Examen du cours MOPSI Quelques éléments de réponses pour la

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Examen du cours MOPSI Quelques éléments de réponses pour la
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&/46,.-d01(pG ,1>$'&)3O4612)-/461-7-/"*^3 & t ≥ 0 i "- (0, 0) '&=-71,q3O46>A,%-W86&[email protected]*^31-/46-=>$#. "*+",%-r&=-`8<\1!$ ]
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R
{ 46*^*+ f '&=-+0 8<, & L (Ω) "-+#. Ω '&=-^2/("‚1!A>$"2 i 1,…2/('&=1!A-`8<-k01†2/("‚1!98<2=>A-/(
∇v =
fv]
"!A!A>A31-=>$#. †*+46,%-=2/q#. ,t3O"1-a0146, D†0146,1, "2^1,‡&/", &^ˆ ∂ u &=12 ∂Ω i D'46*^*+
mn46, D"-=>$46,c01 L (∂Ω) ] u ∈ H (Ω) ]Xo
efTV €&/"1!3O46>A,%-q'&=-†01*+46,%-=2/"2#. c!98‰, 462=*+ k · k D'46,.-=2/Š6!$!98, 462=*+ L ƒ ∃C > 0 i
1313O4‚&/46, &)#. CD'"-=-/r312/46312=>$("-/(, C&/46>A-73;86&)jf("2=>HG;(' !$462/&d46,
∀u ∈ H (Ω) i kuk ≤ C kuk ];‹
3O"1-dD'46, &=-=2=1>A2/1, }&=1>A-/01rmn46, D"-=>$46, & u ∈ H (Ω) -/"!A!$#. ku k = 1 "- ku k] Œ ≤ ] V 8
&=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
p-=2`8<>A2/71, )&/46 &’‘“&/1>A-/ T #% D'46,.jf"2=f7m”8<>A\1!$"*+",%-l0 8<, & "-Xmn462=-/"*+",%-0 8<, & jf"2/&z1, mn46, D"-=>$46, u ∈ H (Ω) ] V ?mn8<>A-Wu#% ku k ≤ *+46,%-=2/^#. H!98&=1>A-/ u '&=-r",cm”8<>A-WL01 { 8< D/[email protected]
0 8<, & H i 0146, D u D'46,%jf"2=f}mn462=-/"*+",%-rjf"2/& u 0 8<, & H ]5o 2!”Y•8<3131!A>$DE8<-=>$46,-=2`86D'a'&=-dD'46,.-=>A,% 0146, D
D'46,%jf"2=f:mn462=-/"*+",%-Cjf"2/& 0 8<, &
46,",–01('01>A{ 46*^*+
#. uu∈|H (Ω) "-W#. ∇u = 0 i 0146, D:u|#. u = 0Li D'(∂Ω)
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V a312/46\1!$_"*+ efT '&=-C0146, Da\1>$",€3O4‚&/(:",€1-=>A!A>$&`8<,%-C!$ V "*^*+^01 V 8—B‘“˜c>A!A‚2`8<* ,€>A,.-/("‚2`8<,%3;8<2)3;8<2=-=>$'& i 46, jf("2=>HG;#.  efT D'462=2/'&=3O46, 0cˆa!98amn462=*}1!98<-=>$46,cj‚8<2=>98<-=>$46,1, "!A!$}0q]Z31™ 2/46\1!$_"*+ƒ
0 8<, & Ω i
−∆u = f
&=12 ∂Ω ]
∂ u = −u/ε
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0
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1
n L2
1
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n ε
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n
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n ε
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Ω
fv +
Z
∂n uv.
∂Ω
ε
n ε
1
n
1
1
n
1
n
2
ε
2
ε
ε
Ω
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1
n
{ 46*^*+
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R
Ω fv
i 46,c46\1-=>$",%-0146, D
∀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
&/46>A-  " ,q1-=>A!A>$&b8<,%-d!”Y[('#.1>Aj‚8<!$", D'01'&), 462=*+'&
ku − uε k2H 1
H1
"-
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k · k1
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Z
Z
ε 2
ε 2
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Z
3
ε 2
ε 2
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≤C
|u − u | ,
4ε ∂Ω
Ω
Z
|∂n u|2 ,
≤ Cε
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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
2 Ω
Z
Z
1
1
|u − vh |2 +
|uh − vh |2
+
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1
2
Z
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1
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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
|∂n u|2 +
∂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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8<,1>AjfE8<c0>$&`D"2/"- i i 46,c('D"2=>A-}ƒ3O4612 α > 0 i
C(u)(h2 + h3 /ε)
Z
1
∂n uh uh
aε (uh , uh ) = k∇uh k2L2 + kuh k2L2 (∂Ω) − 2
ε
∂Ω
1
1
≥ k∇uh k2L2 + kuh k2L2 (∂Ω) − αhk∂n uh k2L2 (∂Ω) −
kuh k2L2 (∂Ω)
ε
αh
1
1
2
−
kuh k2L2 (∂Ω) ,
≥ (1 − Cα)k∇uh kL2 +
ε αh
4 C '&=-)!98D'46, &=-`8<,.-/?01?D'46,%-=>A,%1>A-/(:3O4612d!”Y•8<3131!A>$DE8<-=>$46,-=2`86D'a8<c,1>AjfE8<0>$&/D"2/"-  8<3131!A>$#. (?ˆ
T ƒ ∀v ∈ V i
v =∂ u
kv k
≤C h
kv k
.
,™ † O"- i ",k1-=>A!A>$&`8<,.-l!98}D'46,%-=>A,%1>A-/(r01d!”Y•8<3131!A>$DE8<-=>$46,-=2`86D'  D'46,%-=>A,. T "-l1, )>A, ("%8<!A>A-/(>A,%jf"2/&/ i
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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