XX Cong eso de Ecuaciones Di e enciales y Aplicaciones
X Cong eso de Ma em´
a ica Aplicada
Se illa, 24-28 sep iemb e 2007
(pp. 1–6)
The nume ical analysis o highe -o de nonlinea FE
me hod o ad ec ion domina ed p oblems
S. M. C. Mal a, R. C. C. de Almeida
Labo a ´o io Nacional de Compu a¸c˜ao Cien ´ı ica (LNCC/MCT)
A . Ge ´ulio Va gas,333 - Pe ´opolis,
22561-070, Rio de Janei o, B azil.
E-mails: [email p o ec ed], [email p o ec ed].
Keywo ds: ad ec ion domina ed p oblems, ini e elemen me hod, nume ical analysis
Resumen
The nume ical analysis o nonlinea discon inui y-cap u ing me hods applied o
ad ec ion domina ed p oblems has no been comple ely es ablished ye . Some pa -
icula esul s gi e e y good con ibu ion owa ds he exis ence o disc e e solu ions
al hough no uniqueness esul s a e demons a ed. This pape s udies he condi ions
o he uniqueness o he solu ion when using o he CAU (Consis en App oxima e
Upwind) Pe o -Gale kin o sol ing ad ec ion domina ed p oblems. The main issue
in his analysis is ha i elies on he op imali y p ope y o he CAU solu ion which
does no depend on any es ic ions o he app oxima ion spaces.
1. In oduc ion
I is well known ha nume ical simula ions o ad ec ion-domina ed p oblems p esen
nume ical di icul ies ela ed o he lack o s abili y: because ad ec ion domina es di usion,
classical Gale kin ini e elemen (FE) me hods gene a e uns able app oxima ions, which
usually exhibi spu ious oscilla ions. The SUPG (S eamline Upwind Pe o -Galekin)
me hod, p oposed by B ooks and Hughes [1], was he i s a ia ionally consis en , s able
and accu a e ini e elemen model o ad ec ion domina ed p oblems. Fo egula solu ions
his me hod p esen s quasi-op imal a es o con e gence o he s eamline de i a i e and
was i s analyzed by Johnson e al. [4]. Ne e heless, o non egula solu ions, localized
oscilla ions a e s ill obse ed in he neighbo hood o s eep g adien s meaning ha he
s eamline is no always he app op ia e upwind di ec ion. To o e come his lack o mono-
onici y many discon inui y cap u ing e ms we e designed o enhance s abili y ei he in
1
S. M. C. Mal a, R. C. Almeida
a linea [5] o nonlinea way [2, 3]. Qui e p omising esul s we e ob ained using he non-
linea CAU (Consis en App oxima e Upwind) ini e elemen me hod p oposed in [2]. The
sys ema ic way o upda ing he upwind di ec ion in he CAU me hod esul s in adding o
he SUPG o mula ion a nonlinea discon inui y-cap u ing e m in a consis en way, en-
gende ing an addi ional s abili y in he di ec ion o he app oxima e g adien . The heo y
has been e ined o e he yea s in se e al di ec ions.
The nume ical analysis o he CAU me hod, e en o highe -o de elemen s, has been
discussed in a ecen pape [3]. The s abili y analysis was shown based on a linea ized
i e a i e scheme, which uses he solu ion o he SUPG me hod as an ini ial guess in o de
o sol e he CAU (nonlinea ) me hod. Howe e , his analysis has some open ends [6]
conce ning he sol abili y o he i e a i e scheme. The e o e, ou main goal is o imp o e
he de eloped analysis. In pa icula , we add ess he p oblem associa ed o he con e gence
o he linea ized i e a i e scheme as well as he uniqueness o he solu ion o he nonlinea
app oxima e me hod.
2. Ma hema ical Model
We a e in e es ed in s eady-s a e solu ions o con ec ion-domina ed eac ion-di usion
scala p oblems o he o m
Lφ:= γφ +u·∇φ−²∆φ= in Ω
φ= 0 on Γ,(1)
whe e udeno es a gi en eloci y ield, ²is he gi en (small) posi i e di usion coe icien ,
γis he gi en non-nega i e eac ion coe icien and is a sou ce e m. The p oblem is
de ined in he domain Ω ⊂ Rnwi h bounda y Γ.
The Gale kin o mula ion o p oblem (1) eads: ind φh∈Vp
h⊂H1
0(Ω) such ha ,
B(φh, ηh) = ( , ηh),∀ηh∈Vp
h,
wi h
B(φh, ηh) := ²(∇φh,∇ηh)+(u·∇φh, ηh) + γ(φh, ηh)
and Vp
h={ηh∈C0(Ω); ηh|Ωe∈Pp∀Ωe, ηh|Γ= 0}, whe e Ppis he se o in e pola ion
polynomials o deg ee less o equal pde ined in each ini e elemen Ωewi h cha ac e is ic
elemen leng h deno ed by h.
Fo ad ec ion-domina ed p oblems, when ²is aken e y small in p oblem (1), i is well-
known ha classical Gale kin and s abilized ini e elemen me hods gene a e global and
local uns able app oxima ions, espec i ely. As a emedy, discon inui y-cap u ing a ian s
a e conside ed as an addi ional (bu nonlinea ) s abiliza ion. Qui e p omising esul s ha e
been ob ained using he CAU (Consis en -Upwind Pe o -Gale kin) [2, 3] o mula ion,
which eads as: ind φh∈Vp
hsuch ha
D(φh, ηh) +
Ne
X
e=1
(L(φh)− , τc[u− ]·∇ηh)|Ωe=F(ηh),∀ηh∈Vp
h,(2)
2
The nume ical analysis o highe -o de ...
wi h
D(φh, ηh) := B(φ, η) +
Ne
X
e=1
(Lφh, τsu·∇ηh)|Ωe;
F(ηh) := ( , ηh) +
Ne
X
e=1
( , τsu·∇ηh)|Ωe,
whe e τsand τca e he s abiliza ion pa ame e s om he SUPG [1, 4] and CAU [2]
me hods, espec i ely. The e a e a a ie y o possible designs o hese pa ame e s in he
li e a u e [1, 2, 3, 4, 6, 5]. The auxilia y ec o ield in equa ion (2) is a modi ied eloci y
ield such ha i sa is ies he o iginal pa ial di e en ial equa ion (1) o he app oxima e
solu ion φh∈Vp
hin each elemen . This equi emen implies
[u− (φh)] ·∇φh=L(φh)− =: R(φ).(3)
Then, he ec o ield is de e mined such ha i is he eloci y ield closes o he
eal eloci y ield uin he L2sense. Hence, i is ob ained by sol ing he ollowing local
minimiza ion p oblem: ind u∈L∞(Ω)dsuch ha
ku− kL2(Ωe)≤ ku−mkL2(Ωe),∀m∈Qm,(4)
wi h
Qm={m;m·∇φh−²∆φh+γφh− = 0 in Ωe, e = 1, . . . , Ne}.
The solu ion o (4) oge he wi h (3) leads o
( −u= 0 ,i |∇φh|= 0 ;
u− =R(φh)
|∇φh|2∇φh,o he wise.(5)
Thus, i |∇φh|= 0, he linea SUPG me hod is eco e ed. O he wise, |u− |=|R(φh)|
|∇φh|=:
β(φh), which ensu es ha , in each elemen Ωe, limh7→0 (φh) = u, when limh7→0φh=φ.
Besides, as is selec ed minimizing ku− kL2(Ωe), he ollowing inequali y is e i ied
β(φh) = |R(φh)|
|∇φh|≤|R(ψh)|
|∇ψh|=β(ψh),∀ψh∈Vp
h.(6)
Due o his p ope y, one may conclude ha he app oxima e solu ion o (2) using (5) is
op imal. This is a ema kable ea u e o his me hod. In ac , he p ope y (6) is c ucial
o yield he uniqueness esul p esen ed in nex sec ion.
3. Uniqueness o solu ion
Le he CAU me hod (2) be e-w i en as: ind φh∈Vp
hsuch ha
a(φh;φh, ηh) = F(ηh),∀ηh∈Up
h,(7)
whe e
a(φh;φh, ηh) := D(φh, ηh) +
Ne
X
e=1
c(φh;φh, ηh)|Ωe,(8)
3
S. M. C. Mal a, R. C. Almeida
wi h
c(φh;φh, ηh)|Ωe:= µτc|R(φh)|2
|∇φh|2∇φh,∇ηh¶|Ωe.
The non-symme ic bilinea o m D(·,·) is coe ci e (see, o example, [3]), which implies
he ollowing SUPG-s abili y p ope y
D(wh, wh)≥C(θ)|||wh|||2,∀wh∈Vp
h,(9)
wi h C(θ)=1−1/2√1 + θ, whe e θis de ined in [5]. The bilinea o m D(·,·) is also
con inuous in he ollowing s abilized mesh dependen no m:
|||ηh|||2:≡²k∇ηhk2+γkηhk2+
Ne
X
e=1
τsku·∇ηhk2
Ωe.(10)
yielding uniqueness and exis ence solu ion o he SUPG me hod [4], aking τc= 0 in
p oblem (7). Besides, p o iding ha τc≥0, c(ηh;ηh, ηh)|Ωe≥0 o all ηh∈Vp
h, he
s abili y o he CAU me hod ollows immedia ely om he p e ious SUPG s abili y esul ,
a(ηh;ηh, ηh)≥C0(θ)|||ηh|||2.(11)
Fo a qui e simila me hod, i was assumed in [7] ha he quan i y |β(·)|sa is ies
q0≤ |β(φh)| ≤ q1,(12)
o q0, q1>0, implying coe ci i y and con inui y o he bilinea o m a(ηh;·,·) and a
unique solu ion ha comes om he Lax-Milg am heo em [5, 6, 7]. Howe e , a p oo o
he lowe bounded is no exhibi ed in [7]. La e on, i was applied a a ian o B ouwe ’s
ixed poin heo em o p o e he exis ence o a disc e e solu ion o he CAU nonlinea
p oblem (7) bu a uniqueness esul emained s ill open [5]. The e o e, using only he
op imal-CAU p ope y (6) we will show his ac in he ollowing heo em.
Theo em 1: Assuming ha he minimiza ion p ope y (6) holds, he CAU me hod (7)
has a unique solu ion φh∈Vp
h.
P oo : Conside wo solu ions φhand ˜
φhbelong o Vp
ho p oblem (7). Using he de ini ion
o unc ion β(·), we ha e
D(φh−˜
φh, ηh) +
Ne
X
e=1
c(φh;φh, ηh)−
Ne
X
e=1
c(˜
φh,˜
φh, ηh) = D(φh−˜
φh, ηh)
+
Ne
X
e=1 h(τc[β(φh)]2∇φh,∇ηh)|Ωe−(τc[β(˜
φh)]2∇˜
φh,∇ηh)|Ωei= 0.(13)
F om he op imal p ope y o he CAU solu ion, β(φh)≤β(ψh),∀ψh∈Vp
h, ha is,
β(φh)≤β(˜
φh), since ˜
φh∈Vp
h. Simila ly, β(˜
φh)≤β(φh)as well. Hence,
β(φh) = β(˜
φh) = α, (14)
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The nume ical analysis o highe -o de ...
whe e αis a non-nega i e cons an . No ice ha α= 0 only when R(φh) = 0, ha is, φh
sa is ies he esidual equa ion and he non-linea ope a o anishes. Now, subs i u ing (14)
in o (13) yields
D(φh−˜
φh, ηh) +
Ne
X
e=1 ³τcα2(∇φh−∇ ˜
φh),∇ηh´|Ωe= 0,∀ηh∈Vp
h.(15)
Nex , aking ηh=φh−˜
φh∈Vp
hand using he SUPG-s abili y p ope y (9), he equa ion
(15) is e-w i en as
C(θ)|||φh−˜
φh|||2+α2
Ne
X
e=1 ³τc(∇φh−∇ ˜
φh),∇φh−∇ ˜
φh´|Ωe≤
C1(|||φh−˜
φh|||2+
Ne
X
e=1
τc|∇φh−∇ ˜
φh|2
Ωe)≤0,
whe e C1= max{C(θ), α2}is a posi i e cons an . The e o e, since τc≥0, he abo e
inequali y yields φh=˜
φh.¤
4. Linea ized p oblem
To sol e he nonlinea p oblem (7) i is necessa y o conside i e a i e me hods which
a e in ended o keep he main p ope ies o he CAU me hod a each i e a ion. This can
be achie ed by using he ollowing simple me hod: om he p e ious compu ed i e a e
solu ion φn
h,n∈ N, we ge
c(φn
h;φn+1
h, ηh)|Ωe= (R(φn+1
h), τn
c[u− (φn
h)]·∇ηh)|Ωe=¡τc|β(φn
h)|2∇φn+1
h,∇ηh¢|Ωe,(16)
whe e
τn
c=τc(u− (φn
h)) and (u− (φn
h)) = R(φn
h)
|∇φn
h|2∇φn
h.
Hence, he i e a i e algo i hm consis s in: gi en φn
hand [u− (φn
h)], ind {φn+1
h} ∈ Vp
h
such ha
a(φn
h;φn+1
h, ηh) = F(ηh),∀ηh∈Vp
h,(17)
whe e
a(φn
h;φn+1
h, ηh) = D(φn+1
h, ηh) +
Ne
X
e=1
c(φn
h;φn+1
h, ηh)|Ωe
is a non-symme ic bilinea o m de ined on Vp
h×Vp
h, o a gi en φn
h∈Vp
h. I is coe ci e
and con inuous [3, 5, 6] in he no m (10). Consequen ly, he linea ized p oblem (17) is
uniquely sol able ia he Lax-Milg am heo em. The ze o- h i e a i e solu ion φ0
his he
SUPG solu ion, which can be seen as he ze o o de esidual co ec ion o each τ0
c= 0.
Due o he p ope y (6), any i e a i e solu ion {φn+1
h}o (17) sa is ies
β(φh) = |R(φh)|
|∇φh|≤|R(φn
h)|
|∇φn
h|=β(φn
h).(18)
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S. M. C. Mal a, R. C. Almeida
In addi ion, he sequence o solu ions {φn+1}is also uppe bounded [3], ha is, he e is a
posi i e cons an C, independen o n, such ha ,
|||φn+1
h||| ≤ Ck k,∀n. (19)
F om his ac and assuming ha {φn+1
h}, solu ion o he linea ized p oblem (17), con e ges
o φh, solu ion o he nonlinea CAU me hod (7), when n7→ ∞, we may p o e ha he
addi ional e m c(φh, φh, ηh) does no deg ade he a es o con e gence as long as egula
solu ions a e conce ned [3]. This esul is p esen ed in he ollowing heo em.
Theo em 2: Conside ing ²=O(h) and τs=O(h/p) we a i e a he ollowing a p io i
e o es ima e:
|||φ−φh|||2≤C
Ne
X
e=1 µh
p¶2s+1
|φ|2
s+1,Ωe,(20)
whe e φ|Ωe∈Hk+1(Ωe) o some k≥1, o any 0 ≤s≤m´ın(p, k), p≥1.
The p oo o he p e ious esul is based on an impo an open p oblem: o p o e ha
he sequence {φn+1
h}con e ges o φh, solu ion o he nonlinea CAU me hod (7), when
n7→ ∞. Wo k in his subjec is ongoing and will be add essed in a o hcoming pape .
Acknowledgmen s
The au ho s would like o hank he B azilian Go e nmen , h ough he Agency CNPq,
o he inancial suppo p o ided.
Re e ences
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[4] C. Johnson, U. Na e and J. Pi ka an a, Fini e elemen me hods o linea hype bolic p oblems,
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