Examen du cours MOPSI Quelques éléments de réponses pour la
Transcription
"!$#% '&)("!$("*+",.-/&012/("3546, &/'&)3546127!98:3;8<2=-=>$?8<,;8<!A@B&/C"-DE8<!$D"1!F&/D">$",%-=>HG;#. IJKMLONQP/NRK
] V ^&=@B&/-/_"*+?!A>A, (E8<2=>$&`(:8<1-/4612?01aD'a3O46>A,%-0ZY[('#.1>A!A>A\12/
US TWV :&/"1!X3O46>A,%-C0ZY[('#.1>A!A>A\12/+'&='&=-
(0, 0)
d
x
−1 0
x
.
=
y
0 0
dt y
V 8:!A>A, (E8<2=>$&b8<-=>$46,c, C3O"2=*+"-73;86&d01CD'46, D"!A12/?&=127!98:&=-`8<\1>A!A>A-/(01 (0, 0) ]
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&=@.*^31-/46-=>$#. "*+",%-r&=-`8<\1!$ ]
stLOu7vaw`xzyK
SUTW{ Y['&=-!$?312/46\1!$_"*+:01}|W>A2=>$D/~1!$"-C~ 46*+46f_", :D"!986&/&/>$#% 354612W!$'#% "!M46,35"1-WD'46, &=-=2=1>A2/a1, &/46!A1-=>$46,q3;8<27!98}mn462=*}1!98<-=>$46,j8<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>$"2 i 1,
2/('&=1!A-`8<-k012/("1!98<2=>A-/(
∇v =
fv]
"!A!A>A31-=>$#. *+46,%-=2/q#. ,t3O"1-a0146, D0146,1, "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=f7m8<>A\1!$"*+",%-l0 8<, & "-Xmn462=-/"*+",%-0 8<, & jf"2/&z1, mn46, D"-=>$46, u ∈ H (Ω) ] V ?mn8<>A-Wu#% ku k ≤ *+46,%-=2/^#. H!98&=1>A-/ u '&=-r",cm8<>A-WL01 { 8< D/~.@
0 8<, & H i 0146, D u D'46,%jf"2=f}mn462=-/"*+",%-rjf"2/& u 0 8<, & H ]5o 2!Y8<3131!A>$DE8<-=>$46,-=2`86D'a'&=-dD'46,.-=>A,% 0146, D
D'46,%jf"2=f:mn462=-/"*+",%-Cjf"2/& 0 8<, &
46,",01('01>A{ 46*^*+
#. uu∈|H (Ω) "-W#. ∇u = 0 i 0146, D:u|#. u = 0Li D'(∂Ω)
a#%1>M] '&=-rD'46,%-=2`ku860>$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 8Bc>A!A2`8<* ,>A,.-/("2`8<,%3;8<2)3;8<2=-=>$'& i 46, jf("2=>HG;#. efT D'462=2/'&=3O46, 0ca!98amn462=*}1!98<-=>$46,cj8<2=>98<-=>$46,1, "!A!$}0q]Z31 2/46\1!$_"*+
0 8<, & Ω i
−∆u = f
&=12 ∂Ω ]
∂ u = −u/ε
fTg 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
2
n
1
n
n ∂Ω
1
0
2
∂Ω
n ε
L2
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*^*+
aε (uε , v) =
,312/",;8<,.-
R
Ω fv
i 46,c46\1-=>$",%-0146, D
∀v ∈ H 1 (Ω), aε (u − uε , v) =
Z
∂n uv.
D'46*^*+Cm 46, D"-=>$46,-/'&/-)0 8<, &)!Y[('#.;8<-=>$46,312/('D'('01",%-/ i 46,c8^0146, D
∂Ω
v = u − uε
Z
Z
Z
1
ε 2
ε 2
|u − u | =
|∇(u − u )| +
∂n u(u − uε ),
ε ∂Ω
Ω
∂Ω
Z
Z
1
2
|u − uε |2 ,
≤ε
|∂n u| +
4ε
∂Ω
∂Ω
&/46>A- " ,q1-=>A!A>$&b8<,%-d!Y[('#.1>Aj8<!$", D'01'&), 462=*+'&
ku − uε k2H 1
H1
"-
i "-73O4612
T
k · k1
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 ]
Tg 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<,%-!$X46mn,c8<>A-F",#. 01 ('i 03111>A->$&/#.#. 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[462/02/
10 D'46,.jf"2=f", εD'=},hY['&=-7i 3;86&d4631-=>A*k8<! ]
fT o ,qjf("2=>HG;#. D'"-=-/r, 461jf"!A!$m 462=*+r\1>A!A>A, (E8<>A2/?'&=-)D'46,%-=>A,. i "-7, C3O4&/W31!A &d01r312/46\1!$_"*+
01rD'46, &=>$&=-`8<, D': a (u − u , v) = 0 ]B ,2/"312/",;8<,%-7!$'&DE8<!$D"1!$&312/('D'('0 8<,.-/& i !Y["2=2/"12)'&=-0146, DW",
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!Y8<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!Y8<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
Documents pareils
Adaptateur secteur universel pour ordinateur portable
VGN-BX51XP, VGN-BX541, VGN-BX541B, VGN-BX543, VGN-BX543B, VGN-BX546,
VGN-BX546B, VGN-BX670P, VGN-BX675P, VGN-BX740NW, VGN-BX740P,
VGN-BX740PS, VGN-BX740PW, VGN-BX760NS, VGN-BX760P, VGN-BX760PS, So...
Nonlinear stabilization of magnetohydrodynamics equations and
Nore and Jean-Luc Guermond, who allowed me to enter the world of scientific research.
Their combined experiences in fluids mechanics and mathematics let me adapt and develop
numerical model for a l...
Adaptateur secteur universel pour ordinateur portable
Ordinateur portable Sony, Sony VAIO TR2/B, Sony VAIO U101, Sony VAIO VGN-FZ160E,
Sony VAIO VGN-Y70P, Sony VAIO VGN-Y90PSY, Sony VGN-G1, Sony VAIO PCG-C1
série Picturebook, Sony VAIO série PCG-V505E...
Liste des organisations internationales
International Atomic Energy Agency (IAEA)
International Bank for Reconstruction and Development (IBRD)
International Civil Aviation Organization (ICAO)
International Development Association (IDA)
I...