTª "@& ) /K >P '<

5C4 wx ÆÚÉeʶ<yy

ah(vh,vh) + ju(vh,vh) + jp(qh, qh) ≥ ‖σ12 vh‖

20,Ω

+ 2‖ν12 ε(vh)‖2

0,Ω + ju(vh,vh) + jp(qh, qh) +1

2‖|β · n|

12 vh‖

20,∂Ω

+ ‖γ12ν

h

) 12

vh‖20,∂Ω + ‖γ

12n max

|β|,ν

h

12

vh · n‖20,∂Ω − 〈4νε(vh)n,vh〉∂Ω

ÆÇ¥eÊ

¶*uwyy¨¶ywy[uy¨¹Ú,*uH[ÃH¹Ùy[*º eymFº ®®H[Ã

(β · ∇vh,vh) =1

2〈β · nvh,vh〉∂Ω .

t3uwymy[x« XÆǥʨ Ý"y"mw wy[Ýw uyJMHFueeÍFu¶mÐ%º yTew×¹Ôº¶y[§eÁm6º Ìmy[y¨º yTew×Ã2m

| 〈4νε(vh)n,vh〉∂Ω | ≤cI2γν

‖ν12 ε(vh)‖2

0,Ω +1

2‖γ

12ν

h

) 12

vh‖20,∂Ω.

ÜÝyFuyγν >

cI

2 > 0 H®¶ym Hwyuwym¹wº w¼m " º yTew×Â

ÜÝy ¶y¨uymººPyTw×3¹Ôm3uyyy¨¹Ômx%º §ÆVÉÊsuyw¹¹¶*uwFum "y¹Ômw wº A "y H׸ A  ßã ± ß !#6°T²­ ' Ú° à ,.' %$= J 7 8

C > 0/=9.' /K C4

h /

ν 7)7,

-,.

sup06=(vh,qh)∈W k

h

A((uh, ph), (vh, qh)

)

|||(vh, qh)|||∗≥ C|||(uh, ph)|||∗,

4 (uh, ph) ∈W k

h

ß Û²± !# ,#9 5 -,.7 & & 9 N-,. -,.J/ '#9 =<'%'& ÆVÉÊ K/& N @ '@

>@03A%*# Ò $MP

À1y1uw3uwymºÐyTP¹Ômx%wm ® *y[ ºyP wy"y Hy e*¹5uyº[6*y w< xÍ"yTÂ3t3uywmmxyy[3¹ÔmLuyy[ymºÐ[º ½[y[L

h2/|β|¶*uwy uwyº[Ly m

xCHy[ReK

wºÏm wÕmh3/ν

¶*uy[ ReK

%x2mºbÂܧy¶*ºs w¶ u¶ uw%uC[º wºÌy[<mºxPm5%m"y[my[xPy[º uwyuºu§ÆÚºmÊ*y[ m x%"y*yºx2y¨¶*uy uymº Ïx2u5Ã

(u, p) ∈ [Hk+1(Ω)]d+1 m wÁ §uy2¶ ÆÔmʼ*y w¼ xCHy[y[mx2yw wyPm wFÕymºÓmx2º w[ÂÝÜÝyuy[ ÎÌmyÃ(w §uy A wº Í À1×FFuywmººy[Fu weyÓÆÚyyÃsym w¾ºÇÈ[ËÙÊÃsuw2uyÌmyºy[CuwnÌmymx2m*m Ìymy HyÖy[mݺ ÎuyL2 Í mx ¶*uy[ §uyP[*y[ m x%"y¶Ã5¶*×um% Ïmw׺ w`x2%2Hm wm¹uyÖwººÐTº 5 AºFm¹3mºs¶<y2Ìmyuwy2¹Ômº¶* m"y½¹uy

L2 Í my[m Ým ÝuywyV k

h

öyy2H yCºy x m wu¶ wn¸x2mººPyTw×TÂ

å)(+*5åÙí

Page 14: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

!#"$%'&('*)+-,.0/ ÇÇ

<ß Û² 'Th<7QN% ' ;@ 4& &( , /

l ≥ 1,

K∈Th

‖hl∇πhu‖2l,Ω ≤ C‖u‖2

1,Ω,

-,C > 0

7 7 !/=9.' /K C4h

5C4 t3uwyC¹ Áx2xyTy2 wy[ey Hy%m¹uwyH1 Í mºº¹uy L2 Í y[m ½ mºÖewm×Í º¹Ômx x2y[uy[ÆVy[y¾ ¥ËÔÊ<º x% wº ®¶*ºuºmBº Ìmy[y1yTºxPymà Hx2ymÃ

K∈Th

‖hlK∇πhu‖

2l,K ≤ C‖∇πhu‖

20,Ω ≤ C‖u‖2

1,Ω.

<ß Û² '(u, p) ∈ [Hk+1(Ω)]d+1

;4J &(k ≥ 1

/KJ, 7 @& 9 C4N%7' 6@ 4& 4 , & C, ! ,

|||(u − πhu, p− πhp)||| ≤ C(

σ12 hk+1 + max

ν, ‖β‖0,∞,Ωh

12 hk

)

‖u‖k+1,Ω

+ C maxK∈Th

‖β‖− 1

2

0,∞,Khk+ 1

2 ‖p‖k+1,Ω,

-,C > 0

7 7 $/K89/' γν

γnγβ

γdiv

/γp

5C4 z®uy HF®y[m3yTºxPy¨¹Ôm3uwyL2 Í yT,m ÝÆÚyy2¾ÇnÈ[ËÔʶ<yuHnÌmy

‖σ12 (u − πhu)‖0,Ω ≤ Cσ

12hk+1‖u‖k+1,Ω.

:¼ ½ ¶@uyH1 ͵F×Ám¹3uy L2 Í my[m Ým Óº[ºÏewºÍ º¹Ômx«x2y[uy[ÖÆVy[y½¾ ¥mËÙÊ

¶y¨myTö*×uihy m uy m6º ey["m e* w2F wwFÖ yHm y<yTºxPymÃ

‖ν12∇(ihu − πhu)‖0,Ω ≤ C‖ν

12∇(ihu − u)‖0,Ω ≤ Cν

12 hk‖u‖k+1,Ω.

ÜÝyyT*uwy%"m wwÖy[xP*H 2uwyFmyº wy[ewº§ÆÔ¥Ê3 ÁmxC wº ¶*ºuuy%mHÌyyTºxPy[¹Ô3uy

L2 Í mx H®uyH1 Í yx26 xÃym wÂÃ

‖u− πhu‖20,∂Ω ≤ C

e⊂∂Ω

(h−1

e ‖u− πhu‖20,Ke

+ he‖u− πhu‖21,Ke

)

≤ Ch2k+1‖u‖2k+1,Ω,

ÆÇT£mÊ

¶*uwyyKe

y my[%uwyºx2y¸ÒwFuÕuwe ⊂ ∂Ke ∩ ∂Ω

ÂÓt3uwy eym%"y HºÓyxPCmyyTy[º uwymxy¨¹Úum muy"m ww2yxPÂ`ÜÝyuwnÌy

ju(u − πhu,u − πhu) ≤ ju(ihu− πhu, ihu − πhu)

+ ju(u − ihu,u − ihu).

ÆÇ[ÉÊ

* * æRSTEH

Page 15: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

Çn© "@& ) /K >P '<

t3uwyQ2Hyx« Ïuº wy[ewºÁ Ý"yPy[x2yTÏw uyFmyC yTH×KÆÔ¥eʨm wÁuy ¿+yx2xP¥wÂÇúy[º

ju(ihu − πhu, ihu− πhu) =∑

K∈Th

[

γβ

h2K

‖β‖0,∞,K

∂K

|βh · J∇(ihu− πhu)K|2 ds

+ γdivh2K‖β‖0,∞,K

∂K

J∇ · (ihu− πhu)K2 ds

]

≤ C‖β‖0,∞,Ω

K∈Th

h2K‖∇(ihu− πhu)‖2

0,∂K

≤ C‖β‖0,∞,Ω

K∈Th

(hK‖∇(ihu− πhu)‖2

0,K + h3K‖∇(ihu − πhu)‖2

1,K

)

≤ C‖β‖0,∞,Ωh2k+1‖u‖2

k+1,Ω,

ÆÇnÈÊ

wm3uyyT wÖyx ÓÆÇ[ÉmʶymF 5à ®uyx2y¨¹Úmuº 5Ã

ju(u− ihu,u− ihu) ≤ C‖β‖0,∞,Ωh2k+1‖u‖2

k+1,Ω.ÆÇ[ÅÊ

A wmüuyÓy[wy®x2yx yTyT9w ÒuyÕmx2yÝmx2y em w uyݹÚmuwξ(ReK) ≤ 1

Ât3u3ºy[[Ã

jp(p− πhp, p− πhp) ≤ C maxK∈Th

‖β‖−10,∞,Kh

2k+1‖p‖2k+1,Ω,

¶*uwFuxwºyy[uy¹Â

!#"$%&(')*

µ uw*y[m ®¶yÌmy¼m Ìmy[y wyL ®uyºwºy¨ mxÂ`t3uy[yyTº3my¨mºxPm6º wy"y Íy eÖ¹Auwyºm5*y w3 ewx%"y*¶*uy[ uyy¸,Lºm *@?Öy ex2u+ ßã ± ß # @& Q,

(u, p) ∈ [Hk+1(Ω)]d+1I4 &(

k ≥ 1 Q-,. '@ 4 ÆÇTÊ

/(uh, ph) ∈ W k

h

<7 , '@ C4 ÆVÉÊ ,.'

|||(u − uh, p− ph)||| ≤ C

[ (

σ12 +

‖β‖1,∞,Ω

σ1/2

)

hk+1 + maxν, ‖β‖0,∞,Ωh12 hk

]

‖u‖k+1,Ω

+ C maxK∈Th

‖β‖− 1

2

0,∞,Khk+ 1

2 ‖p‖k+1,Ω,

-,C > 0

7 7 $/K89/' γν

γnγβ

γdiv

/γp

5C4 ¿+y1yTx"y¨uyy[m(u − uh, p− ph)

º ¶H

u− uh = u− πhu︸ ︷︷ ︸

+πhu − uh︸ ︷︷ ︸

−eh

= eπ − eh,

p− ph = p− πhp︸ ︷︷ ︸

+πhp− ph︸ ︷︷ ︸

−yh

= yπ − yh.

å)(+*5åÙí

Page 16: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

!#"$%'&('*)+-,.0/ ÇT¤

*¹Ôº¶L<uy[ uw

|||(u − uh, p− ph)||| ≤ |||(eπ, yπ)||| + |||(eh, yh)|||.¿+yx2xPÏ¥HÂ ©ÏmÌmyT2 ÒyTºxPy¹Ôm

|||(eπ, yπ)|||Ã<uy Hy×Ö@?ÖyT%Õw

|||(eh, yh)|||Â :1º

yFÌeºÖ Humm wmººmÿ5y[x2x2*¤Âô©%m w½©ÂÇöymy

C|||(eh, yh)|||2≤ A

((eh, yh), (eh, yh)

)

= A((eπ, yπ), (eh, yh)

)

= ah(eπ, eh) + bh(yπ, eh) − bh(yh, eπ)

+ ju(eπ, eh) + jp(yπ, yh)

ÆÇ[¦Ê

zÁm Ým[m ϹsuyJMwFueÍFu¶mÐ y[ewmºº½ ÏuyPx2x2yCH¨m¹uyPyyy[ºm"yFm* w y CÖH3 uy Ìmy[Ìmy¼y[x¶ym

ah(eπ, eh) ≤ |||(eπ, 0)||||||(eh, 0)||| + |(eπ,β · ∇eh)|

− 〈2νε(eπ)n, eh〉∂Ω − 〈eπ, 2νε(eh)n〉∂Ω ,¶*uwyymÃ6¹Ôx2ººmÃuwyPH wm½yx ¹Ômx«uyP ym ÏÁHuwHy[y Õ wwyT uwy 2wF1yxEm uy%ºue1uw H½ymÂ*ÜÝy% wyuw¼ ½uymx2y¶n¶<y%uwnÌmyÃwwº wPuyMmwFue͵Fu¶3ÐL y[ewmººmÃCFmyº yTew× w2º[6 eÌyFy¨ y[ewmººmÃ

〈2νε(eh)n, eπ〉∂Ω ≤ C|||(eh, 0)||||||(eπ, 0)|||.wmuy®y[m wÝ"m ww½y[x!¶<yPHyPuy MmwFuÍFu¶3Ð% y[ewmººÁ¹Ôº¶<yTÏÓFmy yTH×®m w n¸ºxPm mx2y e[ÃxÁÆÇ[ÉmʵÍFÆÇnÈÊÃPmº

〈2νε(eπ)n, eh〉∂Ω ≤ Cν12 hk‖u‖k+1,Ω|||(eh, 0)|||.

t3uwy< Ìmy[Ìmy`yx¡A emºyTw Le eÌyFysº yTew×m wuwy<umm wmºº1m¹uyL2 Í y[m 5Ã6¹ÙyuHnÌe y[yT½uwy2m e mH¼Ìyº 2wy β

½ºy[y¶*y2º wy[ ey["m e

βh

Ã

|(eπ,β · ∇eh)| ≤ |(eπ, (β − βh) · ∇eh)| + |(eπ,βh · ∇eh)|

≤∑

K∈Th

‖β‖1,∞,K‖eπ‖0,KhK‖∇eh‖0,K

+ |(eπ,βh · ∇eh − π∗h(βh · ∇eh))|

≤ C‖β‖1,∞,Ω

σ1/2hk+1‖u‖k+1,Ω|||(eh, 0)|||

+ ‖h−12 e

π‖0,Ω‖h12 (βh · ∇eh − π∗

h(βh · ∇eh))‖0,Ω.À1¶¶y®¿+yx2xPP¤ÂÇ1PmF

‖h−12 e

π‖0,Ω‖h12 (βh · ∇eh − π∗

hβh · ∇eh)‖0,Ω ≤ ‖h−12 e

π‖0,Ω‖β‖12

0,∞,Ωju(eh, eh)12

≤ C‖β‖12

0,∞,Ωhk+ 1

2|‖u‖k+1,Ω|||(eh, 0)|||.

* * æRSTEH

Page 17: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

Ç[¥ "@& ) /K >P '<

M<ºy[ y[xP3¶y¨uHnÌmy

ah(eπ, eh) ≤ C|||(eπ, 0)||||||(eh, 0)|||

+ Cν12hk‖u‖k+1,Ω |||(eh, 0)|||

+ C‖β‖1,∞,Ω

σ1/2hk+1‖u‖k+1,Ω|||(eh, 0)|||

+ C‖β‖12

0,∞,Ωhk+ 1

2 ‖u‖k+1,Ω|||(eh, 0)|||.

Æb©ªÊ

wm*uy%y[m wy[x ¶yuwnÌmyÃwº 2uy%umm wmººP¹(uyL2 ÍbyT,º m wy[m

u¶*ºup ÓÆÇT£ÊÃ

bh(yπ, eh) = −(yπ,∇ · eh − π∗h(∇ · eh)) + 〈yπ, eh · n〉∂Ω

≤ ‖h−12 yπ‖0,Ω‖h

12 (∇ · eh − π∗

h(∇ · eh))‖0,Ω

+ C maxK∈Th

min‖β‖−1

0,∞,K ,hK

ν

1

2

‖yπ‖0,∂Ω|||(eh, 0)|||

≤ C maxK∈Th

‖β‖− 1

2

0,∞,K‖h−12 yπ‖0,Ωju(eh, eh)

12

+ C maxK∈Th

min‖β‖−1

0,∞,K ,hK

ν

1

2

hk+ 12 ‖p‖k+1,Ω|||(eh, 0)|||

≤ C maxK∈Th

‖β‖− 1

2

0,∞,Khk+ 1

2 ‖p‖k+1,Ω|||(eh, 0)|||.

Æb©ÇTÊ

µ ½2ºx2m<¹Úuwº 5Ãw¹Ùy* eym e®w3 uyuºFy[xÃm wy¨mº H

bh(yh, eπ) = −(yh,∇ · eπ) + 〈yh, e

π · n〉∂Ω = (∇yh, eπ)

= (∇yh − π∗h(∇yh), eπ)

≤ ‖h12 (∇yh − π∗

h(∇yh))‖0,Ω‖h− 1

2 eπ‖0,Ω

≤ Ch−12 max

K∈Th

ν, ‖β‖0,∞,KhK12 jp(yh, yh)

12hk+ 1

2 ‖u‖k+1,Ω

≤ Cmaxν, ‖β‖0,∞,Ωh12 |||(0, yh)|||hk‖u‖k+1,Ω.

Æb©m©mÊ

A wmÃuwyº ey[m*"y HºPy[xP*¹Ô

ju(eπ, eh) + jp(yπ, yh) ≤ C|||(eπ, yπ)||||||(eh, yh)|||.

Æb©¤ÊÜÝy%m wºwyuy%m¹`mºyT,º 2uyy[ºLm¹%ÆV©mªÊ Í,ÆV©m¤mÊ3 ÒÆÇT¦mÊ*m w½wº 2uywn¸ºÍxPm ¿5yx2xP¥wÂô©Â

®ß ³ ã ß ± ° ß ± ß ¯[¯ +± ßµ uyCÖm+y[m*yTºxPyÌy[t3uwymyx ¥wÂÇuwy%y[wyy[m*1 w¼m emºyT Ï® mxÂ1µ Áu¼y[m 5ÃB¶y%uw¶¡u¶ ®y[Ìmy[1 +m¹`uy

L2 Í xE¹`uy%wyTyy[mTÂ

å)(+*5åÙí

Page 18: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

!#"$%'&('*)+-,.0/ Çn£

ßã ± ß (u, p) ∈ [Hk+1(Ω)]d+1

I4Q '& k ≥ 1

<7Q '@ C4 ÆÇnÊ /(uh, ph)<7 ,# '3@ 4 ÆVÉmÊ ;-,.' -,.' #,. /

‖p− ph‖0,Ω ≤ C(

h+ maxν, ‖β‖0,∞,Ωh12

)

hk(‖p‖k+1,Ω + ‖u‖k+1,Ω

),

-,C > 0

. * /K89/=' 4h/

ν

5C4 wmº¶* ¾ô©Çà M<ºÖ©Â ¥Ë Ãuy[yy¸vp ∈ [H1

0 (Ω)]dwFu®uw

∇ · vp = p− ph, ‖vp‖1,Ω ≤ C‖p− ph‖0,Ω.Æb©¥eÊ

:¼ uw* ½x% wº ®¶*ºuÁÑy·º muwmm wº¶yyTmÖmF

‖p− ph‖20,Ω = (p− ph,∇ · vp)

= (p− ph,∇ · (vp − πhvp)) + 〈p− ph, πhvp · n〉∂Ω

+ ah(u − uh, πhvp) + ju(u− uh, πhvp).

t3uw[à y Cw[Ã×*¹Ôº¶Luw

‖p− ph‖20,Ω = (∇(p− ph),vp − πhvp)

︸ ︷︷ ︸

I

+ ah(u− uh, πhvp)︸ ︷︷ ︸

II

+ ju(u− uh, πhvp)︸ ︷︷ ︸

III

.

Æb©m£mÊ

wm<uy2wFy[xÃwº w%uy¼umm wmºº%m¹5uwyL2 Í my[m 5Ã.MwFueÍFu¶mÐ3 y[ewmººmÃ

M<ºP¤ÂÇÃ+Æb©¥Ê3m wmn¸ºxPº 5ö<yy

I = (∇(p− ph) − πh∇p+ π∗h∇ph,vp − πhvp)

≤(‖h(∇p− πh∇p)‖0,Ω + ‖h(∇ph − π∗

h∇ph)‖0,Ω

)‖h−1(vp − πhvp)‖0,Ω

≤ C(

hk+1‖p‖k+1,Ω + maxν, ‖β‖0,∞,Ωh12 jp(ph, ph)

12

)

‖p− ph‖0,Ω.

M< wy[L ¶uyCyTm Hm wuwºFy[xP* ÛÆV©£Ê :1 Puwy%y2H ׺ ½¹(uwyº y[m*¹ÔmxahÆVÅÊ,Ãw¶<yuwnÌymùÙy* eym e®w3 uym ÌmyT,Ìmy¼y[xÃ

II + III ≤ |||(u − uh, 0)||||||(πhvp, 0)||| + (u− uh,β · ∇πhvp)

− 〈2νε(u − uh)n, πhvp〉∂Ω − 〈u − uh, 2νε(πhvp)n〉∂Ω.

Æb©ÉÊ

wm3uy eÌy[Ìmy¼yx¶yuwnÌmyÃw uwyH1 Fºº®m¹+uy L2 ÍbyT,º H§Æb©¥eÊÃ

(u − uh,β · ∇πhvp) ≤ ‖β‖0,∞,Ω‖u− uh‖0,Ω‖∇πhvp‖0,Ω

≤ C‖β‖0,∞,Ω‖u− uh‖0,Ω‖p− ph‖0,Ω.

* * æRSTEH

Page 19: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

ÇTÉ "@& ) /K >P '<

t3uwy¨"mw wPyx2*mym emºyTÖ uy¹Ôº¶*º w%¹Úum

〈2νε(u − uh)n, πhvp〉∂Ω + 〈u − uh, 2νε(πhvp)n〉∂Ω

≤ ‖(2νh)12 ε(u − uh)‖0,∂Ω‖(2ν)

12 h−

12 πhvp‖0,∂Ω

+ ‖(2νh)12 ε(πhvp)‖0,∂Ω‖(2ν)

12 h−

12 (u − uh)‖0,∂Ω.

µ ½ºm 5öyuwnÌmy

‖(2νh)12 ε(u − uh)‖0,∂Ω ≤ ‖(2νh)

12 ε(u − πhu)‖0,∂Ω

+ ‖(2νh)12 ε(πhu − uh)‖0,∂Ω,

¶*uwyy¼uy 2wF3y[x 2Hy[[ÃeH P yyx2y e¶*y¨y¨º yTew×Ö¹Ômº¶y[®en¸ºxPͺ 5Ã

‖(2νh)12 ε(u − πhu)‖0,∂Ω ≤ Cν

12hk‖u‖k+1,Ω,

m w®uyy[m w6ÃH PCFmy y[ewmººP¹Ôº¶<yT®®6 Ìmy[y¼ yTH×Ã

‖(2νh)12 ε(πhu− uh)‖0,∂Ω ≤ C|||(πhu− uh, 0)|||.

µ uymxy¨¹Úum ¶ym Hwy¨uw

‖(2ν)12h−

12πhvp‖0,∂Ω ≤ C|||(πhvp, 0)|||,

m w‖(2νh)

12 ε(πhvp)‖0,∂Ω ≤ C|||(πhvp, 0)|||.

ÜÝyy H®uyem¹® w uH|||(πhvp, 0)||| ≤ C‖p− ph‖0,Ω.

ß Û²± &Q%Q @& ! '7/ $9=9/ $ 9 ,7 '&

( # u ∈ [H2(Ω)]d

p ∈ H1(Ω)

µ ®uyy1¶<y% w3uwnÌmy*uy¨yTBymwmºCx2y[P Öuy¨y[̺wuyy[x ¶<y¨ºBmyxPyyTºxPy[sº 2uyL¶Î"*y w` x%"y<y[mxyÂAt3uwyLxP Ö Byy Hy3¹Ômxuy"m eL¹`̺y[¶m¹`m w3*uH*uwyy[yFºÐTm yx uw3P"yyTyTmLm wm wy[ eyxÂ`Ü ×uuLx uwy¹Ômº¶* 2ºy[x2x2¶*º+Ìy¼Hy¹Ô<ß Û² ! '

p ∈ H1(Ω)-,.'

jp(πhp, πhp) ≤ C maxK∈Th

min‖β‖−1

0,∞,K ,hK

ν

h‖p‖21,Ω.

å)(+*5åÙí

Page 20: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

!#"$%'&('*)+-,.0/ ÇÈ

5C4 t3uwym¹+¹Ôm¶L<®uyy¨º yTew× º ÌyFy¨º wy[ewº® w¿5y[x2x2¥HºÇÂM<y[m¶*uy[ CuyLLy m` xCHy[`º¶Î¶<y*uwnÌy

jp(πhp, πhp) ≤ Ch2‖p‖21,Ω

 wyp º my[¼uHm¼uy2ymºyTeºy[½¹Ôm

jp(p, p)xPm·my2y[ wyCuwyCyC wmx x%wHy

y[y2w yT6Â(ܧyxwºwyCºy wmx¶*×um*uyy[y¨"y wm×Pyxà wx2y[ºÃ

|||vh|||2u

= ‖σ12 vh‖

20,Ω + ‖ν

12∇vh‖

20,Ω + jv(vh,vh) + ‖|β · n|

12 vh‖

20,∂Ωin

+ ‖(γνν)12h−

12 vh‖

20,∂Ω + ‖γ

12n max

|β|,ν

h

12

vh · n‖20,∂Ω,

¹ÔLvh ∈ [V k

h ]dÂ

<ß Û² ¬ ± ã Û²­ ' Ô° à 'u ∈ [H2(Ω)]d

-,.'+-,.' #,.%/

|||u− πhu|||u ≤ C(

σ12h2 + max

ν, ‖β‖0,∞,Ωh

12 h

)

‖u‖2,Ω,

5C4 y emB2uwym¹¹(¿5y[xxP2¥HÂ ©Â

ßã ± ß ! (u, p) ∈ [H2(Ω)]d ×H1(Ω)

<7Q,Q '@ 4 ÆÇ[Ê /(uh, ph) ∈ W k

h

<7-,. '@ 4 ÆÚÉeÊ ;-,.'

|||u− uh|||u ≤ C

[(

σ12 +

‖β‖1,∞,Ω

σ1/2

)

h2 + maxν, ‖β‖0,∞,Ωh12h

]

‖u‖2,Ω

+ C maxK∈Th

min‖β‖−1

0,∞,K ,hK

ν

1

2

h12 ‖p‖1,Ω.

5C4 ܧy¶*º< §º wyÖuwy®¹*uyyÖºÌ ÏHyT¹Ôw% Õu¶ ½yTuy® wm Ím wy e¼ eym1"y Hºyx ¹Ôm¼uyy[ym A ¼º ÏuyyÌmw¼my%º¼.?Pºy[ e1wÌy1uy eÌymy[ wy¼¹

eh = uh − πhuÂ

:¼ Öy[ºÌºÁÆÚ¿5y[x2x2P¤w ©mÊ3my̺wº®m wyTº CuH

|||(eh, yh)|||2

= |||eh|||2u

+ jp(yh, yh),

¶*uwyyyh = ph − πhp

öyuwnÌmy

C|||(eh, yh)|||2 ≤ ah(eh, eh) + ju(eh, eh) + jp(yh, yh).Æb©ÈÊ

t3uwy¨x2%2wy[½Ñmºy[· umm wmºº·y[uy¹Ômx

ah(uh − u,vh) + bh(ph − p,vh) + ju(uh − u,vh)

− bh(qh,uh − u) = −jp(ph, qh),Æb©ÅÊ

¹ÔL(vh, qh) ∈ W k

h

Â;:¼ P ¶ ÆV©mÅmʺ ÕÆb©ÈʶyuwnÌmy

C|||(eh, yh)|||2≤ ah(eπ, eh) − bh(yπ, eh) + bh(yh, e

π) + ju(eπ, eh)

− jp(πhp, yh).

Æb©¦Ê

* * æRSTEH

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ÇTÅ "@& ) /K >P '<

t3uwy 2wF¼¹Ô¨yxP¨º §uy2ºue¨uwm wÏyCm¹2Æb©¦mʨmyCy[y[Ïm¨º §uwyw¹m¹t3uy[myx¥HºÇmÂt3u3y

ah(eπ, eh) ≤ C

(

|||(eπ, 0)||| + ν12h‖u‖2,Ω +

‖β‖1,∞,Ω

σ1/2h2‖u‖2,Ω

+ ‖β‖12

0,∞,Ωh32 ‖u‖2,Ω

)

|||(eh, yh)|||,

bh(yh, eπ) ≤ Cmaxν, ‖β‖0,∞,Ωh

12h‖u‖2,Ω|||(eh, yh)|||,

ju(eπ, eh) ≤ |||eπ|||u|||(eh, yh)|||.

wm3uyyT wÖuyx ÕÆV©¦Ê¶<y 2wFLy[["uw

(yπ,∇ · eh) ≤ ν−12 ‖yπ‖0,Ω|||(eh, 0)|||

≤ Ch

ν‖p‖1,Ω|||(eh, 0)|||,

m wPuy[ wyÃPx% uyTºxPº Ö¶*ºuPuy¼ yLw uyLy[Fu ey1m¹¨Æb©Ç[Ês¶<y1mF

bh(yπ, eh) ≤ C maxK∈Th

min‖β‖−1

0,∞,K ,hK

ν

1

2

h12 ‖p‖1,Ω|||(eh, yh)|||,

A wmùÔm3uy w wy e3y[x uwnÌmy¨ ÓÆb©¦mʶyuwnÌy

jp(πhp, yh) ≤ jp(πhp, πhp)12 jp(yh, yh)

12

≤ jp(πhp, πhp)12 |||(eh, yh)|||

m w¶<y wwy®m m[º m¹(¿+yx2xPC¥HÂ ¤CyTºxPyjp(πhp, πhp)

12

Ât3uy1¹Ômº¶* %mmm% Px2xyTy¨ wyTey wyL¹6uy¼mHÌyL eÌymy[ wy* HÂ

ã ± ã ' 'Ú²± à (u, p) ∈ [H2(Ω)]d ×H1(Ω)

<7 , '@ C4 ÆÇTÊ (uh, ph) ∈ W k

h

<7 -,. '@ 4 ÆÚÉÊ /

yhdef= ph − πhp

,.' , %$K Q 7 7 C > 0

/=9.' /K 4h )7, -,.

jp(yh, yh)12 ≤ C max

K∈Th

min‖β‖−1

0,∞,K ,hK

ν

1

2

h12 ‖p‖1,Ω

+ C(

1 + maxν, ‖β‖0,∞,Ωh12

)

h‖u‖2,Ω,

jp(ph, ph) ≤ Cjp(yh, yh).À1¶¶yFy2 Ìmymy[ wy¼yTw×3¹Ôm3uyy[wyÂ

ßã ± ß (u, p) ∈ [H2(Ω)]d ×H1(Ω)

*< '@ 4 ÆÇTÊ /(uh, ph)

<7 -,. 7 '6 9/; 8 '%& '; '@ C4 ÆVÉÊ -,.'+-,.' ,. /

‖p− ph‖0,Ω ≤ C maxK∈Th

min‖β‖−1

0,∞,K ,h

ν

1

2

h12

(‖p‖1,Ω + ‖u‖2,Ω

).

å)(+*5åÙí

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!#"$%'&('*)+-,.0/ ÇT¦

5C4 µ uwymxyºº*uy¹A¹At3uwymyx ¥H ©Â;:1º JM<ºC¥HºǨ¹ÔwÌ %uwuyyyy[wy¨mFmy eAx2wLm Ìmy[y[[Â

t3uy®¹Ômº¶* uyy[x/ºÌy[m ÕmºxPL2 Beymy[xPyÖ¹ÔmCÌmy[ººÏ¶*uwy Õuy®Ly m3 xCHy[*3¶Â

ßã ± ß (u, p) ∈ [H2(Ω)]d ×H1(Ω)

<7Q,Q '@ 4 ÆÇ[Ê /(uh, ph) ∈ W k

h

<7-,. '@ 4 ÆÚÉeÊ 7 @&( ,

‖β‖0,∞,Ωh ≤ ν,ÆÚ¤ªÊ

/N,$, '3@ (ϕ, ψ)

C4 , K/' P9 K< &

σϕ− β · ∇ϕ− 2ν∇ · ε(ϕ) −∇ψ = u− uh

Ω,

∇ · ϕ = 0

Ω,

ϕ = 0

∂Ω,

ÆV¤ÇTÊ

<7'%' [H2(Ω)]d × [H1(Ω)]

/ $

‖ϕ‖2,Ω + ‖ψ‖1,Ω ≤ C‖u − uh‖0,Ω.ÆV¤©mÊ

, , %$= &7. C > 0

/=9.' /K C4h )7, -,.

‖u− uh‖0,Ω ≤ Ch2(‖u‖2,Ω + ‖p‖1,Ω

).

5C4 O½ºº Õuy 2wF®y[ewm ¹ÝÆÚ¤ÇTÊPu − uh

wKuy§y[m wX−(p − ph)

à ey[mF Öw*m ww 2uyx232wyTÑmºy[· umm wmººÁÆV©mÅÊÃ×¹A¹Ôm¶Lsuw

‖u− uh‖20,Ω = ah(u − uh, ϕ) + bh(p− ph, ϕ) − bh(ψ,u − uh)

= ah(u − uh, ϕ− πhϕ) + bh(p− ph, ϕ− πhϕ) − bh(ψ − πhψ,u− uh)︸ ︷︷ ︸

I

+ ju(u − uh, ϕ− πhϕ)︸ ︷︷ ︸

II

+ jp(ph, πhψ)︸ ︷︷ ︸

III

.

wmº¶* Cuymx2y e*¹uy¹Ú3m¹At3uy[myxP3¥wÂǨm w¥wÂô©Ãwm w§ÆV¤mªmʶymy

I ≤ |||u − uh|||u|||ϕ− πhϕ|||u + |(u − uh,β · ∇(ϕ − πhϕ))|

− 〈2νε(u − uh)n, ϕ− πhϕ〉∂Ω − 〈u− uh, 2νε(ϕ− πhϕ)n〉∂Ω

≤ C|||u − uh|||u|||ϕ− πhϕ|||u + C|||u − uh|||uh‖ϕ‖2,Ω

+ C‖p− ph‖0,Ωh‖ϕ‖2,Ω + C|||u− uh|||u‖h‖ψ‖1,Ω.

:¼ JMHFueeÍFu¶mм wÓÆÇTÉmÊ Í,ÆÇTÅmÊm wy¨mº H

II ≤ ju(u − uh,u− uh)12 ju(ϕ− πhϕ, ϕ− πhϕ)

12

≤ C‖β‖0,∞,Ω|||u− uh|||uh32 ‖ϕ‖2,Ω.

* * æRSTEH

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©mª "@& ) /K >P '<

A wmÃwº wLM<mmm2¥w ¤ wÓÆÚ¤ªmÊ,ùÔm3uwy3y[x ¶<yuwnÌy

III ≤ jp(ph, ph)12 jp(πhψ, πhψ)

12

≤ Cjp(ph, ph)12 max

K∈Th

min‖β‖−1

0,∞,K ,hK

ν

1

2

h12 ‖ψ‖1,Ω

≤C

νjp(ph, ph)

12h‖ψ‖1,Ω

t3uwyem¹(m wºHy[1ex% Puwy%"ÌmyyTºxPº w*¶*ºuÒÆV¤mªmÊÃ"t3uwymyxP*¥H ¤Pm w¥w ¥ÃM<ºÖ©¦ÃH w®uyx2y[ymwmÐ %"yuHnÌemÆV¤©mÊÂ

¿+y2wCx Òuyy[×Fmº yTÒ½¹ÚTÂÓt3uyy[x!¥H异 wÒt3uyy[x!¥H ©Áu¶ uw¶*uwy 2uwy1["*y s x%"y<suºuÖ wuy¼mº Psy[m¶yL¶*ºHy 2 PxPO(hk+ 1

2 )ym Ìymy Hy%m¹suy2y¼ Ïuy

L2 x ¹Ô¨uy2Ìmy[º×Ám wÁuy2y[ymÂwmºm w¼uHy%y[¼y¼t3uwymyx ¥H ¤uw¶LLuw¶*uy §uwyCy[yC¨

H1(Ω)m wPuy¨Ìmyº2º [H2(Ω)]d

¶y¨my* mºxPmO(h)

y[xPy¼ Öuyy y[ wmx ¶*uwy uymH*y[e wms xCHy[<<º¶Ãe*%"mºxPByTºxPy1m¹

O(h12 )¶*uy Öuy¨º[B*y

xCHy[3umu5Â`t3u3y1uy¼¹Úmuwuwy¨ wm wy[ wy[ ÖuyyTy¼mºÐ[º "myTuyy[ ymÖ wmxy[xPy¨¹Ô3uyÌy׺yTÂ

¬ 'Ô° ß ± A²° ß ¯°T²­ ' m²° ã ²±²% ß ° ß ±¯ ã ±*° ß ®ßeà ã 'ÚäA¯ §­ ß ±3± ß' ßA ¶3n§m¹1mF §m Òy[xPy®¶*uFuÒ%xP3 Ò"uÒuyumuÎmFyCy[mxy wÛuy¶ mFy%y[mxy®¶*uy ÒuyyTyPCm wº

H1 Ïx2º¹ÔÝuyFºÐTm ÕHFx2yyF§ºÌym emº XÝ my[% x ÛuyuºuÎ*y[ m x%"y2ymx2ymÂÕµ Îu2yT,º ¶yÁm"yÏwFu9 9m×y wºÌyÏym¹FºÐ[m wmmx2yy[[Â@ wyÏuyÏyTFu wy[Öm¹w¹¨y®y emÝuey¹1uyyÌmwyT,º Ò¶y ºÕ yuyem¹Ú F2¶<yPuyy[Hy[ wy Hy¨¹5uy¼m"*y[ m< ewx%"yº ®uy¨y[y¼mºÐ[m ®m wP wy[ wyTmy¼uyCmºÐ[m m uwy% wxwyTºwº×m H׺ 5Â*t3uL¶3nPuy%ºwºy x ¶*º"y%x2º wyTe®uy

H1 Í x º uy¶A*y w* x%"yLyºx2ym w®uyH Ì Í wmx«º ÓuyPumuÕ*y[e wm x%"yyºx2y wy

u ∈ [H2(Ω)]d w

p ∈ H1(Ω)¶yPx%H

y¸uwymºÐ[º Á¹suyy[y Ïy[1 e1¹Ô1uyC wm wy[ wÂt+mF xP×K¶*uy[ 9y[xP ÕuyÏwyTyÁÌmy[ººÎmwº Õyx2®uwy§m y®m emLm¹uyP wxwyTºwº×Ý wºm Ý%yTwºy[6ÂPÜ ºuÓu%xÃ+¶yÖ wy[uyy2uy®x%2wyTmºÐ[m yxPùÔ3uymºÐ[m ¹+uy wxwyTºwº× w®uywyTy

K∈Th

γdivhK

∂K

J∇ · uhKJ∇ · vhK ds,

K∈Th

γph3K

∂K

J∇phK · J∇qhK ds.

ÆV¤m¤Ê

ÜÝyCx2º¹ÔÖuwy[º 2¶*ºuh¹Auwy*"m wwPy[xm¹ÆÚÅÊ<F· 2 ¶

⟨γnh

−1uh · n,vh · n

∂Ω.

ÆV¤¥eÊ

å)(+*5åÙí

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!#"$%'&('*)+-,.0/ ©Ç

A% my®uw%uwy® w wy e%wyTyx2Õy[x.uwuy®ºue%mFyTÂÁt3u¹Ôº¶L¿+yx2xP%¥H ¤C¶*ºuuy y[¶mºÐ CHFx2yyT A 5ú Hy

peyT *uwnÌy1uyy[m×

y[ey[2¹Ôjp(p, p)

CxP·y¨y Hy¼uy¨y¨ wmx x%w*Hyy[y'2w y[®º 2x2º<¹Úmum m Áy[m ¥wÂô©Â;DLy[ wy¶ywyy mx¶*ºu*uywyTy¼"y wm×®y[xà wmxy[ºÃ

|||vh|||2u

= ‖σ12 vh‖

20,Ω + ‖ν

12∇vh‖

20,Ω + jv(vh,vh) + ‖|β · n|

12 vh‖

20,∂Ωin

+ ‖(γνν)12h−

12 vh‖

20,∂Ω + ‖γ

12nh

− 12 vh · n‖2

0,∂Ω,

¹ÔLvh ∈ [V k

h ]dÂ

<ß Û² 'u ∈ [H2(Ω)]d

,+, #,.%/

|||u − πhu|||u ≤

[

σ12 h2 + max‖β‖0,∞,Ωh, ν

12h+ γdivh

]

‖u‖2,Ω,

5C4 y[ euwyC¹m¹s¿5yx2xP¥wÂô©Ã"· ® emm e¼uyx232wyTÏmºÐ[º HFx2yy3º uyº Hmx2y[º×ÖmºÐ[º 5Â

À1yuw2uwymn¸ºxPº ÕmFy2m¹¼¿5y[xxP§¥wÂô£ÁP¶3nhÂε ÒuyumuX*y

xCHy[1yºx2ymÃ"¶yCu¶=º ½uy¹Ôº¶* Öyx2xPPuH1uyCºy xEºÌy[1m em+¹‖∇ ·

uh‖0,ΩÂ;DLy[ wy¨uwyn¸ºxPmººPyTº*LmºxPbÂ

<ß Û² -,.&(0/ $7/ =<'%>' &Q ÆÚ¤¤mÊ P,/ ' 4&Q@ ÆÚÉeÊ ' *

‖∇ · uh‖0,Ω ≤ |||(uh, ph)|||u.

5C4 ty[ P¶*ºu(vh, qh) = (0, πh∇ · vh)

ºy[w

‖∇ · uh‖20,Ω ≤ ‖∇ · uh − πh∇ · uh‖

20,Ω + jp(ph, πh∇ · uh) + 〈πh∇ · uh,uh · n〉∂Ω .

wm3uy#2wF3y[xöyuwnÌmy

‖∇ · uh − πh∇ · uh‖20,Ω ≤ ‖∇ · uh − π∗

h∇ · uh‖20,Ω ≤ ju(uh,uh).

wm<uyy[ wPyxÃmº MwFueÍFu¶mÐ* y[ewmººmÃFmy¨ y[ewmººP w Ö ÌmyFy yTH×ö<ym

jp(ph, πh∇ · uh) ≤ jp(ph, ph)12 jp(πh∇ · uh, πh∇ · uh)

12

≤ ε−1jp(ph, ph) + εC‖∇ · uh‖2,

¶*ºuε > 0

m wm e1Hy 2¸y[½y¨m 5 Aº wmºmÃ"¹Ô1uy%"m HyxÃB¶<ymº Ϻ Ïuyx2y¨¹Úmum

〈πh∇ · uh,uh · n〉∂Ω ≤ ε−1‖γnh− 1

2 uh · n‖2∂Ω + εC‖∇ · uh‖

2.

ÜÝy wwy¹AeFuº ε > 0

xP6y wmu5Â

* * æRSTEH

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©© "@& ) /K >P '<

ßã ± ß (u, p) ∈ [H2(Ω)]d ×H1(Ω)

<7 , @ 4 ÆÇTÊ /(uh, ph) ∈ Wh

<7-,. '@ 4 ÆÚÉeÊ ,

|||u − uh|||u ≤ Ch‖p‖1,Ω

+ C

[ (

σ12 +

‖β‖1,∞,Ω

σ1/2

)

h2 + max‖β‖0,∞,Ωh, ν12h+ γdivh

]

‖u‖2,Ω.

5C4 ܧy¹Ôm¶=uy2¹sm¹<t3uyy[xE¥w ¤ÖF·º w® m e¨uyyTw×F¼m¹<¿5y[x2x2¨¥w Ém w¥w ¤Ât3uwöyuwnÌy

bh(yπ, eh) = −(yπ,∇ · eh) + 〈yπ, eh · n〉∂Ω

≤ ‖yπ‖0,Ω‖∇ · eh − π∗h∇ · eh‖0,Ω + C‖yπh

12 ‖0,∂Ω‖γ

12nh

− 12 eh · n‖0,∂Ω

≤ Ch‖p‖1,Ω|||(eh, yh)|||,

jp(πhp, yh) ≤ jp(πhp, πhp)12 jp(yh, yh)

12

≤ jp(πhp, πhp)12 |||(eh, yh)|||

≤ Ch‖p‖1,Ω|||(eh, yh)|||,

¶*uwFuº¶wPmº wy[®uyyTy[y[x2yÂA x2x2y[ym HyTwy wy¨m¹+uy"Ìmy eÌymy[ wy¨ wmº3uy¹Ôº¶* mmm

ã ± ã ' 'Ú²± à yh = ph − πhp

-,.' #,.%/

jp(yh, yh)12 ≤ Ch‖p‖1,Ω + C(ν

12h+ h)‖u‖2,Ω,

jp(ph, ph) ≤ Cjp(yh, yh).

ßã ± ß (u, p) ∈ [H2(Ω)]d ×H1(Ω)

<7 @ 4 ÆÇ[Ê /(uh, ph)

<7Q, 6 9 /; 8 & ; '3@ 4 ÆÚÉeÊ -,.'+-,.' ,%/

‖p− ph‖ ≤ Ch(‖p‖1,Ω + ‖u‖2,Ω)

5C4 µ Ýuy®x2yPºº%muyPem¹m¹3t3uwymyx ¥wÂô©ÃH M<mm¥H ©Á (uyyy¨wyTy¼ºy[ ex2w[Â

9.)6 <A (.*G)+$ :A

t3uwy<x2yuy[y ey[%mHÌyuwAmx2ymx2x2m %¹Ôy[y[¶*×uCÌmº wx%º[y<x2yu¹Ômy<y[x%º wÆÔÄO§ÊAmº ewy[ ¾ô©¥ËbÂAÌmy[ umumwxP %y[Ìmy<¹ÔxCy<2w ºy<yyx2y eAxyu%¶*uFu%AFy`¹ÔAmº*y w5 x%"yF+uy<º ey[wyº m¹+uyy[y¨mºÐ[m xyuwº PÀ1×FFuy"y¶<yT·ºPx2HeyT®"mw w®m wºm mx2yuϹÔm¿5Ï yy[ wÂ(DL¶yÌy¨ º·yuyPÄ OÏÁ¶*uwyyC¶[y[

Vh w

VHy

m wyy[6Ãmº Psy3uy2w w×y1yyx2y[ wyVhyy[y eFAuy¼m %y[mÌmyTCmºyLm wuy

å)(+*5åÙí

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!#"$%'&('*)+-,.0/ ©m¤

wwºy[ ¼ÌeºPmm Ö ÖuwymºyT<uw*my¼ *y[mÌmyTPm VhÂ`ÀLmyCuH3uy

¶y[m·"m wwC wºm ws¶*ºH,sPH wmm H׺ w` 2uy1uºuÖ*y w` x%"yyºx2y H3 ͵ºm w׺ w*¶*uy[ uy"m HÖnmyF3yyTºÌy[ÖÌyÖx%HFu·mym y¶mwݺ·y2uy[x Ͻ Û¿5<BÂA*y[y e§ÁÄOÏݶ3mm"y[Ýw Áy[m Ýx2yuw¹ÔFx2y[¶<·¾ô©m£nË Ât3uyy[¶3mP¶<·P ®¶P Byy e1mºyT

Vhm w

VH wº ewyuy

y wy%x2y(m wºm Ïuy 2w y[mºyTLyy[y ey[½Vh \ VH

Ât3uy%¹ÔxCm ½m¹1¾ô©m£TËyTy emº·y[suy¨¹Ôx ¹ACm wF®Ñmºy[· Ö¹Ômx%wm P¹Ô

uhwyx2y eyTÖ¶*ºuuy

y eLÌeºÖm 2m Öm uwy#2w yy[ uy¹Ôxm¹(m ºm wmBy[x

(νT (I − PH)ε(uh), (I − PH)ε(vh)),ÆV¤£mÊ

¶*uwyyPH

2mx2yx2mÛ¹Ômx 2H ymºyTCÓmymºyT M<mx2w Ï ¶ ¶*×uÒuyyTmymºÐ[m xyu2¶y*¶<Fuweey

H = h wueH`xP·my3uy3wwºy[ <Ìeº%mm

½uwyCy[Luwmy% ¨yTºÌy[m uy2wyVh A º ¿5y[xxP¤ÂǶ<yCºx2x2y[y

yL yº*Hy[ wºÖ eyyFm ¹uyyx/ÆV¤£Êö*×uPH

def= π∗

h

Ã

‖ν12

T (I − π∗h)ε(uh)‖2

Ω ≤∑

K∈Th

∂K

νThKJε(uh)K : Jε(uh)K ds

m w¶<y wwyuwL2"ywmx2yº yx¶<"y

jT (uh,vh) =∑

K∈Th

∂K\∂Ω

νThKJε(uh)K : Jε(vh)K ds

¶*uwyy<uyLFuy<m¹νT w¶Ò`¼x2y[º ym suwmHy3 my[Cuw`uyLFumy

νT = γhKºÌy[*H¼2yxE¶*uFu½¨mex2[ºy[eÌnmºy[ e1ÖuyCy[yCwººÐTº Hy[Lwº uy½¶*uºy½ºy[ e[Â9t3uPx2y[m w2uHm eÎFumy¹

νT¶*ºuXumuwyÖy[Pº

hK¶*º¼Hy

wymmºyxHy[ÓÏuyyBy[,Pm¹Luy½mºÐ Áyx2[ wm2º HF wyÖuwy½x2mm w·ey wysx2yemÌmy[ e

νT = Ch2K |ε(uh)|

y[ º uy*2w y[y[5xCw"y<y¸"y[yTHy®mx2yyÏmx2 wy[ݽuwyÖFºÐ y[xPy¸y¶*uy

|∇uh| ≈ h−1K

 A uy ey[yTº wyÌnm º u¹Ômx2y¶m·`uw¹Ôx@"m mx2wy[myy3uy[y3uy[y*y¸C1 wwmy1m¹ Vh

¶*ºuwn¸xPm PwHy[ºyT¼ÆVF C¶*ºuuwy¼y[Hy[LD1ºy[uÍCM<ºmuÍtFuywmymÃ5yy¾ºÇT¤nËÔÊ, ¨¹Ôº¶L¼uwuyPmm ÝxPn½HyPyTmx2"y[Ϻ e y

C1 wm¶*uwFuÓ ewFuy[§ÁuyÖFºÐ[º yxP wÝ wuwyC0 wm¶*uwFuÝHy[ wºÐ[y[6Âܧym wºHy1uwuwy¼x2yu®y[ uy[y¼ywmm Pm"yPFuwFmy[*¹Ôm3Ìm wm

xCº[yx2yuww6m"y[ P¾ ©¥Ë¨Hº x2mmmFy[5uy[5uw ¨uºy[mFuw(x2y[uyTÂwmx2myymºy[Ówm Ó wy[ wº ÓÆÔmʼyT,m ÓwººÐ[y[+2w ºyPy[ºy[xy[ e Hmy¨y[w®ºxCm y[yP¾ ¤ËbÂ

-/:$M3(#<%ÕA*):,)

µ Óu%y[,º Õ¶<y®y[H%yÌyF` x2y[y¸Hy[x2y euwCuw¶Luy®mÝm Ìmy[y wywHy[ºyT`¹6mwsFºÐ[y[ 2w ºy¼yyx2y esx2yu6Â(µ Pwmm[ö<y*y[Ìmy[(uy1m Ìmy[y wyFy[m yT ½y[m ¥Â

* * æRSTEH

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©¥ "@& ) /K >P '<

) #') ( *"&& )(

¿+yλ = (ν−1 − (ν−2 + 16π2)

12 )/2

Ât3uwy uyy¸,Lmm ÁÆÇTÊ3mÌmy[ ÏÆVy[yP¾ ©eÈ[ËÙÊ

u1(x1, x2) = 1 − eλx1 cos 2π x2,

u2(x1, x2) = λx1

2π eλ sin 2π x2,

p = 12e

2λx + C,

¶*ºuβ = u

Ãσ = 0

m wÓmueuwm wÕy®x2Fu uyÖy¸,%mm 5µ Õmwy¸x2y[[öy®mÁFuy

CÁºÌyPÐy[½x2y[m Õy[ymÂÜÝymÌmy[§u%myx!n¸ºxPÌmyÁ

Ω = (−1/2, 3/2) × (0, 2)Ã5wº wº×ÏwFx2yyF1H FººÁwmmx2yy[

γβ = γp =γ = 1/100

m wγβ = 1/10

Â+ÜÝyy[y e<uyºm wm ®¶<2"y[y[ e3x2yTuyT3º L2wy©C Hǵ 2wy¤2¶yu¶uy Ìmymy[ wy¼¹Ôm

ν = 10−4 Â

Aºy2ÇKE(¿+y¹Ù0Emy¨xyTuÌy׺yTÂ`*mue0E 2H yx2y[uÌyºy[

Amy©@E¿+y¹Ù5Emy¼x2yTuy[ymÂ`*mue0E 2H yx2y[uwyTy

å)(+*5åÙí

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!#"$%'&('*)+-,.0/ ©£

-6 -4 -2 0 2

-11

-10

-9

-8

-7

-6

-5

-4

-3

log( h)

log(

L2-e

rror

)

VelocityPressure

Aºy¤E!M<m Ìmy[y wy1umP¹Ômν = 10−4

) #') ( *"&& )(

ÜÝym Hy< ¶ myx ÆÇTÊsº ®¤» ¶*×u® Íbuwmx2my y[mw`"mw wP wºm w[ÂAt3uwy¼ºueuH wy

fm wuy"m HÖF2yFuwy ® mFyPy wy1uw*uyy¸mLºº m¹

ÆÇTÊ<*mÌmy[ Puy¹Ômº¶* y¸y[m §¾ºÇTÅnË8E

u1(x1, x2, x3) = bea(x1−x3)+b(x2−x3) − aea(x3−x2)+b(x1−x2),

u2(x1, x2, x3) = bea(x2−x1)+b(x3−x2) − aea(x1−x3)+b(x2−x3),

u3(x1, x2, x3) = bea(x3−x2)+b(x1−x2) − aea(x2−x1)+b(x3−x1),

p(x1, x2, x3) = (a2 + b2 + ab)

[

ea(x1−x2)+b(x1−x3)

+ ea(x2−x3)+b(x2−x1) + ea(x3−x1)+b(x3−x2)

]

.

ÆV¤mÉÊ

¶*ºuβ = u

Ãσ = 1

Ãν = 10−4 Ã a = b = 0.75

wΩ = (0, 1)3

uyw ×1"ymÂt3uyPy[º ½ º wmyx ¶ºÌy[Ýn¸ºxPÌmywº uyÖwººÐ[y[ÝÍ

yy¨¹Ômx%wm §ÆVÉÊÃu¶yÌmy[uy"m HÖ wºm w*¶yy mPy ¹Ôy[6 A º+ ewxy[Ímy[2uHnÌmyHy[y XHy[¹Ôx2yTÛwº wÓm ¹Ôx2 Ï y[mÖ wÎHmFJ2w ºy½y[ºy[xy[ eC¹ÔmÌyº HyTymÃH wx2y

P1/P1 w

P2/P2ÆÔx2ºy[x2y eyTÁ ÁÖ¤m»@yTyTFFuyÖ¾ºÇT¦nËÔÊ,Â

t3uwymºÐ[m wFx2yyFº ÌmºÌy[®º uwy3x2w3y[xPÆÇ[ªÊm wÓÆÇmÇTʶ<y[yFuey[ m

γp = γβ = γdiv =

1

8

×¹k = 1,

1

32

×¹k = 2,

* * æRSTEH

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©mÉ "@& ) /K >P '<

0.0001

0.001

0.01

0.1

1

10

0.01 0.1 1

erro

r

h

pressure L

velocity H hh

velocity L

2

21

2

Aºy¨¥ E*M<m Ìmy[y wy1umE(º y[m3yyx2y e(k = 1)

1e−05

0.0001

0.001

0.01

0.1

1

0.01 0.1 1

erro

r

h

erro

r

pressure L

velocity H h

velocity L 21

2

h 5/2h 3

2

Aºy£.E!M<m Ìmy[y wy1umE(ewF¨yyx2y eF(k = 2)

µ 2wyT¥Ým wK£Ï¶yu¶Ãy["y[,ºÌymÃ`uy½Ìmy[º×Ûm wÛy[y Ìmymy[ wy®uy[¹Ô

k = 1m w

k = 2ÂÕÀLmyuw[Ã Ò"uÎyTÃ(uy ewxy[mmº Ûy¸uºmºxPm

m Ìmy[y wy1y[[Ãuy[ wyà mmyy[xy[ e¶*ºu½t3uy[myxP¥wÂÇm w¥wÂô©ÂÜÝy1u¶Ãm 2wmy¼È1uwyLy[y* mF( C¶"y[y[ esx2y[uy[LÆÚ¶*uFu2myLy,y[2

2Hmy1ÉmÊswº %º y[msy[ºy[xy[ e[Â(À1CºwwyTy*ºº wsmyLwyÌmy[5Â+ÜÝy¼y"ms 2Hmy1Åuy¼m e¹Buy¨yT wPx"m wy es¹

uhÃuh2

ö*×uγβ = 1/8

wγβ = 0

Ãmm Öuyº wwm y

x = 0.5 A ºuwmuÁuwyy¸m,mm §xmuÁuy2¹uy2 wmºÐy[

mº XÆÚmue,ʨy¸u×F%ºweºm w[ÂÀ1yPuwuwy®wmHÌmy[ººÁeºm wÆÚmue,Ê<mymx2yy[ºm eºy[®e®uyy[mx2º y͵y[ÌÌmysx2wÆÚºy¹ÙFÊ,Â

å)(+*5åÙí

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Page 31: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

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Page 34: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

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)= ah(uh, πhvp) + ju(uh, πhvp)

+ bh(ph, πhvp)

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bh(ph, πhvp) = −(ph,∇ · πhvp) + 〈ph, πhvp · n〉∂Ω

= −(ph,∇ · (πhvp − vp)) + 〈ph, (πhvp − vp) · n〉∂Ω

+ ‖ph‖20,Ω

= (∇ph, πhvp − vp) + ‖ph‖20,Ω

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20,Ω

≥ −‖∇ph − π∗h(∇ph)‖0,Ω‖πhvp − vp‖0,Ω + ‖ph‖

20,Ω

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20,Ω

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20,Ω

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α1|||(uh, ph)|||

2u

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ah(uh, πhvp) + ju(uh, πhvp) ≥ −C|||(uh, 0)|||‖vp‖1,Ω

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α2|||(uh, ph)|||

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20,Ω,

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hm w

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À1¶Ãeyº (vh, qh)

def= (uh + απhvp, ph), α > 0,

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)= A

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)+ αA

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)

≥ C|||(uh, ph)|||2− αC3|||(uh, ph)|||

2+ αC3‖ph‖

20,Ω

≥ C4|||(uh, ph)|||2∗.

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hm w

ν Aº wmºmùÔx ÆÚ¤ÈÊÃ×*¹Ôº¶Luw

|||(πhvp, 0)||| ≤ C‖ph‖0,Ω,

m w®uy[y¹Ômy

|||(vh, qh)|||2∗ = |||(uh + απhvp, ph)|||

2+ ‖ph‖

20,Ω

≤ 2|||(uh, ph)|||2+ 2α2|||(πhvp, 0)|||

2+ ‖ph‖

20,Ω

≤ C5|||(uh, ph)|||2∗,

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Page 37: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

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Page 38: Edge Stabilization for the Incompressible Navier-Stokes … · 2020-07-04 · ISSN 0249-6399 ISRN INRIA/RR--5349--FR+ENG apport de recherche Thème BIO INSTITUT NATIONAL DE RECHERCHE

Unité de recherche INRIA RocquencourtDomaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)

Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)

ISSN 0249-6399