# Examen du cours MOPSI Quelques éléments de réponses pour la

## Transcription

Examen du cours MOPSI Quelques éléments de réponses pour la
``` "!\$#% '&)("!\$("*+",.-/&012/("3546, &/'&)3546127!98:3;8<2=-=>\$?8<,;8<[email protected]&/C"-DE8<!\$D"1!F&/D">\$",%-=>HG;#. IJKMLONQP/NRK
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efThg 4612 α = 2 i 46,kjf("2=>HG;W#. V '&=-l1, 7mn46, D"-=>\$46,01 V @%8<311, 4Uj ] o ,k3;8<2=-=>\$D"1!A>\$"2 i !\$'&&/46!A1-=>\$46, &
&/46,.-d01(pG ,1>\$'&)3O4612)-/461-7-/"*^3 & t ≥ 0 i "- (0, 0) '&=-71,q3O46>A,%-W86&[email protected]*^31-/46-=>\$#. "*+",%-r&=-`8<\1!\$ ]
stLOu7vaw`xzyK
SUTW{ Y['&=-!\$?312/46\1!\$_"*+:01}|W>A2=>\$D/~1!\$"-C~ 46*+46f_", :D"!986&/&/>\$#% 354612W!\$'#% "!M46,35"1-WD'46, &=-=2=1>A2/a1, &/46!A1-=>\$46,q3;8<27!98}mn462=*}1!98<-=>\$46,j8<2=>98<-=>\$46,1, "!A!\$+B-=2/461jf"2 u ∈i H (Ω) -/"!#. ∀v ∈ H (Ω) i R ∇u ·
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,&/", &^ ∂ 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/6!\$!98, 462=*+ L  ∃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]  ≤ ] V 8
&=1>A-/ u '&=-l\O462=, ('r0 8<, & H  "- Ω '&/-l1,q0146*k8<>A, r\O462=, ( T 0146, Dr46,3O"1-&/131354&/"2  #.1>A-=-/r
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)
a#%1>M] '&=-rD'46,%-=2`ku860>\$D"k-/46>A≤2/k8'jfi 'Da!\$}mn8<>A-r#. 3;8<28<>A!A!\$"12/& i kuk = 1 ]
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, 0ca!98amn462=*}1!98<-=>\$46,cj8<2=>98<-=>\$46,1, "!A!\$}0q]Z31 2/46\1!\$_"*+
0 8<, & Ω i
−∆u = f
&=12 ∂Ω ]
∂ u = −u/ε
fTg 8<2)>A,%-/("2`8<-=>\$46, &W3;8<273;8<2=-=>\$ i 46,qjf("2=>HG;#% 1
0
2
Ω
1
0
2
n
2
1
2
L2
1
n L2
1
n
n ε
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n
1
n
n ∂Ω
1
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n ε
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ε
n
ε
∀v ∈ H 1 (Ω), aε (u, v) =
Z
Ω
fv +
Z
∂n uv.
∂Ω
ε
n ε
1
n
1
1
n
1
n
2
ε
2
ε
ε
Ω
1
n
1
n
{ 46*^*+
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 ,312/",;8<,.-
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
"-
i "-73O4612
T
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ε < 3/4
Z
Z
ε 2
ε 2
|u − u | ,
|∇(u − u )| +
≤C
∂Ω
ZΩ
Z
3
ε 2
ε 2
|∇(u − u )| +
≤C
|u − u | ,
4ε ∂Ω
Ω
Z
|∂n u|2 ,
≤ Cε
"-d0146, D u -/", 0qjf"2/& u  ",, 462=*+ H T #.;8<, 0 ε -/", 0qjf"2/& 0 ]
Tg 4612d-/461-/rmn46, D"-=>\$46, v ∈ V i 46,c8
∂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
+
2ε ∂Ω
2ε ∂Ω
"-d0146, D
1
2
Z
h
|∇(uh − vh )|2 +
1
4ε
Z
|uh − vh |2 ≤ ε
Z
 ,?1-=>A!A>\$&`8<,%-!\$X46mn,c8<>A-F",#. 01 ('i 03111>A->\$&/#.#. u 3O∈461H2 (Ω) i infT
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 .
,312/",;8<,%46,8}0146, DW1, D'46,.jf"2=f", D'C",k, 462=*+ H ", h ]Bo ,jf46>A-#. !Y[462/02/
10 D'46,.jf"2=f", εD'=},hY['&=-7i 3;86&d4631-=>A*k8<! ]
fT o ,qjf("2=>HG;#. D'"-=-/r, 461jf"!A!\$m 462=*+r\1>A!A>A, (E8<>A2/?'&=-)D'46,%-=>A,. i "-7, C3O4&/W31!A &d01r312/46\1!\$_"*+
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e
3/2
1
ε
ε
3/4
"-r'&=-W0146, Da4631-=>A*k8<!\$:",312/",;8<,%- ε 312/463O462=-=>\$46,1, "!X h ] g 4612r!98D'4."2/D">AjB>A-/(
8<,1>AjfE8<c0>\$&`D"2/"- i i 46,c('D"2=>A-}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>\$#. (?
T  ∀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 ∀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, "!r 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<!\$ ]

ε < h/(2C)
 3;8<2
```

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