Examen du cours MOPSI Quelques éléments de réponses pour la
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"!$#% '&)("!$("*+",.-/&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
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