A POSTERIORI ANALYSIS OF ITERATIVE ALGORITHMS FOR A NONLINEAR PROBLEM Christine Bernardi, Jad Dakroub, Gihane Mansour, Toni Sayah To cite this version: Christine Bernardi, Jad Dakroub, Gihane Mansour, Toni Sayah. A POSTERIORI ANALYSIS OF ITERATIVE ALGORITHMS FOR A NONLINEAR PROBLEM. 21 pages. 2013. <hal- 00918226v2> HAL Id: hal-00918226 http://hal.upmc.fr/hal-00918226v2 Submitted on 18 Dec 2013 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 destin´ ee au d´ epˆ ot et ` a la diffusion de documents scientifiques de niveau recherche, publi´ es ou non, ´ emanant des ´ etablissements d’enseignement et de recherche fran¸cais ou ´ etrangers, des laboratoires publics ou priv´ es.
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A POSTERIORI ANALYSIS OF ITERATIVE
ALGORITHMS FOR A NONLINEAR PROBLEM
Christine Bernardi, Jad Dakroub, Gihane Mansour, Toni Sayah
To cite this version:
Christine Bernardi, Jad Dakroub, Gihane Mansour, Toni Sayah. A POSTERIORI ANALYSISOF ITERATIVE ALGORITHMS FOR A NONLINEAR PROBLEM. 21 pages. 2013. <hal-00918226v2>
HAL Id: hal-00918226
http://hal.upmc.fr/hal-00918226v2
Submitted on 18 Dec 2013
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, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.
strt ♣♦str♦r rr♦r ♥t♦rs ♥ st ♥ r♥t ②rs ♦♥ t♦ tr r♠r♣t② t♦ ♥♥ ♦t s♣ ♥ r② ♥ ♦♠♣t♥ s ♦r s t ♣♦str♦r rr♦rst♠t♦♥ ♦r t ♥t ♠♥t srt③t♦♥ ♦ ♥♦♥♥r ♣r♦♠ ♦r ♥ ♥♦♥♥r qt♦♥♦♥sr♥ ♥t ♠♥ts s♦ t srt ♣r♦♠ s♥ t♦ trt ♠t♦s ♥♦♥ s♦♠♥ ♦ ♥r③t♦♥ ♦r ♦ t♠ tr r t② t♦ s♦rs ♦ rr♦r ♥♠② srt③t♦♥ ♥♥r③t♦♥ ♥♥ ts t♦ rr♦rs ♥ r② ♠♣♦rt♥t s♥ t ♦s ♣r♦r♠♥ ♥ ①ss♥♠r ♦ trt♦♥s r rsts t♦ t ♦♥strt♦♥ ♦ ♦♠♣t ♣♣r ♥t♦rs ♦r t rr♦rr ♥♠r tsts r ♣r♦ t♦ t t ♥② ♦ ♦r ♥t♦rs
és♠é s ♥trs rrr ♣♦str♦r ♦♥t été ♦♣ ♦♥sérés ♦rs s r♥èrs ♥♥ésà s rs ♣tés r♠rqs à ♠é♦rr tss t ♣rés♦♥ ♥s rés♦t♦♥ térts ♣r♦è♠s ♥s tr ♥♦tr t st ♣♣qr tt ♠ét♦ ♣♦r ♥ ♣r♦è♠ ♥♦♥ ♥ér♦s ♣r♦♣♦s♦♥s ♦rs ① ♦rt♠s térts rés♦t♦♥ ♣r♦è♠ t ♥♦s ét♦♥s ♦♥r♥ s ♦rt♠s rs s♦t♦♥ ♣r♦è♠ srt ❯♥ ét♣ ♠♣♦rt♥t ♦♥sst à é♠♦♥trrs st♠t♦♥s rrr ♣♦str♦r ♥ st♥♥t s rrrs ♥érst♦♥ t srétst♦♥♦s ♣rés♥t♦♥s ♥♠♥t qqs réstts ①♣ér♥s ♥♠érqs
②♦rs ♣♦str♦r rr♦r st♠t♦♥ ♥♦♥♥r ♣r♦♠s trt ♠t♦s
♥tr♦t♦♥
♥② rsr ♦rs t t ♣♦str♦r ♥②ss ♥ t ♣t ♠sr♥♠♥t ♦r ♥t ♠♥t srt③t♦♥ ♦ ♣t ♣r♦♠s ❬❪ rst ♣♦str♦r ♥②ss ♦♥tr♦s t ♦r srt③t♦♥rr♦r ♦ t ♣r♦♠ ♥ t ♣r♦s rr♦r ♥t♦rs ♥ ♦♠♣t r♦♠ t ♦♠♣t ♥♠rs♦t♦♥ ♥ t ♥ t ♦ t ♣r♦♠ ♥ ts rr♦r ♥t♦rs r ♦♥strt ♣r♦ tr♥② ② ♦♥♥ ♥t♦r ② t ♦ rr♦r s ♥②ss s rst ♥tr♦ ② s❬❪ ♥ ♦♣ ② ❱rurt ❬❪ ♥ ts ♦r r ♥trst ♥ st②♥ t ♦♦♥ ♥♦♥♥r♣r♦♠
t Ω ♥ ♦♣♥ ♣♦②♦♥ ♦ IRd, d = 2, ♦♥sr
−∆u+ λ|u|2pu = f ♥ Ω,
u = 0 ♦♥ ∂Ω,
r λ ♥ p r t♦ ♣♦st r ♥♠rs rt♥ s f ♦♥s t♦ H−1(Ω) t ♦t ♦♦ s♣ H1
0 (Ω) ❯s♥ P1 r♥ ♥t ♠♥ts t srt rt♦♥ ♣r♦♠ ♠♦♥tst♦ s②st♠ ♦ ♥♦♥♥r qt♦♥s tt r s♦ s♥ ♥ trt ♠t♦ ♥♦♥ s♦♠ ♥ ♦♥r③t♦♥ s t♦ s♦rs ♦ rr♦r ♣♣r ♥♠② ♥r③t♦♥ ♥ srt③t♦♥ ♠♥ ♦♦ ts ♦r s t♦ ♥ ts t♦ s♦rs ♦ rr♦r ♥ t t srt③t♦♥ rr♦r ♦♠♥ts t♥ t♥♦♥♥r s♦r trt♦♥s s r r♦r ♦r ♦t s t♦ t ♣♦str♦r rr♦r st♠ts
st♥s♥ ♥r③t♦♥ ♥ srt③t♦♥ rr♦rs ♥ t ♦♥t①t ♦ ♥ ♣t ♣r♦r s t②♣ ♦♥②ss s ♥tr♦ ② ♦ ♥ r ❬ ❪ ♦r ♥r ss ♦ ♣r♦♠s rtr③② str♦♥② ♠♦♥♦t♦♥ ♦♣rt♦rs t ♥ ♦♣ ② ♦ r♥ ♥ ❱♦rí ❬❪♦r ss ♦ s♦♥♦rr ♠♦♥♦t♦♥ qs♥r s♦♥t②♣ ♣r♦♠s ♣♣r♦①♠t ② ♣s♥ ♦♥t♥♦s ♥t ♠♥ts ♥ t t ♠♥ r♥ t♥ ts t♦ ♦rs s tt ♥ ❬❪ t②♦♥sr ♥ trt ♦♦♣ ♦r t ♥r③t♦♥ ♣r♦r ♥ ❬ ❪ t② r♣ t ♥♦♥♥r♣r♦♠ ② s♠♣ ♥r ♠♦ t♦t ♦♥sr♥ ♥② ♣t ♣r♦r
rtr♠♦r ♥ ts ♦r ♣rs♥t t♦ r♥t strts ♦r t ♥r③t♦♥ ♣r♦ss ♥♠② ①♣♦♥t ♦rt♠ ♥ t♦♥ ♦rt♠ ♦t strts r trt ♥ t ♦rt♠ ♥ ♦t♥s ♦♦s
♥ t ♥ ♠s ♣r♦r♠ ♥ trt ♥r③t♦♥ ♥t t st♦♣♣♥ rtr♦♥ s sts t rr♦r s ss t♥ t sr ♣rs♦♥ t♥ st♦♣ s r♥ t ♠s ♣t② ♥ ♦
t♦ st♣ (1)
♥ ♦♠♣r ts t♦ ♦rt♠s t② t t♦♥ trt♦♥ rs♦♥ s str ♦♥r♥rts t♥ t ①♣♦♥t trt♦♥ rs♦♥ t s ♠♦r s♥st t♦ t ♥t s
♥ ♦t♥ ♦ t ♣♣r s s ♦♦s ♥ t♦♥ 2 ♣rs♥t t rt♦♥ ♦r♠t♦♥ ♦ ♣r♦♠ (1.1)(1.2) ❲ ♥tr♦ ♥ t♦♥ 3 t srt rt♦♥ ♣r♦♠ t t ♣r♦r st♠t t♦r♥t ♦rt♠s r st ♥ t♦♥ 4 ♣♦str♦r ♥②ss ♦ t srt③t♦♥ ♦ ♦t ①♣♦♥t ♦rt♠ ♥ t♦♥ ♦rt♠ s ♣r♦r♠ ♥ t♦♥ 5 t♦♥ 6 s ♦t t♦ t ♥♠r①♣r♠♥ts
♥②ss ♦ t ♠♦
❲ sr ♥ ts st♦♥ t ♥♦♥♥r ♣r♦♠ (1.1)(1.2) t♦tr t ts rt♦♥ ♦r♠t♦♥rst ♦ r t ♠♥ ♥♦t♦♥ ♥ rsts s tr ♦♥ ♦r ♦♠♥ Ω ♥♦t ② Lp(Ω)t s♣ ♦ ♠sr ♥t♦♥s s♠♠ t ♣♦r p ♦r v ∈ Lp(Ω) t ♥♦r♠ s ♥ ②
‖ v ‖Lp(Ω)=
(∫
Ω
|v(x)|pd①
)1/p
.
r♦♦t ts ♣♣r ♦♥st♥t② s t ss ♦♦ s♣
Wm,r(Ω) =v ∈ Lr(Ω); ∀|k| ≤ m, ∂kv ∈ Lr(Ω)
,
r k = (k1, k2) s 2t♣ ♦ ♣♦st ♥trs s tt |k| = k1 + k2 ♥
∂kv =∂|k|v
∂xk1
1 ∂xk2
2
.
Wm,r(Ω) s q♣♣ t t s♠♥♦r♠
|v|m,r,Ω =
∑
|k|=m
∫
Ω
|∂kv|rd①
1/r
,
♥ t ♥♦r♠
‖ v ‖m,r,Ω=
(m∑
ℓ=0
|v|rℓ,r,Ωd①
)1/r
.
♦r r = 2 ♥ t rt s♣ Hm(Ω) =Wm,2(Ω) ♥ ♣rtr ♦♥sr t ♦♦♥ s♣
H10 (Ω) = v ∈ H1(Ω), v|∂Ω
= 0,
♥ ts s♣ H−1(Ω)
❲ r t ♦♦ ♠♥s s ♠s ❬❪ ♣tr 3
♠♠ ♦r 1 ≤ j <∞ ♥ d = 2 tr ①sts ♣♦st ♦♥st♥t Sj s tt
∀v ∈ H10 (Ω), ‖ v ‖Lj(Ω)≤ Sj |v|1,Ω.
P ❨ ❱ P
♠r ♦r d = 3 ♥qt② (2.1) t st♥r ♥t♦♥ ♦ H10 (Ω) r♠♥s ♦♥② ♦r j ≤ 6
♥ t ♥trst ♦ ♦r♥ ♥ ♠♥s♦♥ d = 2.
♠♦ ♣r♦♠ (1.1)(1.2) ♠ts t q♥t rt♦♥ ♦r♠t♦♥
♥ u ∈ X s tt
∀v ∈ X,
∫
Ω
∇u∇vd①+
∫
Ω
λ|u|2puvd① = 〈f, v〉,
t X = H10 (Ω).
♦r♠ Pr♦♠ (2.2) ♠ts ♥q s♦t♦♥ u ∈ X.
Pr♦♦ ❲ ss♦t t ♦♦♥ ♥r② ♥t♦♥ t ♣r♦♠ (2.2)
E(u) =1
2
∫
Ω
∇u(x)2d①+λ
2p+ 2
∫
Ω
|u(x)|2pu(x)2d①− 〈f, u(x)〉,
t♥ t st♠t♦♥
E(u) ≥1
2
∫
Ω
|∇u(x)|2dx+λ
2p+ 2
∫
Ω
|u|2p+2− ‖ f ‖−1,Ω‖ u ‖1,Ω .
♥ t rst ♦♦s r♦♠ t ♥r② ♠♥♠③t♦♥ ♦r♦r② s ❬❪ ♣tr 3
❲ ♥♦ ♥tr♦ t ♦♦♥ t♥ ♠♠
♠♠ t ♥ ♣ tr r ♥♠rs ❲ t ♦♦♥ rt♦♥∣∣|a|p − |b|p
∣∣ ≤ p|a− b|(|a|p−1 + |b|p−1
).
Pr♦♦ rst ♦♦s r♦♠ ♣♣②♥ t ♠♥ t♦r♠ t♦ f(x) = xp t x > 0
♠r ♥ t sq ♥♦t ② C C ′ ♥r ♦♥st♥ts tt ♥ r② r♦♠ ♥ t♦ ♥ tr ②s ♥♣♥♥t ♦ srt③t♦♥ ♣r♠trs
♥t ♠♥t srt③t♦♥ ♥ t ♣r♦r st♠t
s st♦♥ ♦ts s♦♠ s ♥♦tt♦♥ ♦♥r♥♥ t srt stt♥ ♥ t ♣r♦r st♠t
t (Th)h rr ♠② ♦ tr♥t♦♥s ♦ Ω ♥ t s♥s tt ♦r h • ♥♦♥ ♦ ♠♥ts ♦ Th s q t♦ Ω
• ♥trst♦♥ ♦ t♦ r♥t ♠♥ts ♦ Th ♥♦t ♠♣t② s rt① ♦r ♦ ♦ ♦ttr♥s• rt♦ ♦ t ♠tr hK ♦ ♥② ♠♥t K ♦ Th t♦ t ♠tr ♦ ts ♥sr r s s♠rt♥ ♦♥st♥t ♥♣♥♥t ♦ hs s h st♥s ♦r t ♠①♠♠ ♦ t ♠trs hK K ∈ Th
t Xh ⊂ H10 (Ω) t P1 ♥t ♠♥t s♣ ss♦t t Th, ♠♦r ♣rs②
Xh =
vh ∈ H1
0 (Ω), ∀K ∈ Th, vh|K∈ P1(K)
,
r P1(K) st♥s ♦r t s♣ ♦ rstrt♦♥s t♦ K ♦ ♥ ♥t♦♥s ♦♥ IR2.
❲ t♥ ♦♥sr t ♦♦♥ ♥t ♠♥t srt③t♦♥ ♦ Pr♦♠ (2.2) ♦t♥ ② t r♥♠t♦
♥ uh ∈ Xh s tt
∀vh ∈ Xh,
∫
Ω
∇uh∇vhd①+
∫
Ω
λ|uh|2puhvhd① = 〈f, vh〉.
♥ ♦rr t♦ ♣r♦ t ①st♥ ♦ s♦t♦♥ t♦ ♣r♦♠ (3.1) t s r s♦♠ rsts ♦♥ t ♥t♠♥s♦♥ ♣♣r♦①♠t♦♥s ♦ ♥♦♥♥r ♣r♦♠s t♦ r③③♣♣③rt t♦r♠ ❬❪ ①t
❯ ❯ ❨
♣♣② t♠ t♦ ♣r♦♠ (3.1) t V ♥W t♦ ♥ s♣s ❲ ♥tr♦ C1 ♠♣♣♥ G : V →W
♥ ♥r ♦♥t♥♦s ♠♣♣♥ S ∈ L(W,V ) ❲ st
F (u) = u− SG(u).
❲ ♦♥sr t ♥t ♠♥s♦♥ ♣♣r♦①♠t♦♥ ♦ s♦t♦♥ u ∈ V ♦ t qt♦♥ F (u) = 0
♦r h > 0 r ♥ ♥t ♠♥s♦♥ ss♣ Vh ♦ t s♣ V ♥ ♥ ♦♣rt♦r Sh ∈ L(W ;Vh).❲ st ♦r uh ∈ Vh :
Fh(uh) = uh − ShG(uh).
♣♣r♦①♠t ♣r♦♠ ♦♥ssts ♦♥ ♥♥ s♦t♦♥ uh ∈ Vh ♦ t qt♦♥
Fh(uh) = 0.
♥ t ♦♦♥ t♦r♠ s ❬❪ t♦♥ 3 ♦r ❬❪ ♣tr 4
♦r♠ ss♠ tt G s C1 ♠♣♣♥ r♦♠ V ♥t♦W t DG ♣st③♦♥t♥♦s SDG(u) ∈L(V ) s ♦♠♣t ♥ DF (u) s ♥ s♦♠♦r♣s♠ ♦ V ♥ t♦♥ ss♠ tt ♦r v ∈ V
limh→0
‖ v −Πhv ‖V = 0,
♦r s♦♠ ♥r ♦♣rt♦r Πh ∈ L(V ;Vh) ♥
limh→0
‖ Sh − S ‖L(W,V )= 0.
♥ tr ①st h0 > 0 ♥ ♥♦r♦♦ O ♦ t ♦r♥ ♥ V s tt ♦r ♥② h ≤ h0 ♣r♦♠(3.3) ♠ts ♥q s♦t♦♥ uh s tt uh − u ♦♥s t♦ O
rtr♠♦r ♦r s♦♠ ♦♥st♥t M > 0 ♥♣♥♥t ♦ h
‖ uh − u ‖V ≤M
(‖ u−Πhu ‖V + ‖ (Sh − S)G(u) ‖V
).
♥ ♦rr t♦ ♣♣② t r③③♣♣③rt t♦r♠ ❬❪ t♦ ♣r♦♠ (3.1) t V = H10 (Ω) ♥
W = H−1(Ω) ❲ ♥tr♦ t ♥r ♦♥t♥♦s ♠♣♣♥
S : W → V
f 7→ Sf = w,
r w s t s♦t♦♥ ♦ t ♣r♦♠
−∆w = f ♥ Ω,w = 0 ♦♥ ∂Ω.
t s r② tt S s t s③ s♦♠♦r♣s♠ ♥ ♥ s♦♠tr② t♥ H−1(Ω) ♥ H10 (Ω).
♠♠ ♦♦♥ stt② ♣r♦♣rt② ♦s ♦r ♥② f ♥ H−1(Ω)
|Sf |1,Ω ≤‖ f ‖−1,Ω .
❲ ♦♥sr ♥♦ t ♦♦♥ C1 ♠♣♣♥
G : V → W
w 7→ G(w) = f − λ|w|2pw
♥ ♦sr tt ♣r♦♠ (2.2) ♥ rtt♥ s ♦♦s
u− SG(u) = 0.
♠♠ r ①sts r ♥♠r L > 0 ♥ ♥♦r♦♦ V ♦ u ♥ H10 (Ω) s tt t
♦♦♥ ♣st③ ♣r♦♣rt② ♦s
∀w ∈ V, ‖ S(DG(u)−DG(w)
)‖L(H1
0(Ω))≤ L|u− w|1,Ω.
Pr♦♦ ❲
‖ S(DG(u)−DG(w)
)‖L(H1
0(Ω))≤‖ DG(u)−DG(w) ‖L(H−1(Ω)) .
P ❨ ❱ P
❲ ♦sr tt ♦r ♥② z ∈ H10 (Ω),
DG(u).z −DG(w).z = 2λp(|w|2p−1w − |u|2p−1u
)z − λ
(|w|2p − |u|2p
)z.
❯s♥ ♠♠ 2.4 ♥ ♦♠♥♥ (3.7) t (3.8) ②s t sr ♣r♦♣rt②
♥ t ♦♣rt♦r SDG(u) ∈ L(V ) s ♦♠♣t t ♦♦s r♦♠ t r♦♠s tr♥t tt DF (u) s♥ s♦♠♦r♣s♠ ♦ V t qt♦♥
DF (u).w = 0 t w ∈ V
s ♦♥② t ③r♦ s♦t♦♥ s s♥
DG(u).w = −(2λp|u|2p−1uw + λ|u|2pw
),
♦♥sr t ♦♦♥ ♣r♦♠
−∆w + 2λp|u|2p−1uw + λ|u|2pw = g ♥ Ω,w = 0 ♦♥ ∂Ω,
t g ∈ H−1(Ω), λ > 0.
♥♦ ♥tr♦ t ♦♦♥ ♠♠ tt ♣r♦s t ♥q♥ss ♦ t s♦t♦♥ w = 0.
♠♠ qt♦♥(I − SDG(u)
).w = 0 ♠ts ♥q s♦t♦♥ w = 0.
Pr♦♦ ①st♥ ♥ ♥q♥ss ♦ s♦t♦♥ t♦ (3.9) s s② sts t♦ ①r♠t♦r♠ ♥ t rst ♦♦s r♦♠ t t tt w = 0 s ♦♦s② s♦t♦♥ ♦ ♣r♦♠ (3.9).
❲ ♥♦t ② Sh t ♦♣rt♦r ss♦t t ♥② f ♥ W t s♦t♦♥ uh ♦ t srt ♥r♣r♦♠
Sh : W → Vhf 7→ Shf = wh,
r wh stss
∀vh ∈ Vh,
∫
Ω
∇wh∇vh d① =
∫
Ω
fvh d①.
❲ r ♥♦ ♥ ♣♦st♦♥ t♦ stt t ♦♦♥ ♦r♦r② rs ♦♥ ♦r♠ 3.1 ♥ ♣r♦s t ♣r♦r rr♦r st♠t
♦r♦r② t u t s♦t♦♥ ♦ (2.2) r ①st ♥♦r♦♦ ♦ t ♦r♥ ♥ V ♥ r♥♠r h0 > 0 s tt ♦r h ≤ h0 ♣r♦♠ (3.1) s ♥q s♦t♦♥ uh t uh − u ♥ ts♥♦r♦♦ ♦r♦r t ♦♦♥ ♣r♦r rr♦r st♠t ♦s
‖ uh − u ‖V ≤M
‖ u−Πhu ‖V + ‖ (Sh − S)G(u) ‖V
,
r M s ♦♥st♥t ♥♣♥♥t ♦ h
♥ t♦♥ u ∈ H2(Ω)
‖ uh − u ‖1,Ω≤ Ch ‖ u ‖2,Ω .
trt ♦rt♠s
♥ ts st♦♥ ♥ ♦rr t♦ s♦ ♦r ♥♦♥♥r srt ♣r♦♠ ♣r♦♣♦s t♦ r♥t ♦rt♠s♥♠② t ①♣♦♥t ♦rt♠ ♥ t t♦♥ ♦rt♠ ♥ ♦ t ♠♥ ♥ts ♦ t ①♣♦♥t ♦rt♠ s tt t ♦♥rs t♦ t ♥q ①♣♦♥t ♦ t ♥t♦♥ ♦r ♥② strt♥ ♣♦♥t♦r t ♥t ss s s♥t② ♦s t♦ t t♦rt s♦t♦♥ t♦♥ trt♦♥ rs♦♥ st♦ ♠ str ♦♥r♥ rts t♥ t ①♣♦♥t trt♦♥ rs♦♥ ❲ strt ② ♥tr♦♥ t①♣♦♥t ♦rt♠
❯ ❯ ❨
①♣♦♥t ♦rt♠ t u0h ♥ ♥t ss ❲ ♥tr♦ ♦r i ≥ 0 t ♦♦♥ ♦rt♠
♥ ui+1h ∈ Vh s tt
∀vh ∈ Vh, (∇ui+1h ,∇vh) + λ(|uih|
2pui+1h , vh) = 〈f, vh〉.
t s r② tt ♣r♦♠ (4.1) s ♥q s♦t♦♥ tt ♣♥s ♦♥t♥♦s② ♦♥ f
❲ ♥♦ st♠t t s♦♥ tr♠ ♥ t rt♥ s ♦ (4.5) ②r③ ♥qt② s(|uih|
2p(uh − uhi+1), vh
)≤ ‖ uih
2p‖L4(Ω)‖ u
i+1h − uh ‖L4(Ω)‖ vh ‖L2(Ω)
≤ ‖ uih ‖2pL8p(Ω)‖ ui+1h − uh ‖L4(Ω)‖ vh ‖L2(Ω) .
♣♣②♥ (2.1) ♥ (4.2) ♦t♥
λ(|uih|
2p(uh − uhi+1), vh
)≤ λS2S4S
2p8p ‖ f ‖2p−1,Ω |ui+1
h − uh|1,Ω|vh|1,Ω.
♦♦s♥ vh = ui+1h − uh ♥ ♦t♥ (4.7) ♥ (4.8) ♣r♦ t sr st♠t
P ❨ ❱ P
t♦♥ ♦rt♠ trt♥ r♦♠ ♥ ♥t ss u0h ♦♥strt t sq♥ (uih) ♥ Xh stt ♦r i ≥ 0
ui+1h = uih − [DFh(uh)]
−1.F (uih).
♣♣②♥ (4.9) t♦ ♣r♦♠ (3.3) ♦t♥ t ♦♦♥ ♣r♦♠
♥ ui+1h ∈ Xh s tt
∀wh ∈ Xh (∇ui+1h ,∇wh) + λ(2p+ 1)
((uih)
2pui+1h , wh
)= 2λp
((uih)
2p+1, wh
)+ 〈f, wh〉.
①st♥ ♥ ♥q♥ss ♦ s♦t♦♥ t♦ (4.10) s sts t♦ t ①r♠ t♦r♠
♥ ♦rr t♦ ♣r♦ t ♦♥r♥ ♦ t t♦♥s ♦rt♠ ♣♣② ❬❪ ♣tr 4 ♦r♠ 6.3 s
♦r♠ ♦♥r♥ ♦r♠ r ①st α > 0 s tt ♦r h ≤ h0 ♥ ♥ ♥tss u0h ♥ t t ♥tr uh ♥ rs α t t♦♥s ♦rt♠ (4.10) tr♠♥s ♥qsq♥ (uih) ♥ ts tt ♦♥rs t♦ t s♦t♦♥ uh ♦ ♣r♦♠ (3.3) rtr♠♦r t ♦♥r♥s qrt
‖ ui+1h − uh ‖X≤ C ‖ uih − uh ‖2X .
♣♦str♦r rr♦r ♥②ss
❲ strt ts st♦♥ ② ♥tr♦♥ s♦♠ t♦♥ ♥♦tt♦♥ s ♥ ♦r ♦♥strt♥ ♥♥②③♥ t rr♦r ♥t♦rs ♥ t sq
♦r ♥② tr♥ K ∈ Th ♥♦t ② E(K) ♥ N (K) t st ♦ ts s ♥ rts rs♣t② ♥ st
Eh =⋃
K∈Th
E(K) ♥ Nh =⋃
K∈Th
N (K).
❲t ♥② E ∈ Eh ss♦t ♥t t♦r n s tt n s ♦rt♦♦♥ t♦ E ❲ s♣t Eh ♥ Nh
♥ t ♦r♠
Eh = Eh,Ω ∪ Eh,∂Ω ♥ Nh = Nh,Ω ∪ Eh,∂Ω
r Eh,∂Ω s t st ♦ s ♥ Eh tt ♦♥ ∂Ω ♥ Eh,Ω = Eh \ Eh,∂Ω s♠ ♦s ♦r Nh,∂Ω
rtr♠♦r ♦r K ∈ Th ♥ E ∈ Eh t hK ♥ hE tr ♠tr ♥ ♥t rs♣t② ♥♠♣♦rt♥t t♦♦ ♥ t ♦♥strt♦♥ ♦ ♥ ♣♣r ♦♥ ♦r t t♦t rr♦r s é♠♥ts ♥tr♣♦t♦♥ ♦♣rt♦rRh t s ♥ Xh ♦♣rt♦r Rh stss ♦r v ∈ H1
0 (Ω) t ♦♦♥ ♦ ♣r♦①♠t♦♥♣r♦♣rts s ❱rürt ❬❪ ♣tr 1
‖ v −Rhv ‖L2(K) ≤ ChK |v|1,∆K,
‖ v −Rhv ‖L2(E) ≤ Ch1/2E |v|1,∆E
,
r ∆K ♥ ∆E r t ♦♦♥ sts
∆K =⋃
K ′ ∈ Th; K ′ ∩K 6= ∅
♥ ∆E =
⋃ K ′ ∈ Th; K ′ ∩ E 6= ∅
.
❲ ♥♦ r t ♦♦♥ ♣r♦♣rts s ❱rürt ❬❪ ♣tr 1
Pr♦♣♦st♦♥ t r ♣♦st ♥tr ♦r v ∈ Pr(K) t ♦♦♥ ♣r♦♣rts ♦
C ‖ v ‖L2(K) ≤ ‖ vψ1/2K ‖L2(K) ≤ ‖ v ‖L2(K) ,
|v|1,K ≤ Ch−1K ‖ v ‖L2(K) .
r ψK s t tr♥ ♥t♦♥ q t♦ t ♣r♦t ♦ t r②♥tr ♦♦r♥ts ss♦tt t ♥♦s ♦ K
❯ ❯ ❨
♥② ♥♦t ② [vh] t ♠♣ ♦ vh r♦ss t ♦♠♠♦♥ E ♦ t♦ ♥t ♠♥ts K,K ′ ∈ Th❲ ♥♦ ♣r♦ ♣rrqsts t♦ sts ♥ ♣♣r ♦♥ ♦r t t♦t rr♦r t ui+1
h ♥ u t s♦t♦♥ ♦ t trt ♣r♦♠ ♥ t ♦♥t♥♦s ♣r♦♠ rs♣t② ② sts② t ♥tt②
∫
Ω
∇(ui+1h − u)∇vd① =
∫
Ω
∇ui+1h ∇vd①+ λ
∫
Ω
|u|2puvd①−
∫
Ω
fvd①.
❲ ♥♦ strt t ♣♦str♦r ♥②ss ♦ t ①♣♦♥t ♦rt♠
①♣♦♥t ♦rt♠ ♥ ♦rr t♦ ♣r♦ ♥ ♣♣r ♦♥ ♦ t rr♦r rst ♥tr♦ ♥♣♣r♦①♠t♦♥ fh ♦ t t f s ♦♥st♥t ♦♥ ♠♥t K ♦ Th ♥ st♥s tsrt③t♦♥ ♥ ♥r③t♦♥ rr♦rs ❲ rst rt t rs qt♦♥∫
Ω
∇u∇vd①+ λ
∫
Ω
|u|2puvd①−
∫
Ω
∇ui+1h ∇vd①− λ
∫
Ω
|uih|2pui+1
h vd①
=
∫
K
(f − fh)(v − vh)d①+∑
K∈Th
∫
K
(fh +∆ui+1h − λ|uih|
2pui+1h )(v − vh)d①
−1
2
∑
E∈Eh,Ω
∫
E
[∂ui+1
h
∂n](v − vh)dτ
,
r τ ♥♦ts t t♥♥t ♦♦r♥t ♦♥ ∂K
❲ t t ∆ui+1h r ♥ trrs ♦r ttr ♥rst♥♥ ♥ s♦ ♥ ♦ t ①t♥s♦♥ t♦
r ♦rr ♥t ♠♥ts t t ♥ss s♥ r ♦r♥ t ♣s ♥ ♥t♦♥s
② ♥ ♥ strt♥ λ
∫
Ω
|ui+1h |2pui+1
h vd① ♦t♥
∫
Ω
∇u∇vd①+ λ
∫
Ω
|u|2puvd①−
∫
Ω
∇ui+1h ∇vd①− λ
∫
Ω
|ui+1h |2pui+1
h vd①
=
∫
K
(f − fh)(v − vh)d①+∑
K∈Th
∫
K
(fh +∆ui+1h − λ|uih|
2pui+1h )(v − vh)d①
−1
2
∑
E∈Eh,Ω
∫
E
[∂ui+1
h
∂n](v − vh)dτ
+ λ
∫
Ω
(|uih|
2p − |ui+1h |2p
)ui+1h vd①.
❲ ♥♦ ♥ t ♦ ♥r③t♦♥ ♥t♦r η(L)K,i ♥ t ♦ srt③t♦♥ ♥t♦r η
(D)K,i ②
η(L)K,i = |ui+1
h − uih|1,K ,(η(D)K,i
)2= h2K ‖ fh +∆ui+1
h − λ|uih|2pui+1
h ‖2L2(K) +∑
E∈Eh,Ω
hE ‖ [∂ui+1
h
∂n] ‖2L2(E) .
ss♠♣t♦♥ s♦t♦♥ ui+1h ♦ ♣r♦♠ (4.1) s s tt t ♦♣rt♦r Id + SDG(ui+1
h ) s ♥s♦♠♦r♣s♠ ♦ H1
0 (Ω)
♠r ♥ t♦ t ♦♥r♥ ♦r♠ 4.1 r C1 > 0 ♥ C2C−11 < 1 ss♠♣t♦♥ 5.2 s
s② r r♦♠ t t tt Id+ SDG(u) s ♥ s♦♠♦r♣s♠ ♥ h s s♠ ♥♦
❲ ♥ ♥♦ stt t rst rst ♦ ts st♦♥
♦r♠ ❯♣♣r ♦♥ t ui+1h ♥ uh t s♦t♦♥ ♦ t trt ♣r♦♠ (4.1) ♥ t
st♠ts ♦♥ ♦r ♠♦ ♣r♦♠ ❲ ♦♥sr t ♦♠♥ Ω =]−1, 1[2 t ①t s♦t♦♥ u = −100(x2+y2)
♥ t s♠ trt ♦rt♠ s ♥ t rst tst s ❲ ♥ t♦ r♥t st♦♣♣♥ rtr
ηL ≤ η∗,
ηL ≤ γη(D)i ,
t γ ♣♦st ♣r♠tr ♥ t t s♠ ♥t♦♥ ♦r η(D)i s ♣r♦s②
r 11 ♦♠♣rs t ①t s♦t♦♥ t t st♠t s♦t♦♥ ♦r λ = 10 ♥ p = 1
r ①t s♦t♦♥ t ♥ ♥♠r s♦t♦♥ rt tr t ♠s r♥♠♥t ♦r λ = 10 ♥ p = 1
P ❨ ❱ P
rs 12 t♦ 15 s♦ t ♦t♦♥ ♦ t ♠s s♥ t t♦♥ ♦rt♠ t t ♥ st♦♣♣♥
rtr♦♥ (ηL ≤ γη(D)i )
r s r♥♠♥t rts
r s r♥♠♥t rts
r s r♥♠♥t rts
r s r♥♠♥t rts
r 16 ♣rs♥ts t rr♦r r ♦r ♥♦r♠ r ♥ ♣t ♠s r♥♠♥t s♥ t ♥st♦♣♣♥ rtr♦♥ t λ = 10 p = 1 ♥ γ = 10−3 ❲ ♥♦t tt t rr♦r s♥ ♥ ♣t ♠s s♠ s♠r t♥ t rr♦r s♥ ♥ ♥♦r♠ ♠s
♥② r 17 strt t ♣r♦r♠♥ ♦ ♦r ♥ st♦♣♣♥ rtr♦♥ ♦r t t♦♥ trt♦♥ ②♦♠♣r♥ t t♦ ♠♦r ss st♦♣♣♥ rtr♦♥
❯ ❯ ❨
Log
(Err
or)
Uniform Error
Adapt Error
Number of vertices
r rr♦r r s ♥t♦♥ ♦ t rts ♥♠r♦r λ = 10 ♥ p = 1 ❯♥♦r♠rr♦r t♦♣ ♣t rr♦r ♦tt♦♠
Nu
mb
er
of
ite
rati
on
s
New
Class
Refinement level
r ♠r ♦ trt♦♥s s ♥t♦♥ ♦ t r♥♠♥t ♦r λ = 10 ♥p = 1 ss rtr♦♥ t♦♣♥ rtr♦♥ ♦tt♦♠
♦♠♣rs♦♥ ♦ t ♦rt♠s ♥ ts st♦♥ ♦r ♣r♣♦s s t♦ ♦♠♣r t ①♣♦♥t ♦rt♠ t t t♦♥ ♦rt♠ s♥ ♣t② r♥ ♠ss r 18 strt t ♣r♦r♠♥♦ t t♦♥ ♦rt♠ t ♥ ♥t ss u0h = 0.03 ❲♥ ♦♠♣r♥ t t♦ ♠t♦s t♦♥s♠t♦ rqrs ♦♥② t♦ trt♦♥s ♦r ♦♥sr r♥♠♥t rs t ①♣♦♥t ♦rt♠rqrs 10 trt♦♥s ♥ t rst r♥♠♥t ♥ ♦s ♥♦t ♦ ♥t 4 ♦r t rst s t s rtt t ♠t♦ ♦ t♦♥ ♦s ♣r♦r♠♥ ♥ ①ss ♥♠r ♦ trt♦♥s
Nu
mb
er
of
ite
rati
on
s
FPA
NA
Refinement level
r ♠r ♦ trt♦♥s ♦r t ♥ st♦♣♣♥ rtr♦♥ ♦♥ ♣t② r♥♠ss t γ = 0.001. P ♦rt♠ t♦♣ t♦♥s ♦rt♠ ♦tt♦♠
① ♦rt♠ s s ♥ ts st♦♥ t ♠♥ ♥t ♦ t t♦♥s ♦rt♠ stt t ♦♥rs str t♥ t ①♣♦♥t ♦rt♠ ♦r t ♦ ♦ t ♥t t ♥ t♦♥s♠t♦ s qt ♠♣♦rt♥t ♥ ♥t ss u0h tt s ♥♦t ♦s t♦ t t♦rt s♦t♦♥ ♥ t♦ tr♥ ♦ ♦r ♣r♦♠ s ♥ ♦rr t♦ ♠♥t ts r ♣r♦♣♦s ♠① ♦rt♠ ♥t ♥t r♦♠ t ♥ts ♦ ♦t ♦rt♠s ❲ strt t rst trt♦♥ ② t ①♣♦♥t♠t♦ ♥ ♦♥t♥ t trt ♣r♦ss s♥ t t♦♥s ♦rt♠ ♦♦♥ t ♣r♦st P❯ t♠ ♦ t ♦♥r♥ ♦ ♦rt♠
P ❨ ❱ P
♥t t u0h t♦♥ ♦rt♠ ① ♣♦♥t ♦rt♠ ① ♦rt♠ s s s s s s s s s s s s s s s s s
P❯ t♠ ♥ s♦♥s ♦ t ♦♥r♥ ♦ ♦rt♠
♦♥s♦♥ ❲ ♣rs♥t ♥ ts ♦r ♥ ♣♣ trt♠♥t ♦ ♣♦str♦r rr♦r st♠t♦♥♦r ♥t ♠♥t ♣♣r♦①♠t♦♥ ♦ t ♥♦♥♥r ♣r♦♠ (1.1) ♥ ♦rr t♦ s♦ t srt ♣r♦♠ ♣r♦♣♦s t♦ r♥t ♠t♦s t ①♣♦♥t ♦rt♠ ♥ t t♦♥ ♦rt♠ s t♦ s♦rs♦ rr♦r ♣♣r t ♥r③t♦♥ rr♦r ♥ t srt③t♦♥ rr♦r ♥♥ ts t♦ rr♦rs s r②♠♣♦rt♥t ♥ t t ♦ ♣r♦r♠♥ ♥ ①ss ♥♠r ♦ trt♦♥s s ts ♥②ss ♥ ♣♣ ♦r ♠♥② ♦tr ♥♦♥♥r ♣r♦♠s rtr♠♦r ♦♠♣r t t♦ r♥t trt♦♥♦rt♠s ♥ t t t♦♥s ♦rt♠ ♦♥rs str t♥ t ①♣♦♥t ♦rt♠ ♦rt ♦ ♦ t ♥t t ♥ t♦♥s ♠t♦ s qt ♠♣♦rt♥t rsts r ♣rs♥t ♥ 2