Examen du cours MOPSI Quelques éléments de réponses pour la

Transcription

Examen du cours MOPSI Quelques éléments de réponses pour la
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R
{ 46*^*+ f '&=-+0 8<, & L (Ω) "-+#. Ω '&=-^2/("‚1!A>$"2 i 1,…2/('&=1!A-`8<-k01†2/("‚1!98<2=>A-/(
∇v =
fv]
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∀u ∈ H (Ω) i kuk ≤ C kuk ];‹
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&=1>A-/ u '&=-l\O462=, ('r0 8<, & H  "- Ω '&/-l1,q0146*k8<>A, r\O462=, ( T 0146, Dr46,†3O"1-Ž&/131354‚&/"2  #.1>A-=-/rˆ
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0 8<, & Ω i
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&=12 ∂Ω ]
∂ u = −u/ε
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0
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1
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n
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ε
n ε
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n
1
1
n
1
n
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2
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ε
Ω
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1
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R
Ω fv
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∀v ∈ H 1 (Ω), aε (u − uε , v) =
Z
∂n uv.
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∂Ω
v = u − uε
Z
Z
Z
1
ε 2
ε 2
|u − u | =
|∇(u − u )| +
∂n u(u − uε ),
ε ∂Ω
Ω
∂Ω
Z
Z
1
2
|u − uε |2 ,
≤ε
|∂n u| +
4ε
∂Ω
∂Ω
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ku − uε k2H 1
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"-
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Z
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ε 2
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3
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ε 2
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Ω
Z
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∂Ω
ε
1
h
aε (uh − vh , uh − vh ) = aε (uh − u, uh − vh ) + aε (u − vh , uh − vh )
Z
=−
∂n u(uh − vh ) + aε (u − vh , uh − vh )
∂Ω
Z
Z
1
2
≤ε
|uh − vh |2
|∂n u| +
4ε
∂Ω
∂Ω
Z
Z
1
1
2
|∇(u − vh )| +
|∇(uh − vh )|2
+
2 Ω
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Z
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1
1
|u − vh |2 +
|uh − vh |2
+
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Z
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Z
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Ch kuk i
ε < 1/4
Ω
∂Ω
2
3/2
H2
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∂Ω
vh ∈Vh
R
1
2
Z
|∇(u − vh )|2 +
Ω
∂Ω |u−vh |
1
2ε
Z
|u − vh |2 .
∂Ω
≤ C inf vh ∈Vh ku−vh kH 1/2 ≤
ku − uh k2H 1 (Ω) ≤ Ckuk2H 2 ε + h2 + h3 /ε .
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e
3/2
1
ε
ε
3/4
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C(u)(h2 + h3 /ε)
Z
1
∂n uh uh
aε (uh , uh ) = k∇uh k2L2 + kuh k2L2 (∂Ω) − 2
ε
∂Ω
1
1
≥ k∇uh k2L2 + kuh k2L2 (∂Ω) − αhk∂n uh k2L2 (∂Ω) −
kuh k2L2 (∂Ω)
ε
αh
1
1
2
−
kuh k2L2 (∂Ω) ,
≥ (1 − Cα)k∇uh kL2 +
ε αh
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T ƒ ∀v ∈ V i
v =∂ u
kv k
≤C h
kv k
.
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46,8†ƒ ∀v ∈ V i
h
n h
h
h
h L2 (∂Ω)
h
1/2 −1/2
h L2 (Ω)
h
kvh kL2 (∂Ω) ≤ c0 kvh kH 1/2 (Ω) ≤ Ch−1/2 kvh kL2 (Ω) .
, 312/",;8<,%- = 1/(2C) "- ε #%1>)'&/-+312/4635462=-=>$46,1, "!rˆ h "-#%1>7jf("2=>HG;
p™ 1"*^31!$ ε = αh/(4C)
T i 46,c8^D'4B"2/D">Aj.>A-/(}"-)D'46,%jf"2=f", D'a4631-=>A*k8<!$ ]
š
ε < h/(2C)
 3;8<2

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