Analyse numerique et optimisation : Une introduction a la modelisation mathematique et a la simulation numerique French

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L'ouvrage complet est disponible auprès des Éditions de l'École Polytechnique Cliquez ici pour accéder au site des Éditions

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∂u ∂2u ∂u +V −ν 2 =0 ∂t ∂x ∂x

; &<

8 . 8 V

unj+1 − unj−1 ∂u (tn , xj ) ≈ V ∂x 2∆x

,? 8 H . @ 8 ( ) J A 9 8 L M 9 A ( ) − un−1 un+1 ∂u j j (tn , xj ) ≈ ∂t 2∆t

E U IQ

# ? n j ; :+4%9. &) '+4.0&:,*<

H C E T

N

un+1 − un−1 unj+1 − unj−1 −unj−1 + 2unj − unj+1 j j +V +ν = 0. 2∆t 2∆x (∆x)2

Y L O P

; -<

3 L M . 6 A +) :+4%9. ):1 '*+.3./-) &) +.-+5-)0 &): :,-51',*: .330,+4%): , . 6 ( ; &< ;.

LE O ÉC

!! !

N H EC

E U IQ

&

, 2 "< V / I )

? * , A 6 : . ( ) .

T Y POL

( ) ; Y :+4%9. &!5-)0 0%10,(0.&)<

LE O ÉC

unj − un−1 ∂u j (tn , xj ) ≈ ∂t ∆t

unj − un−1 unj+1 − unj−1 −unj−1 + 2unj − unj+1 j +V +ν = 0. ∆t 2∆x (∆x)2

; <

E U IQ

# J ( ) . ; . Y :+4%9. &!5-)0 30,(0)::'=<

N H EC

− unj un+1 ∂u j (tn , xj ) ≈ ∂t ∆t

ÉC

E L O

T Y POL

un+1 − unj unj+1 − unj−1 −unj−1 + 2unj − unj+1 j +V +ν = 0. ∆t 2∆x (∆x)2

;<

( ; < '9 3-'+'1) 8 # , . (unj )j∈Z 8 . (un−1 )j∈Z A ;< )> j )j∈Z 8 (unj )j∈Z 3-'+'1) , . (un+1 j , n ; < ;< , . , , . ;< 8 n−1 n−1 un−1 −un−1 unj − un−1 − un−1 j j+1 − uj−1 j−1 + 2uj j+1 +V +ν = 0. ∆t 2∆x (∆x)2

E U IQ

1 . ) A # 0 G n J . (u0j )j∈Z ) A A u0j = u0 (j∆x) R u0 , . 6 ( ; &< + L . M ; -< # : J n = 1 0 P . (u1j )j∈Z , 8 ; A , <

LE O ÉC

Y L O P

H C E T

N

-

LE O ÉC

T Y POL

N H EC

E U IQ

" W

N H EC

E U IQ

0 . ν∆t = 0.1(∆x)2

T Y POL

" # 'O 8 R V = 0 ν = 1A ,6?6 -!,* 0%:,51 *59%0';5)9)*1 -!%;5.1',* &) -. +4.-)50C @ 8

ÉC

E L O

u0 (x) = max(1 − x2 , 0).

* . . ;
LE O ÉC

Y L O P

H C E T

N

E U IQ

!! !

N H EC

E U IQ

∆tA '*:1./-)A ,6 ?6 0 , . ∆x ∆t ' ; , < 2 " 8 ;5)- ;5) :,'1 -) +4,'> ∆t ∆xA 0. ; # A 0 G< @ 0 I ) . ;. <

LE O ÉC

T Y POL

ÉC

E L O

T Y POL

$ W

N H EC

E U IQ

. ν∆t = 2(∆x)2

3 , A ; < L0M , ;5)- ;5) :,'1 ∆t ;. 2 $< 9 A ,0. I , ∆t ∆x @ 0 ' ;< J 8 . ∆t 6 ;. 2 &< 0 8 ? . J ∆t ∆x 2%0'@)*1 2ν∆t ≤ (∆x)2 ;<

H C E T

N

E U IQ

0A ;< , . )A 6 0 @ 0 0 ;< 5*) &): 0)9.0;5): -): 3-5: :'93-): )1 -): 3-5: 30,=,*&): &) -!.*.-B:) *59%0';5) 9 8 . - ;. , 6

LE O ÉC

Y L O P

N H EC

E U IQ

V< ' A 2 A C# 9

+,*&'1',* ,5 +,*&'1',* &) ,50.*1H 0')&0'+4:H )1 )IB

LE O ÉC

T Y POL

#

ÉC

& W

E L O

T Y POL

N H EC

E U IQ

. ν∆t = 0.4(∆x)2 ; < ν∆t = 0.51(∆x)2 ;0<

E U IQ

/ I ) 0 . 0 ; # 6 ( < + . 8 ν∆t n ν∆t ν∆t n n+1 uj = uj−1 + 1 − 2 u . ; < unj + 2 2 (∆x) (∆x) (∆x)2 j+1

Y L O P

H C E T

N

0 '2 . )A ; < un+1 j . . unj−1 , unj , unj+1 ; :

LE O ÉC

!! !

N H EC

E U IQ

0 ; < 8 . < 9 A u0 0 m M

T Y POL

m ≤ u0j ≤ M j ∈ Z,

8 H .

LE O ÉC

m ≤ unj ≤ M j ∈ Z n ≥ 0.

;<

;< H , 0 J 0 '2 ;< J , , . &':+0)1 +,*1'*5 . . ? +

'2 . )A ,6?6 2ν∆t > (∆x)2 .

N H EC

E U IQ

3A 0 ; H 0 L M J A u0 ≡ 0 V< * ) u0j = (−1)j

T Y POL

0 8 0 n ν∆t unj = (−1)j 1 − 4 (∆x)2

ÉC

E L O

ν∆t . , ) n . , ) 1 − 4 (∆x) 2 < −1 0 '2 , 8

>)0+'+) CC % 2/(

V = 0( 6% ( $ + 2/ 1 ≤ j ≤ J un0 = unJ+1 = 0 n ∈ N ) m ≤ 0 ≤ M m ≤ u0j ≤ M 1 ≤ j ≤ J $ un+1 unj " n ≥ 0 j * m ≤ unj ≤ M 1 ≤ j ≤ J ∆t ∆x

E U IQ

. ? 0 A ,. . A ,6?6 ? 6 / ?

+ 0 A 0 GA . A : ' , . ) 6 . A ,6?6 , .

LE O ÉC

Y L O P

H C E T

N

LE O ÉC

T Y POL

N H EC

E U IQ

- W

N H EC

E U IQ

. ν∆t = 0.4(∆x)2 . . ∆x

T Y POL

. 0 ; . . ) 6 , 0< 2 - . ) A , : , ∆x ; . 0.5A 0.1A 0.05< ∆t ν∆t/(∆x)2 ; 0 '2
∆t < ' F2%0'@+. 1',* *59%0';5) &) -. +,*2)0()*+)G I ? , 8 ;,6?6 ,# . 0 : <

ÉC

E L O

" # #$

9( , -!%;5.1',* &) +,* 2)+1',*&'A5:',* ; &< . . V = 1 /

E U IQ

H . ν∆t = 0.4(∆x)2 / , D . ( 6 ν ; . 0 *< 0 2 0 ν = 1A 0 ν = 0.01A . ν = 0.1A 0 0 ( @ 0 , .

0 * ν A . 8 ( 8 * A '2 ;
LE O ÉC

Y L O P

H C E T

N

!! !

ÉC

OLE

T Y POL

E U IQ

#

T Y POL

ÉC

N H EC

E L O

N H EC

E U IQ

Y L O P

H C E T

N

E U IQ

W , . 6 ( . ν∆t = 0.4(∆x)2 V = 1 9 A ν = 1A ν = 0.1A 0 ν = 0.01

LE O ÉC

N H EC

E U IQ

A .0 8 ? ν * A -!%;5.1',* &!.&2)+1',* ,06 ? ν = 0 + ,0 '2 ;< 8 ν = 0 ; ∆t ∆x
LE O ÉC

T Y POL

#

E L O

T Y POL

N H EC

E U IQ

W , , . . ∆t = 0.9∆xA V = 1A ν = 0

ÉC

* , , . ;,6?6 ; &< . ν = 0 0 ; # .0 V < 0< * V > 0A ? &%+)*10)9)*1 .2.- J 0 unj+1 − unj ∂u (tn , xj ) ≈ V V ∂x ∆x un+1 = j

LE O ÉC

Y L O P

H C E T

N

E U IQ

!! !

N H EC

"

E U IQ

L, 8 M . ' ? . L M

T Y POL

un+1 − unj unj+1 − unj j +V =0 ∆t ∆x

LE O ÉC

;"<

0 3 &%+)*10)9)*1 .9,*1 ;,6?6 ? V > 0
;$<

2 @ . ) ;$<

ÉC

E L O

T Y POL

N H EC

E U IQ

W , , . . ∆t = 0.9∆xA V = 1

E U IQ

0 . '2 ; ( '2 ;<< |V |∆t ≤ ∆x. ;&<

Y L O P

H C E T

9 (A ;$< 8 V ∆t n V ∆t u un+1 = + 1 − unj , j ∆x j−1 ∆x

LE O ÉC

N

$

N H EC

E U IQ

0 . A ;&< 8 A unj−1 unj * A ;$< . ) A P 0 , &%+)*10) 9)*1 .9,*1 ):1 5*) .510) '&%) 9.<)50) &) -!.*.-B:) *59%0';5) 9 0 D R 8 ,0 . ; 53I'*&'*(A ,6?6 .
LE O ÉC

T Y POL

>)0+'+) CC " ( 07 & $ ( %

$ $ u0j = (−1)j

N H EC

E U IQ

>)0+'+) CC 8 % $ . $ - $ $ 9 * + 7* ' ( % *( $ : 07 ∆t ≤ ∆x

ÉC

E L O

T Y POL

0 . D 5 ,0 A L 0M 8 A , 8 @ 0 0 ; . < ,# J , ,H ,# 0I 8 9)A L0M . 0 ; A A < ; . < # , . * *) 3)51 &,*+ 3.: =.'0) -!%+,*,9')

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H C E T

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Y L O P

/ 0 ) ,# . .0 # . ,#

LE O ÉC

"

N H EC

% & &

&

E U IQ

$ )) 30,/-?9) .5> -'9'1): $ "$ & - ) " $ $ "$ & " """ $"+ $ " ), %@*'1',* CC

LE O ÉC

T Y POL

* A ; < 0 3 A , (

dy dt = f (t, y) 0 < t < T ;-< y(t = 0) = y0 , 0 , (0, T )A . 0 < T ≤ +∞A , L 0 M , t = 0 ; t = T <

$ )) 30,/-?9) &) .5+4B $ "$ & - ) " .' ) $ $ -/ 0$$" ") t' $ "$ 1 / 2 $" $ "$ $" " $ )"$" $ / t = 0' " ) $ t = T , %@*'1',* CC

N H EC

E U IQ

* A , ( ;-< 0 ' #A ; < ; . 0 , x ? 8 I X < 1 0 ? 8 0 0 ' # 3 A , ;-< 0 ' # ? . 0 t 0 ? . 0 , x 5 0

ÉC

E L O

T Y POL

8 , 0 ' # 0 , , , , L0M ,6 /,* 9,&?-) , # #

A ' . . . H . # N = ) , L0M A 30,/-?9) /')* 3,:% ; 0 , 0 0 < @ f ; 0A A A
Y L O P

H C E T

N

E U IQ

$ " )/+ ,3 " /')* 3,:% ) " " $$ " $ "$ $ u' " "" "$ u )$ $"$4$" $$ f , %@*'1',* CC

f

LE O ÉC

-

N H EC

E U IQ

9 ) = J 8 , 0 0 * A 8 ,

J , 0 ? ? V 1 A 8 J , A , A . , . 8 A . . L M L M ;. L M 8< . A , 0 L 0M ) * A 0 . J

X X . A HA . * ? /C Z%A . .# . 8 X E

A X , I A . 6 = ) # 0 J ) A ; , . L 6 0 M I ?< A I 80 ',

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LE O ÉC

ÉC

T Y POL

E L O

>)0+'+) CC

T Y POL

N H EC

Y L O P

E U IQ

H C E T

N

E U IQ

0 %:

) Ω = (0, 1) × (0, 2π)

LE O ÉC

"

N H EC

E U IQ

0 %: x y ⎧ ∂2u ∂2u ⎪ ⎪ ⎪ ⎨ − ∂x2 − ∂y 2 = 0 u(x, 0) = u(x, 2π) = 0 ⎪ ⎪ √ ⎪ ⎩ u(0, y) = 0, ∂u (0, y) = −e− n sin(ny) ∂x

O P LE O ÉC −

T Y L

√

Ω 0 < x < 1

0 < y < 2π

n

u(x, y) = e n sin(ny)sh(nx) " x = 0 * ;( ( x > 0( u(x, y) n $ 0

' ( ) $

! " " " " #! " " $ % & %' t & %' x " " " ( )! " " )! % ' % " %

'% " % * %' u(x, y) (( '

& + " ! , "

ÉC

E L O a

T Y POL

N H EC

E U IQ

∂2u ∂u ∂2u ∂2u ∂u + c +e + f u = g. + b +d ∂x2 ∂x∂y ∂y 2 ∂x ∂y

# !

" a, b, c, d, e, f

b

− 4ac < 0 b − 4ac > 0 % ' ' - " " . // " 0 + ' " "

b − 4ac = 0 2

2

2

ax2 + bxy + cy 2 + dx + ey + f = 0

N

E U I Q

' " * ! b − 4ac < 0 ' b2 − 4ac = 0 (' b2 − 4ac > 0 1 " . // % " % % " ( 2 (x, y) %' (t, x) ! " " " % + ! " " , "

LE O ÉC

Y L O P

H C E T

2

E U " " % (' " " IQ " 1 3 " " 0 ' N ! C H " " ' % ' " (' " E T 4 % " Y ' ' " L % " " ' 5 ( O ' " % ' " P E (' " 6

+ L " O ÉC

" ' " (' " " # ! ( %' 1 (x, y) → (X, Y ) ( %' & " 7 ' J = Xx Yy − Xy Yx Zz % Z & z! , " # ! % A

∂2u ∂u ∂2u ∂2u ∂u +C +E + F u = G, +B +D 2 ∂X ∂X∂Y ∂Y 2 ∂X ∂Y

%

A C = aYx2

E U IQ

= aXx2 + bXx Xy + cXy2 B = 2aXx Yx + b(Xx Yy + Xy Yx ) + 2cXy Yy + bYx Yy + cYy2 % " B 2 − 4AC = J 2 (b2 − 4ac) 0

N H EC

( %' " " % # ! * 8 " 9 " " ∂ ∂ + ∂Y " ' " & " ( ∂X ∂ ∂ ∂ ∂ − " (' " & " ∂X − ∂Y • ∂X ∂Y 2

2

2

T Y POL 2

2

2

2

E L 'O : " ' C %

% % É % " % # ! " 2

2

2

$ ' " " " ∂2u ∂x2

= 1

a = 1

' " ' " b2 − 4ac = 0!

" %' y ; <! 6 * * % " 8 9 • b=c=d=e=f =0

LE O ÉC

Y L O P

H C E T

N

E U IQ

T Y POL

N H EC

E U IQ

LE O ÉC

T Y POL

N H EC

E U IQ

1 # ( ) / ) :1./'-'1% +,*:':1.*+) , A . ? : A 0 0 ? , +,*2)0()*+)

ÉC

E L O

. , ' , , , . 0 8 ,# ( ) 0 . . ? ? 8 ? . 5 ( )

A , ; 6 < D 0 * , A D 0A 8 . ) / A 0 ( ) . 3 A 6 ' - 0 ) ( ) ( ) . 0 .

LE O ÉC

Y L O P

H C E T

N

E U IQ

!! !

N H *$ ) C E T Y L O P LE O ÉC

E U IQ

! "

/ ? , .# ? 6 $ , / , 0 (0, 1) ⎧ ⎨ ∂u ∂2u − ν 2 = 0 (x, t) ∈ (0, 1) × R+ ∗ ; < ∂t ∂x ⎩ u(0, x) = u0 (x) x ∈ (0, 1). * (0, 1)×R+ A , ∆x = 1/(N +1) > 0 ;. N 8< ∆t > 0A ) ,

E U IQ

(tn , xj ) = (n∆t, j∆x) n ≥ 0, j ∈ {0, 1, ..., N + 1}.

N H EC

@ unj . , (tn , xj )A u(t, x) ; <

T Y POL

u0j = u0 (xj ) j ∈ {0, 1, ..., N + 1}.

E L O

; < . H #A , . ) A 1

ÉC

u(t, 0) = u(t, 1) = 0 t ∈ R+ ∗

un0 = unN +1 = 0 n > 0.

* A ? . ? . (unj )1≤j≤N 8 . RN / 0 , ; < 5 ) N ; xj A 1 ≤ j ≤ N < / . unj 3

' . I? :+4%9. )>3-'+'1) un+1 − unj −unj−1 + 2unj − unj+1 j +ν =0 ∆t (∆x)2

C E T Y L PO

n ≥ 0 j ∈ {1, ..., N }A :+4%9. '93-'+'1)

n+1 −un+1 un+1 − unj − un+1 j j−1 + 2uj j+1 +ν = 0. ∆t (∆x)2

LE O ÉC

HN

E U IQ ; <

; <

!! ! #

N H EC

E U IQ

8 . ) ; < ( . 0 ) A ,6 8 unj J (A 8 . ?6 , . un+1 j N ⎛ ⎞ 1 + 2c −c 0 ⎜ −c ⎟ 1 + 2c −c ⎜ ⎟ ν∆t ⎜ ⎟ , ; < ⎜ ⎟ . c = ⎜ ⎟ (∆x)2 ⎝ −c 1 + 2c −c ⎠ 0 −c 1 + 2c

LE O ÉC

T Y POL

. ) ) 8A . 0 9 8 0 . ; < ;
E U IQ

E GA . ; < θ = 0A ; < θ = 1 θ6 ; "< θ = 0 * . θ = 1/2A 0 :+4%9. &) 0.*D'+4,-:,* A ? A

T Y POL

N H EC

n n 5(un+1 un+1 − unj ) un+1 j+1 − uj+1 j j−1 − uj−1 + + 12∆t 6∆t 12∆t

ÉC

E L O

n+1 −un+1 − un+1 −unj−1 + 2unj − unj+1 j−1 + 2uj j+1 +ν + ν = 0. 2(∆x)2 2(∆x)2

; $<

>)0+'+) CC " % 2 $ $ θ% θ = 1/2 − (∆x)2 /12ν∆t

5 E &)5> *'2).5> 8 . @ 0 G . J ? . 9 ; 0< + . ' A :+4%9. &) 5,010.*D)−unj−1 + un+1 un+1 − un−1 + un−1 − unj+1 j j j j +ν = 0, 2∆t (∆x)2

:+4%9. &) #).0

H C E T

n+1 −un+1 3un+1 − 4unj + un−1 − un+1 j j j−1 + 2uj j+1 +ν = 0. 2∆t (∆x)2

Y L O P

; &<

N

E U IQ ; -<

/ . ? 8 0 V 9 6 , . V 0 ,# . H . A GA 0

LE O ÉC

!! !

N H EC

E U IQ

)9.0;5) CC , # 0 f (t, x) , ;
LE O ÉC

T Y POL

un+1 − unj −unj−1 + 2unj − unj+1 j +ν = f (tn , xj ). ∆t (∆x)2 •

)9.0;5) CC 6 A ,6?6 , 8 . 0 ) A A . unj (n , j ) . , (n, j) :1)*+'- ; , # :533,01< 9 A A G : ? ; ? L( 0 MA ,6?6 R (n , j ) < •

N H EC

E U IQ

)9.0;5) CC @ 1 ; < / A 0 ; < 'O ( /

ÉC

E L O

T Y POL

∂u ∂u (t, 0) = 0 (t, 1) = 0. ∂x ∂x

5 ,0 A

un − unN un1 − un0 = 0 N +1 =0 ∆x ∆x

, . un0 unN +1 N . (unj )1≤j≤N ' / , A 0 ', ; < un − unN un1 − un−1 = 0 N +2 =0 2∆x 2∆x

E U IQ

A ,I L ) 8M x−1 xN +2 @ . un−1 unN +2 A ? ) 8A N +2 . ? A ? . (unj )0≤j≤N +1 1, A , .

H C E T

N

u(t, x + 1) = u(t, x) x ∈ [0, 1], t ≥ 0.

Y L O P

9 un0 = unN +1 n ≥ 0A unj = unN +1+j •

LE O ÉC

!! ! #

N H EC

'

"

E U IQ

E GA 8 6 J , , . 5# ,. ' * 8 , . ( ) A +,*:':1.*+) 30%+':',* E , , ;
8 u . (t, x) 1 ( ) ) A 0 n, j A 8 n+m F∆t,∆x {uj+k }m− ≤m≤m+ , k− ≤k≤k+ = 0 ; <

LE O ÉC

T Y POL

R m− , m+ , k − , k + ) ;. + <

E U # & 5$ *$ , " " $"$" Q- ' ' ) " " "$N I 6$" "$ & - )" H 0 + "" "$' #' *$ ) C E T Y L ' " "$ $" - 7 "$ - 7' $$" ) ))" O P $ )$ $", E 8 ) ' O $ "L # " ) $ ) " $ ") É C "$" , "$ - 7 " "$ $" - 7, %@*'1',* CC

F (u) = 0

u(t, x)

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k− ≤k≤k+

∆t

p

; <

,

∆x

O (∆x) + (∆t) p

q

q

∆t

∆x

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? ) 9 (A I , 8 : . ∆t ∆x ? , . F ' , V * n+m }) = 0 . . A I 8 F∆t,∆x ({uj+k A 8 u(t, x) , , A , , •

N

E U IQ

n+m ' , , O uj+k 8 ; < u(t + m∆t, x + k∆x) ' 1) A .

Y L O P

H C E T

)99) CC # &)" , " $"$"' ) $ ") " $ ), 8 ) ' $ #" 0 $"$" ))" ν∆t/(∆x)2 = 1/6' # " ) $ ") " 9 $ ),

LE O ÉC

!! !

$

N H EC

E U IQ

)9.0;5) CC 1 , $ 6 ) ) ) ∆t ∆x , . F ' ,. , 9 A 0. (6 . 0 ∆t ∆x •

LE O ÉC

T Y POL

%9,*:10.1',*C v(t, x) 8 C 6 * . 5#

(t, x)A , ; <

−v(t, x − ∆x) + 2v(t, x) − v(t, x + ∆x) v(t + ∆t, x) − v(t, x) +ν ∆t (∆x)2 ∆t ν(∆x)2 vtt − vxxxx + O (∆t)2 + (∆x)4 , = vt − νvxx + 2 12

E U IQ

R vt , vx . v v , ;
ÉC

E L O

; <

'%6/ ; "< ;. θ = 1/2< θ6 ; "< ;. θ = 1/2<

? $ ; $< 1 262% ; &< 4 ; -<

T Y POL

N H EC

9 O ∆t + (∆x)2 O ∆t + (∆x)2 O (∆t)2 + (∆x)2 O ∆t + (∆x)2 O (∆t)2 + (∆x)4 ∆t 2 O ( ∆x ) + (∆x)2 O (∆t)2 + (∆x)2

0

0 L2 L∞ '2 2ν∆t ≤ (∆x)2 0 L2 L∞ 0 L2 0 L2 '2 2(1 − 2θ)ν∆t ≤ (∆x)2 0 L2 0 L2 0 L2

E U IQ

W 9 0 . ,

Y L O P

H C E T

N

>)0+'+) CC - % % ) 22( $ : < 2 % 67 7 =

LE O ÉC

!! ! #

N H EC

& +

&

E U IQ

1 ' . . 0 ( ) ) 5 . A 6 A , 0 8 0 ) 0 6 * . 0 ) un = (unj )1≤j≤N / RN , ∆x J ⎞1/p ⎛ N un p = ⎝ ∆x|unj |p ⎠ 1 ≤ p ≤ +∞, ; <

LE O ÉC

T Y POL j=1

R p = +∞ H un ∞ = max1≤j≤N |unj | + ) ∆x ? . ? . , N ∆x = 1/(N +1) 4Q ? ∆xA un p ? Lp (0, 1) 8 6 . [xj , xj+1 [ [0, 1] .A , L Lp M 9 . p = 2, +∞

E U IQ

N H E :$ # & 5$T *$ "C " ) $ ' *$ ) ,' &" $ $"$" $ )$ $" " Y - "$ $" - 7 " O L P E ) " " L O C" $$ $" , É , $ ) ) " "$" "$ $0"' $

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:$ # & 5$ *$ " " -'*%.'0) n+m n+m F∆t,∆x ({uj+k }) = 0 *$" " $ ) ))" 0 $" uj+k , %@*'1',* CC

H C E T

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0 , ? . 8 ? 9 (A ? . , 8 un+1 = Aun , ; <

LE O ÉC

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R A ; A , < RN RN * A ; < A . ⎛ ⎞ 1 − 2c c 0 ⎜ ⎟ c 1 − 2c c ⎜ ⎟ ν∆t ⎜ ⎟ , ; < ⎜ ⎟ . c = ⎜ ⎟ (∆x)2 ⎝ ⎠ c 1 − 2c c 0 c 1 − 2c

T Y POL

LE O ÉC

; < A , . ; < 3 , , A un = An u0 ; A An n6 A
E U IQ

0 ;. 1) < M =

M u , u∈RN ,u=0 u sup

T Y POL

0 . ?

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min

0≤j≤N +1

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≤ unj ≤ max 0,

max

0≤j≤N +1

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!! ! #

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1 0 . ) L0M * 6?A 8 . ) 0 L∞ L2 H 0 ? , 0 Q ? , ,# 2 * 8 A , +,*&'1',*: .5> -'9'1): &) 3%0',&'+'1%A , . u(t, x+ 1) = u(t, x) x ∈ [0, 1] t ≥ 0 * A un0 = unN +1 n ≥ 0A unj = unN +1+j ? N + 1 . unj 3 . un = (unj )0≤j≤N 8 un (x)A A A ) [0, 1]

E L O

T Y POL

N H EC

E U IQ

un (x) = unj xj−1/2 < x < xj+1/2

. xj+1/2 = (j + 1/2)∆x 0 ≤ j ≤ N A x−1/2 = 0A xN +1+1/2 = 1 3 ) A 8 un (x) ? L2 (0, 1) @A , ,# 2 A 8 L2 (0, 1) 2 ;. " T$U< * un (x) = u ˆn (k) exp(2iπkx), ; $<

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+ H un 8 A : u ˆn (k) 2 8 2 8 . J v n (x) = un (x + ∆x)A vˆn (k) = uˆn (k) exp(2iπk∆x) 9 , ; < 3. A A 0 ≤ x ≤ 1A

Y L O P

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−un (x − ∆x) + 2un (x) − un (x + ∆x) un+1 (x) − un (x) +ν = 0. ∆t (∆x)2

LE O ÉC

!! !

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* 8 2 A . ν∆t n+1 uˆ (k) = 1 − (− exp(−2iπk∆x) + 2 − exp(2iπk∆x)) u ˆn (k). (∆x)2

T Y POL

3

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uˆn+1 (k) = A(k)ˆ un (k) = A(k)n+1 u ˆ0 (k) . A(k) = 1 −

4ν∆t (sin(πk∆x))2 . (∆x)2

* k ∈ ZA : 2 u ˆn (k) 0 n . , ) 8 , ) . ) |A(k)| ≤ 1A ,6?6 ; -<

2ν∆t(sin(πk∆x))2 ≤ (∆x)2 .

'2 ;
k∈Z

k∈Z

T Y POL

N H EC 0

E U IQ

, , 0 L2 '2 , 8 A 0 9 (A : ∆x ;. 6 : < k0 ; : < #

2 u ˆ0 (k0 ) = 0 . πk0 ∆x ≈ π/2 ; π < |A(k0 )| > 1 @ .

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−un+1 (x − ∆x) + 2un+1 (x) − un+1 (x + ∆x) un+1 (x) − un (x) +ν = 0, ∆t (∆x)2

Y L O P

8 2 ν∆t uˆn+1 (k) 1 + (− exp(−2iπk∆x) + 2 − exp(2iπk∆x)) =u ˆn (k). (∆x)2

LE O ÉC

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N H EC

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n+1 0

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@ ,( , . R [0, 1] ; ( ? 2 X ,? 2 < / A , R 0 ) . unj ? n , 0 ) . 0 L2 1 8 ,# 2 * A 8 2 0 . , ; < . un . un+1 •

N H EC

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unj = A(k)n exp(2iπkxj ) . xj = j∆x,

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; <

0 > / 8 ;. . ∆t ∆x
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1/2 ≤ θ ≤ 1( 07 2(1 − 2θ)ν∆t ≤ (∆x)2 0 ≤ θ < 1/2

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"

LE O ÉC

∂2u ∂u − ν 2 = cu ∂t ∂x

(t, x) ∈ R+ × R,

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n+1

n

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= Au ,

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E L O

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u ˜n+1 = A˜ un + ∆tn .

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n

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k=1

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@A 0 . un = An u0 ≤ Ku0 A ,6?6 An ≤ K R K n 1, A e0 = 0A ; < en ≤ ∆t

n

H C E T

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k=1

Y L O P

, ; < . CT = T KC ; <

LE O ÉC

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)9.0;5) CC 5 8 .0 . R ? . . 0 + . . ; < 9)A 0 ; < , .0 . 0 [0, T ] , 0 N •

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E L O

T Y POL

N H EC

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, N A A1 A2 A A1 A2 n+1 n U = AU = U n, Id 0

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∀ n ≥ 1.

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LE O ÉC

!! ! #

N H EC

8 2

3 A n+1 8ν∆t 2 − (∆x) (k) u ˆ 2 (sin(πk∆x)) ˆ n+1 (k) = U = n u ˆ (k) 1

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E U IQ

8ν∆t (k) + (sin(πk∆x))2 uˆn (k) − u ˆn−1 (k) = 0. (∆x)2

T Y POL

uˆ

n+1

"

1 0

ˆ n (k) = A(k)U ˆ n (k), U

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A(k)n U 2 ≤K U 2 U∈R2 ,U=0 sup

∀ n ≥ 1,

; $<

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R U 2 R2 * A , ; $< . 2 k ∈ ZA 8 * |ˆ u0 (k)|2 + |ˆ un 22 = |ˆ un (k)|2 ≤ K u1 (k)|2 = u0 22 + u1 22 , k∈Z

T Y POL k∈Z

N H EC

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ÉC

E L O

N

E U IQ

)9.0;5) CC ,# 2 . , . ? . ;. + -< , I 2 A 0 n+1 1 uj uj n = A(k) exp(2iπkxj ) u0j unj

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!! !

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R A(k) 9.10'+) , ) ; 8 < @ +,*&'1',* &) :1./'-'1% &) ,* )59.**

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ρ(A(k)) ≤ 1 k ∈ Z,

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E U IQ

( ) , : 0 6 , ' , , ; , , 6 A < Ω = (0, 1) × (0, L) . 1 ⎧ ∂2u ∂2u ⎪ ∂u ⎪ ⎨ − ν 2 − ν 2 = 0 (x, y, t) ∈ Ω × R+ ∗ ∂t ∂x ∂y ; -< u(t = 0, x, y) = u0 (x, y) (x, y) ∈ Ω ⎪ ⎪ ⎩ + u(t, x, y) = 0 t ∈ R∗ , (x, y) ∈ ∂Ω.

LE O ÉC

Y L O P

H C E T

N

!! ! #

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&

E U IQ

* ΩA , ∆x = 1/(Nx + 1) > 0 ∆y = L/(Ny + 1) > 0 ;. Nx Ny 8< 3. ∆t > 0A ) , ;. 2 <

T Y POL

(tn , xj , yk ) = (n∆t, j∆x, k∆y) n ≥ 0, 0 ≤ j ≤ Nx + 1, 0 ≤ k ≤ Ny + 1.

LE O ÉC

@ unj,k . , (tn , xj , yk )A u(t, x, y) ; -<

y

(x j, y k)

k ∆y

ÉC

E L O

T Y POL

N H EC

E U IQ x

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1 A n > 0A un0,k = unNx +1,k = 0, ∀ k, unj,0 = unj,Ny +1 = 0, ∀ j.

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E U IQ

n ≥ 0A j ∈ {1, ..., Nx} k ∈ {1, ..., Ny } ( 0 .

8 . '2

H C E T

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B $ 07 ν∆t ν∆t 1 + ≤ . 2 2 (∆x) (∆y) 2

LE O ÉC

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N H EC

E U IQ

/ ; < ; I . < 2 . 6 ( , L0M ; : ( . 0.01 . (1., 0.)<

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Y L O P +ν

H C E T

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LE O ÉC

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. unj,k ,

LE O ÉC

T Y POL

un = (un1,1 , ..., un1,Ny , un2,1 , ..., un2,Ny , ..., unNx ,1 , ..., unNx ,Ny ).

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T Y POL

N H EC

E U IQ

R 0 Dj Ny ⎛ 1 + 2(cy + cx ) −cy 0 ⎜ −c 1 + 2(c + c ) −c y y x y ⎜ ⎜ Dj = ⎜ ⎜ ⎝ −cy 1 + 2(cy + cx ) −cy 0 −cy 1 + 2(cy + cx )

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H C E T

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T Y L

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n+1/2

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2(∆x)2

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n+1/2

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zone où l’on veut de la précision

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#

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/ , , . , 6 0 (0, 1) . . V > 0 ⎧ ∂u ∂u ⎪ ⎨ +V = 0 (x, t) ∈ (0, 1) × R+ ∗ ∂t ∂x ; < u(t, x + 1) = u(t, x) (x, t) ∈ (0, 1) × R+ ⎪ ∗ ⎩ u(0, x) = u0 (x) x ∈ (0, 1).

LE O ÉC

Y L O P

H C E T

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@ I , . ∆x = 1/(N +1) > 0 ;N 8< . ∆t > 0A (tn , xj ) = (n∆t, j∆x) n ≥ 0, j ∈ {0, 1, ..., N + 1}A unj . , (tn , xj )A u(t, x) ; < un0 = unN +1 n ≥ 0A unj = unN +1+j * A , 6 ? . un = (unj )0≤j≤N ∈ RN +1 / 0 , , . ; < 3 ' . I? . :+4%9. )>3-'+'1)

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T Y POL

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un+1 − unj unj+1 − unj−1 j +V =0 ; < ∆t 2∆x n ≥ 0 j ∈ {0, ..., N } 0 ) .

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E L O

T Y POL

@ . ) 8 , ) I A 2 V ∆t 2 |A(k)| = 1 + sin(2πk∆x) ≥ 1, ∆x . 2k∆x , 1 0 @ . . 0 J , :+4%9. '93-'+'1) +)*10% n+1 un+1 un+1 − unj j j+1 − uj−1 +V = 0. ∆t 2∆x

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2un+1 − unj+1 − unj−1 unj+1 − unj−1 j +V =0 2∆t 2∆x A 0 A

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T Y POL

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u(tn , xj+1 ) − u(tn , xj−1 ) 2u(tn+1 , xj ) − u(tn , xj+1 ) − u(tn , xj−1 ) +V = 2∆t 2∆x (∆x)2 (∆x)4 (V ∆t)2 2 . ; $< (ut + V ux ) (tn , xj ) − (t , x ) + O (∆x) + u 1− xx n j 2∆t (∆x)2 ∆t ' , O (∆x)2 /∆t A ,

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LE O ÉC

ˆ 1 (k) . (λ1 , λ2 ) A(k) Uˆ n+1 (k) = A(k)n U #X λ2 −

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* 2 0 A 1 1 v (s)φ (x) ds dx = φ (x) dx v (s) ds = −

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s

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1A w 0 .0 8 0 ) w 1 1 1 − w (x)φ(x) dx = w(x)φ (x) dx = − φ(s)v (s) ds, 0

0

0

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0

? y 1 |v(x)| ≤ |v(y)| dy + 0

≤ 0

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!

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•

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Ω " $ -" /$ 0 C 1 ' /$ Ω = RN+ ' $ Ω = RN ' Cc∞ (Ω) " $ $ H 1 (Ω), 4%,0?9) CC $&) &)*:'1%6

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5 " ; , < . ? + RN + 6 {x ∈ RN xN > 0}

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)9.0;5) CC , Cc∞ (Ω) H 1 (Ω) 86

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)9.0;5) CC , . 16

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Y L O P

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$

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, ;R ,# < , . L. M ;. ( • <

T Y L 1 P HO (Ω) LE O ÉC 1 0

1) 0. 6 H 1 (Ω) 0 . 1

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@ . ;. ' $< H01 (Ω) 8 6 H 1 (Ω) =,*+1',*: ;5' :!.**5-)*1 :50 -) /,0& ∂Ω 8 Cc∞ (Ω) 9 A H01 (Ω) :10'+1)9)*1 3-5: 3)1'1 H 1 (Ω) Cc∞ (Ω) 6 :10'+1 Cc∞ (Ω) ;. 5 " + $< R Ω = RN J (A Ω = RN = Ω 5 " : Cc∞ (RN ) H 1 (RN )A H01 (RN ) = H 1 (RN ) ' , RN , 0

T Y POL

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E ) " 9,% LB $ " $O ) @/", ÉC

0,3,:'1',* CC H01 (Ω)

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|v| dx

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, * . 8

. H01 (Ω)

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. 1, A , * $ ?

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v2H 1 (Ω) ≤ (C + 1)

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2 0 -

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/ . . , N ≥ 2 8 H 1 (Ω) ;. 6 ,9 < ' 8 0A . , 8 v ∈ H 1 (Ω) L M Ω 9 A , . ) L. 0 MA LM v 0 ∂Ω ∂Ω 0 0 2 0 A # H # ) v|∂Ω , 8 H 1 (Ω) ' A A .

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" Ω $ -" /$ 0 C 1 ' /$ , $ *$" ))"$ " γ0

4%,0?9) CC $&) 10.+)6 Ω=

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;"" ))"$ γ0 )$0 ) $"$ " $ $ ))"$ $ $"$ H 1 (Ω) $ L2 (∂Ω)' $" $ γ0 , $ )" ' &" $ $"$" C > 0 " ' ) " " $"$ v ∈ H 1 (Ω)' $

T Y L

N H EC

; <

vL2 (∂Ω) ≤ CvH 1 (Ω) .

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)9.0;5) CC 4Q 5 . , 8 H 1 (Ω) 0 ∂Ω ' 0 , . 8 L2 (Ω) ;. ,9 < • %9,*:10.1',*C 1 6 Ω = RN + = {x ∈ RN , xN > 0} v ∈ Cc∞ (RN + ) 3. x = (x , xN )A

|v(x , 0)|2 = −2

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0

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0

2 ∂v (x , xN ) dxN . |v(x , xN )| + ∂xN

2

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2 ∂v (x) dx, |v(x)| + ∂xN 2

Y L O P RN +

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N

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∞ 1 N N ,6?6 vL2 (∂RN ) ≤ vH 1 (RN ) * Cc (R+ ) H (R+ )A 06 + +

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!

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* . 0 C A 0 Ω = RN + / ;F < H * 6 1

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5 8 H 1 (Ω) 8 4 0 8 C 1 '

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" Ω $ -" /$ 0 u " v $" $"$ H 1 (Ω)' -*$"

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@ ? n → +∞ ;< ∂vn ∂u n un ∂u ∂xi ; .A vn ∂xi < . . u ∂xi ; .A v ∂v ∂x < L2 (Ω) * ? ;
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? Cc∞ (Ω) 0 G A , γ0 @ H01 (Ω) 6 H 1 (Ω) 8 , 0 ∂Ω , 0 ;. . * < ; ? . 5 < / .# ? T&UA T"U

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Y L O P

$

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)9.0;5) CC ' $ : # , γ0

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4Q ' $ . * ? , * ' . , L .M 0 8 9 (A 0 . , * A ( , . 3 . , A , , 0 ? )

510) &%9,*:10.1',* &) -. 0,3,:'1',* CC C @ , , C > 0 A 8 v ∈ H01 (Ω)A |v(x)|2 dx ≤ C |∇v(x)|2 dx, Ω

Ω

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. , vn ∈ H01 (Ω) 1= |vn (x)|2 dx > n |∇vn (x)|2 dx.

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9 A ;"< vn 0 H01 (Ω) * 5 + A 6 vn . L2 (Ω) 1 A ;"< ∇vn . . F L2 (Ω) ; < * A vn ' # H01 (Ω)A

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A . "A v 6 Ω 7 v 0 ∂Ω ; . ' $
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V = {v ∈ H 1 (Ω) v = 0 ∂ΩD }.

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* 8 4 5 "A . 8 , 8 ? H 1 (Ω) ' 8 , 6 6 ) H 1 (Ω)

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" Ω $ -" /$ 0 C 1 , " (ωi)1≤i≤I $ )""$ 0 + Ω' " # ωi " $ -" 0 C 1 ' ωi ∩ ωj = ∅ i = j ' " Ω = ∪Ii=1 ωi , " v $ $"$ $" ""$ # ωi' vi = v|ω ' ))"$" H 1 (ωi ), v " $"$ Ω' v ))"$" H 1 (Ω),

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i

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%9,*:10.1',*C ' . 8 0 v J φ ∈ Cc∞ (Ω)A 8 4 ωi

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∂φ v(x) (x) dx ∂xj Ω

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=

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I ∂vi (x)φ(x) dx + vi (x)φ(x)nij (x) ds ∂xj ∂ω i i=1

∂vi (x)φ(x) dx, ∂xj

0 , ? 9 (A Γ = ∂ωi ∩ ∂ωk 0 . ωi ωk nij (x) = −nkj (x) A v φA vi (x)φ(x)nij (x) ds + vk (x)φ(x)nkj (x) ds = 0. Γ

Γ

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@ v .0 8 0 ∂v ∂vi = . ∂xj ωi ∂xj

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T Y POL Ω

∂ΩN

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# $ 6 * * % Ω

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E L O

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N H EC

y=xr

Ω x

W 9 , .

>)0+'+) CC <% $ $ Ω $ * ) $ Ω ⊂ R2 0 < x < 1 0 < y < xr r > 2 7* 2 ) v(x) = xα " v ∈ H 1 (Ω) 2α + r > 1( v ∈ L2 (∂Ω) 2α > −1 0 B <% ! - 2 *

Y L O P

H C E T

N

E U IQ

>)0+'+) CC $ :

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L (Ω)( $ + $ $ C > 0 2

!

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( v ∈ L (Ω)(

2

T Y POL

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&

v|∂Ω L2 (∂Ω) ≤ CvL2 (Ω) .

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/ 6 ? , , + I X 0 ;. ' &< 0 ,. + ,0 A = 0 ) A , . A 0A 6 . ; ) A . ,9 "<

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Ω " $ -" /$ 0 C 1 ' 1 " " " /$ H (Ω) $ ) " &" $ " $-0$" $ L2(Ω) $ " $>"$ $$ H 1 (Ω) $ L2 (Ω) " )",

4%,0?9) CC $&) )--'+46

T Y POL

N H EC

5 H 8 , . Ω , 0 * A Ω = RN A , I H 1 (RN ) L2 (RN ) , * , . : un (x) = u(x + ne) R e . u 8 H 1 (RN ) ; u e< , 6 un . L2 (RN )

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E L O

)9.0;5) CC , H 1 (Ω) H01 (Ω)A 56

+ . A , , . Ω •

5 ; 8 ? 8 2 A ,3 Y . A A T&UA T"U< / LM N = 1

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%9,*:10.1',*C @ , N = 1 A

A Ω = (0, 1) (un )n≥1 0 H 1 (0, 1)A ,6?6 , K > 0

H C E T

un H 1 (0,1) ≤ K ∀ n ≥ 1.

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1, A x, y ∈ [0, 1]

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|un (x)| ≤ CK |un (x) − un (y)| ≤ CK

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|x − y|.

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;$<

$

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E U IQ

;@ , ' #6 CF ? ;-<< ; 00< (xp )p≥1 [0, 1] . ; A Q ∩ [0, 1]< * p )A un (xp ) 0 R ? ;$< @ 6 . R @ ,0 ? un (x1 ) 0 6 un1 (x1 ) . R * 6 A n1 A . 6 A n2 A un2 (x2 ) . R ; A 0 GA un2 (x1 ) . < 9 # . 6 A A 6 A np A unp (xp ) . A unp (xk ) 1 ≤ k ≤ p . A 6 unp L M ? p * . , , L? MA , 3 A 6 ; < 0 6 A ,6?6 , 6 un1 A un2 A p6 p6 unp 0 um . A p ≥ 1A um (xp ) . R x ∈ [0, 1] > 0 ; < ' (xp )p≥1 [0, 1]A xp |x − xp | ≤ * A um (xp ) . R m . , ) A m0 A m, m ≥ m0 A |um (xp ) − um (xp )| ≤ * A 0

LE O ÉC

|um (x) − um (x)|

ÉC

T Y POL

T Y POL

N H EC

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≤ |um (xp ) − um (x)| + |um (xp ) − um (xp )| + |um (xp ) − um (x)| √ ≤ + 2CK

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. um (x) ' # RA . x ∈ [0, 1] * . 0 A um . L2 (0, 1)

>)0+'+) CC ) Ω = (0, 1) un (x) = sin(2πnx) " un L2 (Ω)( $ $ * - ( *F + * ( ( φ ∈ Cc∞ (Ω)(

1

lim

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un (x)φ(x) dx = 0, 0

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un * L2 (Ω) ? + H 1 (Ω) un

1 H m (Ω)

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N

@ 1) , 0. H 1 (Ω) 8 m ≥ 0 8 .0 8 0 'O . , 0 α = (α1 , ..., αN ) 95-1''*+'&)A ,6?6

LE O ÉC

!

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. ? N . αi ≥ 0 @ |α| = 8 v A ∂ |α| v ∂ α v(x) = (x). α1 N ∂x1 · · · ∂xα N

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N i=1

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$ $" m ≥ 0' ) /- H m (Ω) " *$ )

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$

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LM 5 " m A 8 H m (Ω) A ,6?6 .0 ; : , . 5 " ? 8 v ∈ H m (Ω) ? . ∂ α v ∈ H m−|α| (Ω)< ' H 1 (Ω)A 8 H m (Ω) ; , . Ω Y . 1) "< 5 " 8 ? H m (Ω) / O .

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Ω " $ -" /$ 0 C m ' /$ Ω = RN+ ' " $ $ H m (Ω),

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H 2 (Ω) ∩ C 1 (Ω) → L2 (∂Ω) ∩ C(∂Ω) ∂v v → γ1 (v) = , ∂n

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∂v = ∇u · n, ;"" ))"$ γ1 )$0 ) $"$ " $ $ ))"$ - ∂n $ $"$ H 2 (Ω) $ L2 (∂Ω), $ )" ' &" $ $"$" C > 0 " ' ) " " $"$ v ∈ H 2 (Ω)' $

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∂v ≤ CvH 2 (Ω) . ∂n 2 L (∂Ω)

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# ! 1 Ω % ' C 2

D( #CC $ γ0 H 2 (Ω) ∩ C(Ω) v

→ →

H 1 (∂Ω) ∩ C(∂Ω) γ0 (v) = v|∂Ω ,

H C E T

N

E U IQ #!

" H 2 (Ω) H 1 (∂Ω)

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C2

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v ∈ H (Ω)

u ∈ H 2 (Ω)

1

∆u(x)v(x) dx = − Ω

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%9,*:10.1',*C ' ; < . 8 C 2

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N H EC

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' H

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* 2 4

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ρn (x) = n ρ(nx)

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L2 (RN )

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L (R ) ζ ∈ Cc∞ (RN ) ζn (x) = ζ nx %

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0≤ζ≤1

ζ(x) = 1

|x| ≤ 1 ζ(x) = 0 |x| ≥ 2 $ J #C! " vn v˜n (x) = vn (x)ζn (x) $ % * " v˜n % ∇˜vn ! % % v % ∇v! L2 (RN ) D( #C/ % "

% ' % D( ## RN

E U Q

I N H

' " Ω C Ω = R 1

N +

! P H (Ω) H (R ) v ∈ H 1 (Ω) P v|Ω = v # P vL (R ) ≤ CvL (Ω) $ P vH (R ) ≤ CvH (Ω)

% C > 0 ! Ω D ' Ω = RN+ $ x = (x , xN ) % x = (x1 , ..., xN−1 ) 1 v ∈ H 1 (RN+ ) $

v(x , xN ) xN > 0 P v(x) = v(x , −xN ) xN < 0. $ % " 1 ≤ i ≤ N − 1 ∂v (x , xN ) xN > 0 ∂P v ∂x (x) = ∂v ∂xi (x , −xN ) xN < 0, ∂x " ∂v (x , xN ) xN > 0 ∂x ∂P v (x) = ∂v ∂xN − ∂x (x , −xN ) xN < 0. 5 % v * ; " v %' " *' 4

*! " ; *

L P v M % FC/H ! $ √ P % C = 2 Ω = RN + ! 1 Ω % ' C 1

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O P E L ÉCO 2

N

1

N

2

1

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(ωi )0≤i≤I

(θi )0≤i≤I

Cc∞ (RN )

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I

θi (x) = 1

Ω.

i=0

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Pv =

I

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i=0

: (" Pi ωi 5

θ0 v & Ω P0 (θ0 v)

θ0 v E ( Ω (" i ∈ {1, ..., I} φi " * ωi * Q % . C/ J ##!

+) wi = (θi v) ◦ ( φ−1 i Q

% Q+ = Q ∩ RN+ .

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H C E T

N

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LE O ÉC

$

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L2 (Ω)

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E U IQ

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T Y POL

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##!

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H(div) *

5

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1

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N +

, H(div) " (

* O % " 1 ' % H 1 (Ω) 0 + σ & H(div) 8 <9 " ' %

H 1 (Ω)! & σ · n ∂Ω 5

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N H C " E YT L " * @( O B'! $

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vH 1/2 (∂Ω) = inf φH 1 (Ω)

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C1

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H(div) ∩ C(Ω) σ = (σi )1≤i≤N

→ →

H −1/2 (∂Ω) ∩ C(∂Ω) γn (σ) = (σ · n)|∂Ω

n = (ni )1≤i≤N

∂Ω H(div)

φ ∈ H 1 (Ω)

σ ∈ H(div)

γn H −1/2 (∂Ω)

divσφ dx +

Ω

Ω

σ · ∇φ dx = σ · n, γ0 (φ)H −1/2 ,H 1/2 (∂Ω) .

1 Ω % C 0 C * *

% %

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1

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E U IQ

∂Ω

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* Im(γ0 ) H 1/2 (∂Ω) , *

* #/! γn (σ) '

H −1/2 (∂Ω)

E L O

T Y POL

N H EC

$ ' - " % H01 (Ω) H(div) $ H0 (div) H(div)

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* H(div)

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1 W m,p (Ω) W m,p (Ω) m ≥ 0

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|α|≤m

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(Ω)

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p

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m,p

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Ω = RN

Ω

⎧ ⎨

W 1,p (Ω) ⊂ Lq (Ω) ∀ q ∈ [1, p∗ ] W 1,p (Ω) ⊂ Lq (Ω) ∀ q ∈ [1, +∞[ W 1,p (Ω) ⊂ C(Ω),

pN

⎩

" u ∈ W

1,p

1/p∗ = 1/p − 1/N

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! W

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E U IQ

0! 1 u ∈ W N,1 (RN ) uL (R ) ≤ uW (R ) . #I! 1 u ∈ Cc∞ (RN ) x = (x1, ..., xN ) N,1

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u(x) =

N

T Y POL

x1

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xN

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$ Cc∞ (RN ) W N,1 (RN )

D( ## H 1 (RN )! . ' #I! u ∈ W N,1 (RN ) * Cc∞ (RN ) L∞ (RN ) Cb (RN ) * ! . #I! " " * W N,1 (RN ) ' * A " V B' V ' * V D( ) AE L2 (Ω) & L2 (Ω) $ 1 ' % 0 H01 (Ω) ; <

L, φH −1 ,H 1 (Ω) = L(φ)

L ∈ H −1 (Ω)

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N

Y ! 1 L O P OLE

$ " 0

H C E T

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H01 (Ω) H01 (Ω)

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−1

(Ω)

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−1

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f = v0 +

2

v0 , v1 , ..., vN ∈ L (Ω)

∂xi

i=1

φ∈

!

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H01 (Ω)

H01 (Ω)

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L(φ) =

N

Ω

i=1

.

L ∈ H −1 (Ω)

∂φ vi ∂xi

dx

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v ∈ H 1 (Ω) v

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∂v ∂xi

3 L2 (Ω)

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v Ω

∂φ dx. ∂xi

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−1

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•

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ÉC

E L O

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N H EC

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4 T ∈ D (Ω) ! D(Ω) &

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Tf , φ =

f φ dx. Ω

H C E T

N

E U IQ

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lim Tn , φ = T, φ.

n→+∞

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∈ D (Ω)

∂T ∂φ , φ = − T, ∀ φ ∈ D(Ω). ∂xi ∂xi ∂T ∂xi

f

@ - = % *' . #C % ' . Lp (Ω) 1 ' % H m (Ω) ' D (Ω) 0 % " % " % ' " * ! 0 * % " %

" ! " '

ÉC

E L O

T Y POL

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N H EC

Y L O P

E U IQ

H C E T

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T Y POL

; . 8 0<

Ω

*

LE O ÉC

N H EC

H 1 (Ω) = 0 u, v = (∇u · ∇v + uv) dx Ω

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5 ; < 5 " ;8 4<

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5 ; + < 5 ;8 4<

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u → u|∂Ω H 1 (Ω) L2 (∂Ω)

N H EC

E U IQ

∀ u, v ∈ H 1 (Ω) ∂v ∂u u dx = − v dx + uv ni ds Ω ∂xi Ω ∂xi ∂Ω

T Y POL

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Ω

W * P

LE O ÉC

Y L O P

∂Ω

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0. , 8 0

H C E T

N

E U IQ

T Y POL

N H EC

E U IQ

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E U IQ N H O P E L ÉCO

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LE O ÉC

Y L O P

H C E T

N

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1.3)

7 0 , 8 .

1 8 8 . 6 0 ;"
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E U IQ

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C E T LY

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O P E L ÉCO

' u 8 1 A u = 0 ∂ΩA = 0 V 8 v ∈ V . ) v = 0 ∂Ω 1 A , ;"< . ∇u(x) · ∇v(x) dx = f (x)v(x) dx. ;"< Ω

Ω

E U IQ

* ;"< : ∇u ∇v ? L2 (Ω) ;
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Ω

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N H EC

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LE O ÉC

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1.3) 7 + 8 .

1 . ) 8 . ;""< * 5 67 . ) # . a(u, v) = ∇u(x) · ∇v(x) dx L(v) = f (x)v(x) dx. Ω

Ω

@ . 8 , ' #6 CF a 8 0 H01 (Ω) L 8 H01 (Ω) 1 A . , * ;. ' Y 0 , . Ω 0 a(v, v) = |∇v(x)|2 dx ≥ νv2H 1 (Ω) ∀ v ∈ H01 (Ω).

T Y POL Ω

E L O

N H EC

E U IQ

0

' H01 (Ω) = 0 ;. *
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)9.0;5) CC / . ' A A

8 0 a # A 56 67 9 (A 8 . 6 , , ) J(v) =

1 a(v, v) − L(v) 2

∀ v ∈ H01 (Ω)

;. * " &< * A J A 0 8 . •

1.3) 7 . . ,

H C E T

N

E U IQ

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Y L O P

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E U IQ

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ÉC

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T Y POL Ω

Ω

v = 0 0 ∂Ω @ (∆u + f ) v dx = 0 ∀ v ∈ Cc∞ (Ω), Ω

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N H EC

u = 0 ∂Ω.

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Ω

' Cc∞ (Ω) ⊂ H01 (Ω)A ;"-< , , , , . 8 0 σ L2 (Ω) ;. 1) $ &< . )A v ∈ Cc∞ (Ω)A σ · ∇v dx = − divσv dx. Ω

@

Ω

(divσ + f ) v dx = 0 Ω

H C E T

∀ v ∈ Cc∞ (Ω),

Y L O P

N

E U IQ

A . ' A −divσ = f L2 (Ω) * 6 divσ = ∆u ? L2 (Ω) ; div∇ = ∆
LE O ÉC

"

N H EC

E U IQ

"

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LE O ÉC

T Y POL

" Ω $ -" /$ RN , " f ∈ L2 (Ω), &" $ $ "$ u ∈ "$ -"$$ =,=, 8 ) ' u -* −∆u = f ) )" " $ Ω, " u ∈ H01 (Ω). ;"< $ )) $ ) Ω " 0 C 1 ' u " "$ )/+ & " =, $ . −∆u = f ) )" " $ Ω, u = 0 ) )" " ∂Ω. 4%,0?9) CC

H01 (Ω)

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N H EC

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H C E T

N

E U IQ

* 0 ;"< 0 ; = Y . 1) "
LE O ÉC

Y L O P

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R

Ω

f ∈ L (Ω) 2

C>0

f ∈ L2 (Ω) u ∈ H01 (Ω) 2 L (Ω) H 1 (Ω) 2 f ∈ L (Ω)

N

;"<

uH 1 (Ω) ≤ Cf L2 (Ω) .

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Ω

N H EC

E U IQ

@ I ? , , ' #6 CFA . 8 0

T Y POL

νu2H 1 (Ω) ≤ f L2 (Ω) uL2 (Ω) ≤ f L2(Ω) uH 1 (Ω) ,

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/ . I? 8 . . 6 # ;,A A . . < 9 8 A 8 . ;""< , ; # < . *

0,3,:'1',* CC

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|∇v| dx − 2

Ω

;"<

f v dx. Ω

E U IQ

" u ∈ H01 (Ω) "$ $ "$ -"$$ =,=, u " $ )$" $ $0' " J(u) =

min J(v). 1

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H C E T

N

C) $"' u ∈ H01 (Ω) " $ )$" $ $0 J(v)' u " "$ $ "$ -"$$ =,=,

LE O ÉC

"

N H EC

E U IQ

&

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LE O ÉC

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>)0+'+) CC G $ $ % $ $

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V · ∇u − ∆u = f u=0

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f v dx. Ω

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N

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Y L O P

"

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, , H −1 (Ω) ; Ω< ' : ? # 6 * , 0 #6 8 X 0 Ω Γ # 8 ; . N − 1< Ω * ΓA f˜ ∈ L2 (Γ) ) f ∈ H −1 (Ω) f, vH −1 ,H01 (Ω) = f˜v ds,

LE O ÉC

T Y POL

Γ

0 8 •

H01 (Ω)

T Y POL

E U IQ

Q 5

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0 8

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H01 (Ω)

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LE O ÉC

N

E U IQ •

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R Ω . ; 0< , RN A f ∈ L2 (Ω) g ∈ L2 (∂Ω) , ;"$< . R . I

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LE O ÉC 1.3)

T Y POL

7 0 , 8 .

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HN

(∇u(x) · ∇v(x) + u(x)v(x)) dx −

= Ω

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= Ω

Y L O P

E U IQ

∂u (x)v(x) ds ∂n

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/ . / ;"&< , , , = 0 V * ;"&< : V = H 1 (Ω) ;

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1.3) 7 + 8 .

N

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"

N H EC

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1 . ) 8 . ;"-< * 5 67 . ) # . a(u, v) = (∇u · ∇v + uv) dx L(v) = gv ds + f v dx.

LE O ÉC

T Y POL Ω

∂Ω

Ω

9 , ' #6 CF ? , 5 A . a 8 0 H 1 (Ω) L 8 H 1 (Ω) * A 8 0 a 8 . ;, , I , F < a(v, v) = v2H 1 (Ω) ∀ v ∈ H 1 (Ω).

' H 1 (Ω) = 0 ;. *
E U IQ

)9.0;5) CC * # 0 ;"$< R g = 0A

N H EC

H , / , ∂v = 0 V , 0 V = {v ∈ H 1 (Ω), ∂n = 0 ∂Ω} A ∂v 1

8 v ∈ H (Ω) ∂n , ∂Ω 9 (A ∇v , , 8 L2 (Ω) ; < , ,# ∂v ∂Ω 8 L2 (Ω) @ V = {v ∈ H 2 (Ω), ∂n = 0 ∂Ω} 2 0 6 8 H (Ω) . 5 - 7 . . : J 8 0 a , . V 5 67 ,# # / • , = 0

ÉC

E L O

T Y POL

1.3) 7 . . , / 8 . ;"-< . ) , 0 0 ;"$
E U IQ

" Ω $ -" 0 C 1 RN , " f ∈ L2 (Ω) " g " ∂Ω $ $"$ H 1 (Ω), "$ u "$ -"$$ =, ))"$" H 2 (Ω),

)99) CC

Y L O P

H C E T

N

4Q " 8 4 5 ∂u (x)v(x) ds. ∆u(x)v(x) dx = − ∇u(x) · ∇v(x) dx + ;"< Ω Ω ∂Ω ∂n

LE O ÉC

"

N H EC

E U IQ

.0 u ∈ H (Ω) v ∈ H (Ω) + , 0 ;"< 0 ? 5 - : ∂u u ∈ H 2 (Ω) . ∂n L2 (∂Ω) @ ;"-< 1 ;"< A v ∈ H (Ω)A ∂u (∆u − u + f )v dx = ;" < g− v ds. ∂n Ω ∂Ω 2

LE O ÉC

1

T Y POL

, v ∈ Cc∞ (Ω) ⊂ H 1 (Ω) ;"
@ , H 1 (Ω) , L2 (∂Ω) ;. ∂u &
N H EC

E U IQ

" Ω $ -" 0 C 1 RN , " f ∈ L2 (Ω) " g " ∂Ω $ $"$ H 1 (Ω), &" $ $ "$ u ∈ H 1 (Ω) "$ -"$$ =,, 8 ) ' u ))"$" H 2 (Ω) " " "$ =,% $ . 4%,0?9) CC

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E ) )" " $ L O

−∆u + u = f

ÉC

Ω,

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) )" " ∂Ω.

>)0+'+) CC 6 $ u ∈ H 1 (Ω)

!. $ $ *

uH 1 (Ω) ≤ C f L2(Ω) + gL2 (∂Ω) ,

4 C > 0 u, f g ' 6 A 0 ; H ? 8 < , u ∈ H 1 (Ω) 8 . ;"-< , :,-51',* =.'/-) 0 ;"$<

E U

* " I Q N " * % /)! " % " *

H " C E 0 /)! " T Y L O P E L O ÉC ( , " % Ω " g 2

1

u ∈ H (Ω)

L (∂Ω)

−∆u + u = f σ = ∇u ∈ L2 (Ω)N

Ω

v ∈ Cc∞ (Ω)

σ · ∇v dx ≤ uv dx + f v dx ≤ CvL2 (Ω) Ω

Ω

Ω

"

E U " " % I*' Q % . #N

#G! $ N H C E T Y L # " O 5 " " %

P $ % 4

*' E L % ! * & O 1 ## " D( C É

L2 (Ω)

σ

divσ ∈ L2 (Ω)

∀ v ∈ Cc∞ (Ω),

(divσ − u + f ) v dx = 0

Ω

L2 (Ω)

−divσ = −∆u = f − u

g H(div) = σ ∈ L2 (Ω)N

H(div) divσ ∈ L2 (Ω) .

##G " Ω % C 1 * %

divσv dx + Ω

Ω

/!

σ · ∇v dx = σ · n, vH −1/2 ,H 1/2 (∂Ω) ,

v ∈ H 1 (Ω) σ ∈ H(div) /! H 1/2 (∂Ω) H −1/2 (∂Ω) 1 σ v

* 8%9 " ' ∂Ω vσ · n ds 1 * /! & σ · n ∂Ω

H −1/2 (∂Ω)! σ ∈ H(div) 1 " & /)! % σ = ∇u! " 4

%

H −1/2 (∂Ω) g ∈ L2 (∂Ω) ⊂ H −1/2 (∂Ω)! 5

' " " " * % 4

(' & . ∂u " ∂n = g ∂Ω •

ÉC

T Y POL

E L O

N H EC

E U IQ

' 6 A ;"$< + A g = 0A , ;" < H ;"< ) . 1 / A H (A ? . H01 (Ω) H 1 (Ω) * . *

0,3,:'1',* CC

" J(v) $0 *$ ) v ∈ H 1 (Ω) )

1 J(v) = 2

|∇v|2 + |v|2 dx −

Ω

f v dx − Ω

;"

gv ds. ∂Ω

<

E U IQ

" u ∈ H 1 (Ω) "$ $ "$ -"$$ =,, u " $ )$" $ $0' " J(u) =

min J(v). 1

Y L O P v∈H (Ω)

H C E T

N

C) $"' u ∈ H 1 (Ω) " $ )$" $ $0 J(v)' u " "$ $ "$ -"$$ =,,

LE O ÉC

"

N H EC

E U IQ

>)0+'+) CC Ω * C 1 G $

$ % $ $

7

T Y POL

LE O ÉC

−∆u = f ∂u ∂n + u = g

Ω ∂Ω

;" <

4 f ∈ L2 (Ω) g ∂Ω $ H 1 (Ω) $ * * - vL2 (Ω) ≤ C vL2 (∂Ω) + ∇vL2 (Ω) ∀ v ∈ H 1 (Ω).

>)0+'+) CC Ω G $ $ % $ $ B ⎧ ⎨ −∆u = f Ω ∂u ;" < =0 ∂ΩN ⎩ ∂n u=0 ∂ΩD

N H EC

E U IQ

4 f ∈ L2 (Ω)( (∂ΩN , ∂ΩD ) ∂Ω ∂ΩN ∂ΩD 7* H C .

T Y POL

/ . . 0 ; I , , F ;"$<< 0

−∆u = f Ω ;" "< ∂u ∂Ω ∂n = g

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R Ω . 0 , RN A f ∈ L2 (Ω) g ∈ L2 (∂Ω) : . ;" "< ? ;"$< , , f g . ) +,*&'1',* &) +,93.1'/'-'1% 9 (A 8 . , u ∈ H 2 (Ω)A , Ω ;

0 8 4 ; << f (x) dx + g(x) ds = 0. ;" $< Ω

∂Ω

E U IQ

+ u u + C A . C ∈ RA 9 8 A ;" $< : , , H 1 (Ω)A ? , , + A , . Ω , A 8 ;" $< Ω , . ? , , ;. ) . .0<

Y L O P

H C E T

N

)9.0;5) CC *# A 0 ;" $< , +,*&'1',* &!%;5'-'/0) J f ? . A g ?

LE O ÉC

"

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N H EC

D 0 * , , 0 ;,6?6

;" "<
T Y POL

" E L "

$O C É "$ /

Ω $ g ∈ L2 (∂Ω) u ∈ H 1 (Ω)

4%,0?9) CC f ∈ L2 (Ω)

-" /$ $$& 0 C 1 RN , " -*$" $ "$ )"/" =,%, &" =,=' $ "$ $ $"$" )+,

%9,*:10.1',*C * . 8 . , ;"$< ? ∇u · ∇v dx = gv ds + f v dx Ω

∂Ω

Ω

E U IQ

8 v * ? A H 1 (Ω) = 0 V A . 8 0 ' : 8 A u A u + C * . . A . , . . ,. 8 # 3 A

V = v ∈ H 1 (Ω), v(x) dx = 0

ÉC

E L O

T Y POL

N H EC

Ω

8 . ;" "< J . u ∈ V ∇u · ∇v dx = Ω

f v dx ∀ v ∈ V.

gv ds +

∂Ω

;" &<

Ω

@ V = H 1 (Ω)/RA ,6?6 H 1 (Ω) ; H 1 (Ω)/R 8 H 1 (Ω) ?

< * . 5 67 ? 8 . 6 ;" & 0 A v ∈ H 1 (Ω)A

v dx v − m(v)L2 (Ω) ≤ C∇vL2 (Ω) . m(v) = Ω ;" -< . Ω dx

Y L O P

H C E T

N

E U IQ

, ;" -< 6 * ; < ' m(v) = 0 v ∈ V A ;" -< . ∇vL2 (Ω) V A

LE O ÉC

"

N H EC

"

E U IQ

. ? vH 1 (Ω) A 8 0 . V 2 A ;" &< 0 0 ;" "
LE O ÉC

T Y POL

@A w ∈ H 1 (Ω)A 8 v = w − m(w) ? V 9

8 ;"
∂u dsA ;" < ∂Ω ∂n ∂u (∆u + f )w dx = g− w ds ∀w ∈ H 1 (Ω). ∂n Ω ∂Ω

E U IQ

@ ,0 u . ) 0 0 ;" "<

N H EC

>)0+'+) CC 6 $ * - D * !2.

T Y POL

>)0+'+) CC Ω * ) f ∈ L2 (Ω)

I u ∈ H 1 (Ω)

ÉC

E L O

Ω

∇u · ∇v dx +

u dx v dx = f v dx

Ω

Ω

∀ v ∈ H 1 (Ω).

Ω

6 $ $ J

K ' ( Ω f dx = 0( 5+ K

' 4 $&

1 6 . 0 , @ 8 0 ? A E +,)J+')*1: 2.0'./-): ' # 0 , , ; '
−div(k∇u) = f Ω ;"< u=0 ∂Ω

Y L O P

H C E T

N

E U IQ

R Ω . 0 , RN A f ∈ L2 (Ω) E GA k(x) ≡ 1A . 8 5 "

LE O ÉC

"

$

E U " $ -" /$ , " I Q , $ )) 6$" " $ $"$ / "H N &" & $"$" "" $" )"- " C E) )" " T Y L / O "$ =,3, ' &" $P

$ E L O ÉC 0,3,:'1',* CC

R

Ω k(x) 0 < k− ≤ k+

f ∈ L (Ω)

N

2

0 < k − ≤ k(x) ≤ k +

;"<

x ∈ Ω.

u ∈ H01 (Ω)

%9,*:10.1',*C * . 8 . , ;"< 8 v 8 4 ,9 divσ(x)v(x) dx = − σ(x) · ∇v(x) dx + σ(x) · n(x) v(x) ds, Ω

Ω

∂Ω

. σ = k∇u * 1 H01 (Ω) = 0A . 8 . ;"< J k∇u · ∇v dx = f v dx ∀ v ∈ H01 (Ω). ;" < . u ∈ H01 (Ω) Ω

N H EC Ω

E U IQ

4Q ? ,# ;"< . ) 8 0 ;" < k∇u · ∇v dx ≤ k + ∇uL2 (Ω) ∇vL2 (Ω) ,

E L O Ω

, .

ÉC

T Y POL

k∇u · ∇u dx ≥ k −

Ω

Ω

|∇u|2 dx ≥ νuH01 (Ω) ,

. ν > 0 Q ? , * @ 5 67 A , , 8 . ;" < * . 0 6 0 ;"
)9.0;5) CC , ;"< 8 . J ) , −div(k∇u) = −k∆u − ∇k · ∇u,

H C E T

N

E U IQ

8 , : k .0 3 A , ;"< H k •

Y L O P

9 8 A , ;"< 8 . A L 8 0M , 8 * 6 A . 8 0 ,

LE O ÉC

"

N H EC

&

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+,*&'1',*: .5> -'9'1): &) 10.*:9'::',* 6 . ( ' R (Ω1 , Ω2 )

Ω k(x)

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k(x) = ki > 0 x ∈ Ωi , i = 1, 2.

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;"<

@ Γ = ∂Ω1 ∩ ∂Ω2 , 8 ; Ω< Ω1 Ω2 ;. 2 "
Γ Ω2 Ω1

N H EC

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" W 8 6

T Y POL

#!)"#+ =,33 )/+ =,3 " -$" ⎧ −ki ∆ui = f $ Ωi, i = 1, 2, ⎪ ⎪ ⎨ u1 = 0 ∂Ω, ;"< u = u Γ, ⎪ 1 2 ⎪ ⎩ k1 ∇u1 · n = k2 ∇u2 · n Γ. & $+ 0$ =,39 $" )) +,*&'1',*: .5> -'9'1): &) 10.*: 9'::',* $" Γ,

)99) CC

ÉC

E L O

%9,*:10.1',*C u ∈ H01 (Ω) ;"
A . u1 = u2 Γ σ = k∇u σi = σ|Ωi = ki ∇ui ? Ωi A σ A . A ? L2 (Ω) 3A . 5 &A σ · n Γ . σ1 · n = σ2 · n Γ + A 8 . ;"< 6 , [ . u ;"< @ ui ∈ H 1 (Ωi ) Ωi H 8 v ∈ H01 (Ω) 9 0 ∂u1 ∂u2 ∇u1 ·∇v dx+ ∇u2 ·∇v dx+ + k2 f v dx+ f v dx. k1 v ds = ∂n1 ∂n2 Ω1 Ω2 Γ Ω1 Ω2 ;""< ' n1 = −n2 , , 8 Γ P ? 1, A u ) u1 Ω1 u2

LE O ÉC

Y L O P

H C E T

N

E U IQ

"

-

N H EC

E U IQ

Ω2 A u1 = u2 Γ u ∈ H 1 (Ω) .

* A ;""< , , 8 . 6 ;" <

T Y POL

>)0+'+) CC ) Ω K RN

Ω Ω \ K * ) f ∈ L2 (Ω) Ω 4 K ( $ + $ u ( K $ $: K 0

LE O ÉC

⎧ −∆u = f ⎪ ⎪ ⎪ ⎪ u=C ⎨ ∂u ds = 0 ⎪ ⎪ ⎪ ∂n ⎪ ⎩ ∂K u=0

Ω \ K ∂K ∂K

E U IQ

∂Ω,

N H C % $ " & E $ " * T % % " ! LY& " " " P O E L /CN! O C É 4 C + < $ $ $ (u, C) A(x) = (aij (x))1≤i,j≤N Ω Ω A(x)ξ · ξ =

N

A

α>0

aij (x)ξi ξj ≥ α|ξ|2

ξ ∈ RN ,

i,j=1

" * ' & " β > 0 " " Ω |A(x)ξ| ≤ β|ξ| ξ ∈ RN . /CG! $ −div(A∇·) = −

N ∂ i,j=1

∂xi

aij (x)

/C)!

∂ · , ∂xj

'

E U IQ

Ω, /CI! ∂Ω. % (" & /CI! . ( " %

! " % A(x) & ! A(x) " & + % 0 + A = −div(A∇u) = f u=0

LE O ÉC

Y L O P

H C E T

N

" A +A " A

" s

a

s

E U " IQ N H C T E %

E L O É C/0

Aa = (A−At )/2

t

= (A+A )/2

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−div(A∇u) = −div(As ∇u) + V · ∇u

Vj (x) =

N 1 ∂(aji − aij ) i=1

2

∂xi

(x),

: V

% % f ∈ L2 (Ω) $ .( /CI! 4

" /C)! u ∈ H01 (Ω)

N ∂u ∂u = A(x)∇u · n = aij (x) ni = 0 ∂nA ∂xj

∂Ω,

i,j=1

E U IQ

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% (" " L σ = A∇u " L ' σ · n = 0 ∂Ω A

T Y POL

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/0 f ∈ L (Ω) g ∈ L (∂Ω) ! ! 2

E L O

2

−div(A∇u) + u = f ∂u =g ∂nA

É C! $

Ω, ∂Ω.

1 6 . 6 . 1 1 6 A Ω . 0 RN f ∈ L2 (Ω) 5 " 8

u ∈ H01 (Ω)

−∆u = f Ω ;"< ∂Ω. u=0

0'*+'3) &5 9.>'959

H C E T

N

E U IQ

/ O . . ' ; Q ? 8 N = 1 Y . + < '

Y L O ,P

4%,0?9) CC $0'*+'3) &5 9.>'9596

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u ≥ 0 ) )" " $ Ω

f ≥ 0

) )" " $ Ω'

"

N H EC

E U IQ

)9.0;5) CC 8 6 8 . # J , 6 A ( ;f ≥ 0
LE O ÉC

T Y POL

%9,*:10.1',*C @ 8 . ;""< ;"< . v =

u− = min(u, 0) 0 ? H01 (Ω) . " ; u = u+ + u− < @ − − f u dx = ∇u · ∇u dx = 1u<0 ∇u · ∇u dx = |∇u− |2 dx ≥ 0. ;"< Ω

Ω

Ω

Ω

−

N H EC

E U IQ

7 u ≤ 0 f ≥ 0 Ω * A ;"< A u− ∈ H01 (Ω) u− = 0A ,6?6 u ≥ 0 Ω

T Y POL

v ∈ H01 (Ω)' v+ = max(v, 0) ))"$" H01 (Ω) " ∇v + = 1v>0 ∇v ) )" " $ Ω, . 1v>0 (x) " $"$ - " . v(x) > 0 " ,

)99) CC

ÉC

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)9.0;5) CC " F

; H < 9 6 , A : , 8 ) L MA ? H 1 6 8 6 8 ? H 1 9 (A ? v 6 v > 0 ? 6 v < 0A I 0 , 8 6 ; ) v = 0< , ; 1) "< H v 8 •

E U I Q

? ' " v ∈ H (Ω) G(t) * 1 0

N

R R C " G(0) = 0 G (t) ' R ∇(G(v)) = G (v)∇v H01 (Ω) * " % % v H 1 (Ω) vn % " Ω % v ∇v % M % CCC 1

$

LE O ÉC

Y L O P

|G(vn ) − G(v)| ≤

H C E T

sup |G (t)| |vn − v| , t∈R

G(v) ∈ H01 (Ω) vn ∈ Cc∞ (Ω)

∇vn FNH! /#!

"

E U % % . IQ N H /#C! C E T Y % " % ! 5

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; " & ( P % E ' % ( CG FNH! * L% ; /#C! % % % O C

5 ( " B É ' K % % % " G(vn )

G(v)

2

1≤i≤N

L (Ω)

∂G(vn ) ∂v ∂v ∂v ∂vn − G (v) ≤ G (v ) − G (v) + sup |G (t)| − . n ∂xi ∂xi ∂xi ∂xi ∂xi t∈R ∂v |G (vn ) − G (v)| ∂x i 2 (sup |G (t)|) ∂v L2 (Ω) ∂xi

L2 (Ω) G(vn )

2

L (Ω)

H01 (Ω)

w ∈ H01 (Ω) ∂w ∂v = G (v) ∂x ∂xi i

L2 (Ω) " L2 (Ω) : 4 ( * t → max(t, 0) * Gn (t) " v + & H01 (Ω) 1 G(t) * C 1 (R) " w = G(v)

t ≤ 21 , 0 ≤ G (t) ≤ 1 12 ≤ t ≤ 1, G (t) = 1 1 ≤ t. $ Gn (t) = G(nt)/n n ≥ 1

" Gn (v) ∈ ∂v H01 (Ω) ∂G∂x(v) = Gn (v) ∂x . % " G(t) = 0

n

i

i

T Y POL +

N H EC +

|Gn (v) − v | ≤ sup |Gn (t) − t | ≤ t∈R

E U IQ

1 , n

Gn (v) % % v+ L2 (Ω) $ 1 ≤ i ≤ N

ÉC

E L O

∂Gn (v) ∂v ∂v ∂v − 1 = |G (v) − 1 | ≤ 1 , v>0 v>0 0

100 ∂x∂v L2 (Ω) $

" Gn (v) % % v+ H01 (Ω) " ∇v+ = 1v>0 ∇v n

i

i

>)0+'+) CC " $ v → v+ L2 (Ω)

B( H 1 (Ω) B ∇u = 0 $ u−1 (0)

%(5-.0'1%

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4%,0?9) CC $&) 0%(5-.0'1%6 RN

C m+2 m+2 H (Ω) H m+2 (Ω)

f ∈ H m (Ω)

m≥0

f →u

C>0

uH m+2 (Ω) ≤ Cf H m (Ω) .

Ω u ∈ H01 (Ω)

H m (Ω)

"

N H EC

E U IQ

* 5 " $ 5 " ; 8 H m (Ω)
LE O ÉC

T Y POL

Ω " $ -" /$ RN C m+2 ' f ∈ H m (Ω)' " m > N/2' "$ -"$$ u ∈ H01 (Ω) =,9 " $ "$ " ))"$" C 2 (Ω), $ )" ' Ω " $ -" /$ RN C ∞ ' " f ∈ C ∞ (Ω)' "$ u ∈ H01 (Ω) =,9 " $ C ∞ (Ω),

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)9.0;5) CC 0 9 ∆uA 0 . uA ? 8 A 1,51): . u ? H V E GA . N = 1 [ . . , 8 . •

T Y POL

N H EC

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? : . 0 ; 6 " < 1 A 8 I ? ,# 0 . : A ,# : C m+1 Ω ; Ω 0 C m+2 f ∈ H m (Ω)< •

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E L O

4 D( /N " A " /C

! $ Ω = RN f ∈ L2 (RN ) ' −∆u + u = f RN . /##! 6 0 /##! " ' 4 & 0 f L2 (RN ) % ( " (( u H 1 (RN ) & " u(x) 8 % E 9 & " R |u|2 dx % , * % /##! K % u ∈ H 1 (RN ) " N

(∇u · ∇v + uv) dx = RN

Y L O P RN

H C E T

f v dx ∀ v ∈ H 1 (RN ).

N

E U IQ /#/!

, D( ? CC % " " u ∈ H 1 (RN ) /#/! 0 " & " % ;& * ( " * %

LE O ÉC

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E U " % /##! 4 % I Q N H C ' #$ " E 5 ' !6! LY T O 8 ( P 9 L E " + O ÉC 2

f ∈ L2 (RN ) f ∈ H m (RN )

N

H (R )

u ∈ H 1 (RN ) m≥0 u

H m+2 (RN )

h ∈ RN h = 0

Dh v(x) =

v(x + h) − v(x) |h|

" ' & H 1 (RN ) v ∈ H 1 (RN ) $ % " ∇(Dh v) = Dh (∇v) " v, φ ∈ L2 (RN ) 8* 9

(Dh v)φ dx = RN

v(D−h φ) dx. RN

E U IQ

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|∇(Dh u)|2 + |Dh u|2 dx =

T Y L

RN

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O P E L $ /#)! O C É

N H EC

f D−h (Dh u) dx.

RN

Dh u2H 1 (RN ) ≤ f L2 (RN ) D−h (Dh u)L2 (RN ) .

D−h (Dh u)L2 (RN ) ≤ ∇(Dh u)L2 (RN ) ≤ Dh uH 1 (RN ) .

. Dh uH

1 (RN )

≤ f L2 (RN )

1 ≤ i ≤ N

∂u Dh ∂xi

L2 (RN )

≤ f L2 (RN ) ,

∂u " "

/C " ∂x & H 1 (RN ) & 2 N " u ∈ H (R ) ∂u " 1 " f ∈ H 1 (RN ) ? " ∂x H 1 (RN ) ∂f −∆ui + ui = RN . /#N! ∂xi 1 % " ∂u & H 2 (RN ) & " u ∈ H 3 (RN )

% U% * ∂x ∂φ % /#/! % * ∂x φ ∈ Cc∞ (RN ) i

i

i

LE O ÉC RN

Y L O P i

∇u · ∇

∂φ ∂φ +u ∂xi ∂xi

H C E T

dx =

f RN

∂φ dx. ∂xi

N

E U IQ

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% ' " P E /#N!

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1

u ∈ H (R )

f ∈ H (R )

N

∇

RN

N

∂u ∂u · ∇φ + φ dx = ∂xi ∂xi

RN

∂f φ dx ∂xi

φ ∈ H 1 (RN )

1

∂u ∂xi

N

= ui

H (R )

f ∈ H m (RN ) ⇒ u ∈ H m+2 (RN ) m=1

m

# . v ∈ L (R ) h ∈ R h = 0 & 7 2

N

N

Dh v(x) =

v(x + h) − v(x) ∈ L2 (RN ). |h|

" v ∈ H 1 (RN ) ! Dh vL2 (RN ) ≤ ∇vL2 (RN ) .

N H EC

E /#)! U IQ

8 ! v ∈ L2 (RN ) 9 C h = 0

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2

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RN

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Dh vφ dx ≤ CφL2 (RN ) .

v(D−h φ) dx,

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"

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∂φ v dx ≤ CφL2 (RN ) , ∂xi

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θi (x) = 1

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Ω.

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% ω0 & Ω * ω0 ⊂ Ω! " % ωi i ≥ 1 % ' ∂Ω 1 u ∈ H01 (Ω) " /# ! u = Ii=0 θi u % ( θi u θ0 u u 6

" θ0 u

" H 1 (RN ) −∆(θ0 u) + θ0 u = f0 RN , % f0 = θ0 (f − u) − 2∇θ0 · ∇u − u∆θ0 " & L2 (RN ) /C " θ0 u ∈ H 2 (RN ) 5 f0 % /C " f ∈ H m (Ω) " " θ0 u ∈ H m+2 (Ω) θi u i ≥ 1 * ' 89 ' Ω = RN + 6 * " /C Ω = RN+ 5 ' ( " % & %'

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. 0 < R < +∞ 0 < Θ ≤ 2π ; x1 = r cos θ x2 = r sin θ< @ Γ0 0 Ω R r = RA Γ1 R θ = 0 Γ2 R θ = Θ ' . Ω L MA , ; 0 Γ1 Γ2 < 0 6 *# A , Θ < π 8 , ( A Θ > π ? ; H ? @::50) Θ = 2π < * k ≥ 1A 0 . ⎧ ⎨ −∆u = 0 Ω u = cos kπθ Γ0 ;""< Θ ⎩ ∂u = 0 Γ ∪ Γ 1 2 ∂n

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%9,*:10.1',*C / 0 ? 0 ;""< . / ;, 0 ;""< H < @ L.M 6 ) 8 u0 ∈ H 1 (Ω) 0 [ . @ . ) kπθ r2 u0 (r, θ) = 2 cos R Θ

? A ,6?6 u = u0 + v R v , 0 ⎧ ⎨ −∆v = ∆u0 Ω Γ0 v=0 ;""< ⎩ ∂v Γ1 ∪ Γ2 . ∂n = 0 + . 0 Γ0 0 . Γ1 Γ2 A ,6?6 . θ ; . < , θ = 0A Θ ' ∆u0 ? L2 (Ω)A ;""< H 1 (Ω)A ;""< H 1 (Ω) > ) ;"" < +

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E L O

T Y POL

∆φ(r, θ) =

N H EC

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1 ∂2φ ∂ 2 φ 1 ∂φ + + . ∂r2 r ∂r r2 ∂θ2

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Y L O P

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N

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8 Ω RN , u ; < 8 Ω RN 8 . 8 A e(u)A 8 ? . ,0 # 1 ∂u 1 ∂uj i e(u) = + , ∇u + (∇u)t = 2 2 ∂xj ∂xi 1≤i,j≤N

N H EC

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σ ; 8 ? . ,0 # < ? e(u) =%

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µ > 0 2µ + N λ > 0.

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N

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. fi ui A 1 ≤ i ≤ N A f u 0 RN 9 I 1 A . A 0

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"

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/ . , ,

# ,

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f ∈ L2 (Ω)N ,

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N ∂ui j=1

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N N ∂ui ∂vi ∂uj ∂vi 1 ∂ui ∂uj ∂vj + = + + = 2e(u) · e(v). ∂xj ∂xi ∂xj 2 i,j=1 ∂xj ∂xi ∂xj ∂xi i,j=1

T Y POL

N H EC

' H01 (Ω)N = 0A 0 8 . 6 J . u ∈ H01 (Ω)N 2µe(u) · e(v) dx + λ divu divv dx = f · v dx ∀ v ∈ H01 (Ω)N . ;""&< Ω

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Ω

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. ν = min(2µ, (2µ + N λ)) > 0 * A 0 J A · B = N i,j=1 aij bij # A # A 8 A = Ad + Ah . Ad = A −

1 1 trA Id Ah = trA Id, N N

H C E T

Ad · Ah = 0 |A|2 = |Ad |2 + |Ah |2 @

Y L O P

N

E U IQ

2µ|A|2 + λ( trA)2 = 2µ|Ad |2 + (2µ + N λ)|Ah |2 ≥ ν|A|2

. ν = min(2µ, (2µ+N λ))A A = e(u) 8 ν > 0 , J #

LE O ÉC

" #

N H EC

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µ > 0 (2µ + N λ) > 0 H 1 A

, B ; X A . " 6 < C > 0 |e(v)|2 dx ≥ C |∇v|2 dx

LE O ÉC

T Y POL Ω

Ω

v ∈ H01 (Ω)N 5 A , * ; A . * < C > 0 A v ∈ H01 (Ω)N A |v|2 dx ≤ C Ω

|∇v|2 dx. Ω

3 A ? . 2 2µ|e(v)| dx + λ|divv|2 dx ≥ Cv2H 1 (Ω) . Ω

Ω

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" Ω $ -" RN , " " $"$ v ∈ H01 (Ω)N ' $

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|e(v)| dx =

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R = v(x) = b + M x b ∈ RN , M = −M t : .

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" v ∈ H 1 (Ω)N e(v) = 0 Ω v ∈ R

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6 vn . L2 (Ω)N , B "

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N

vn − vp 2H 1 (Ω) ≤ Cvn − vp 2L2 (Ω) + e(vn ) − e(vp )2L2 (Ω) ,

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v∞ . ) e(v∞ )L2 (Ω) = 0 ' , ,

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"

N H EC

E U IQ

v∞ = 0A . 8 vn H 1 (Ω) = 1 , 8 . ;"$< . , ;"$< 00 ? . 8 5 " . / 2 A , 6 (f, g) → u * , L2 (Ω)N × L2 (∂Ω)N H 1 (Ω)N A v = u 8 . ;"$< 9 6 . 8 0 I 8 A 0

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-!):1'9.1',* &!%*)0(')

;"$$<

Cu2H 1 (Ω) ≤ f L2(Ω) uL2(Ω) + gL2 (∂ΩN ) uL2 (∂Ω) .

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N

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−div(µ∇u) − ∇((µ + λ)divu) = f Ω.

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>)0+'+) CC : $ $ : %* Ω * L > 0 ω ( 4 ω * RN −1 M λ µ G ( Ω = ω × (0, L)( x ∈ Ω( x = (x , xN ) 0 < xN < L x ∈ ω ⎧ −div (2µe(u) + λ tr(e(u)) Id) = 0 ⎪ ⎪ ⎨ σn = g =0 u ⎪ ⎪ ⎩ (σn) · n = 0

Ω ∂ω × (0, L) ω × {0, L} ω × {0, L}

N H EC

E U IQ

;"$<

4 ( v = (v1 , ..., vN )( v = (v , vN ) v ∈ RN −1 vN ∈ R g : @ A( $ + g = (g1 , ..., gN −1 ) = 0 gN x " ! / u = (0, ..., 0, uN ) 4 uN (x )

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ω ∂ω

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. D u 8 0A 6 (u·∇)u u . 0 @ 0 % 8 . ? , . # • .

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H C E T

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Y L O P

µ∇u · ∇v dx −

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Ω

Ω

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N H EC

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. ∇u · ∇v = i=1 ∇ui · ∇vi 3) , 6 0 divu = 0A = 0 6 . H01 (Ω)N V = v ∈ H01 (Ω)N divv = 0 Ω , ;"&<

T Y POL

0 = 0 6 8 H01 (Ω)N @ . 8 . J µ∇u · ∇v dx = f · v dx ∀ v ∈ V, ;"&< . u ∈ V

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Ω

Ω

V @ . ) ;"&< 0 , 5 67 ? 8 . ;"&< 0 9 A . 8 0 . H01 (Ω)N ; Q ? , * < V 6 H01 (Ω)N ;"&< 0

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Ω

E L O

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(µ∆u + f ) = 0 , V L2 (Ω)N , . V 9 (A . 8 φ 8 A A v ∈ V A ∇φ · v dx = − φdivv dx = 0,

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Ω

Ω

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0 8 V * A p ∈ L2 (Ω)A ? A L(v) = p divv dx ∀ v ∈ H01 (Ω)N . H01 (Ω)N

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σ = µ∇u − p Id ? L2 (Ω)N A σ · ∇v dx = f · v dx ≤ CvL2 (Ω) ,

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N

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. σ . 8 0 L (Ω) A −divσ = f * A

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2

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∇p − µ∆u = f Ω.

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divu = 0 Ω.

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L(v) =

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p divv dx ∀ v ∈ H01 (Ω)N .

8 ) p " $ $ $"$" "- )+,

ÉC

)9.0;5) CC * % , ) / , . µ

∂u − pn = g ∂Ω, ∂n

;"&&<

R g 8 ;? . . < L2 (∂Ω)N ' , A / ;"&&< , g 8 0 •

)9.0;5) CC

* # % ;"&< H 6

# , ;. 5 " $< * A % # A ,# ;, H # , < *# A ? , 0 D 0 9 (A , % # ; < A 0 A . D . . , # ; . ,0 • 8 <

LE O ÉC

Y L O P

H C E T

N

E U IQ

"

"

N H EC

E U IQ

' # , , ; &'::'3.1',* 2':;5)5:)< %

T Y POL

>)0+'+) CC ) V $ % + * !& ) J(v) $ * v ∈ V

ÉC

OLE

J(v) =

1 2

µ|∇v|2 dx −

Ω

;"&-<

f · v dx. Ω

) u ∈ V !& " u $ $ *( $ + J(u) = minv∈V J(v) C ( ( u ∈ V $ * J(v)( u !& 1 , . . %

>)0+'+) CC

E U IQ

) = * ( - ) Ω = ω × (0, L) 4 L > 0 * ω ( * RN −1 - x ∈ Ω( x = (x , xN ) 0 < xN < L x ∈ ω

ÉC

T Y POL

⎧ ∇p − µ∆u = 0 ⎪ ⎪ ⎪ ⎪ ⎨ divu = 0 u=0 ⎪ ∂u ⎪ pn − µ ∂n = p0 n ⎪ ⎪ ⎩ ∂u pn − µ ∂n = pL n

E L O

N H EC

Ω Ω ∂ω × (0, L) ω × {0} ω × {L}

;"&<

4 p0 pL " !&/ p(x) = p0 + xLN (pL − p0 )( u = (0, ..., 0, uN ) 4 uN

−p0 ) −µ∆ uN = − (pLL uN = 0

ω ∂ω

4 ∆ x ∈ RN −1

E U IQ

>)0+'+) CC ? $' ! > ) =

!&2

LE O ÉC

Y L O P

H C E T

N

T Y POL

N H EC

E U IQ

LE O ÉC

.

N H EC

# %

,

T Y POL

E U IQ

1 %-%9)*1: @*': 6 8 0 A 0 #0 . -!.330,+4) 2.0'.1',* *)--) . , 0 ) , = 0 V 8 . 6 Vh ) 0 LM Vh ? , # A 9.10'+) &) 0'('&'1% * A Vh ? 6 Vh 0 V uh Vh 8 . F30,+4)G u V = A ) + ' ; < A .A A ,: "6$ ; < 3 A 0 . 8 ) . ', 0 6 R ( I

8 8 ? ; . <

ÉC

E L O

LE O ÉC

Y L O P "

H C E T

N

E U IQ

' !

"

N H EC

E U IQ

. 1 &!.330,>'9.1',* 2.0'.1',**)--) '*1)0*) $ ) , RA .0 A @ ;0 A 8 A < ;. A , < $ ) ;N ≥ 2< @ 9.'--.() ; < &)(0%: &) -'/)01% 8 , ) @ I? 0 N = 1 5 , ( ) , ) A A 0 , , . 6 . ) A , ) 8 ;
LE O ÉC

T Y POL

N H EC

E U IQ

. )

T Y POL

/ ? . 8 . ' = 0 V A 8 0 . a(u, v)A 8 L(v)A 8 . J

ÉC

E L O

. u ∈ V a(u, v) = L(v) ∀ v ∈ V,

;$<

, 5 67

!.330,>'9.1',* '*1)0*) ;$< ? , = 0 V

6 ) Vh A ,6?6 ? J . uh ∈ Vh a(uh , vh ) = L(vh ) ∀ vh ∈ Vh .

;$ <

, ;$ < 8 .

E U IQ

" V $ ) @/" ' " Vh $ ) $$ *$, " a(u, v) $ /$ $"$ " - V ' " L(v) $ $ $"$ V , ))&"$ $"$ %, " $ $ "$, "" "$ ) " /"$ $ -$" $ !"+ $ " *$ )"- " !" a(u, v) " !" ,

)99) C C

Y L O P

H C E T

N

%9,*:10.1',*C , , uh ∈ Vh A ;$
LE O ÉC

5 67 ? Vh * 0

' $

E U IQ

"

h 8 A 0 (φj )1≤j≤Nh Vh uh = N j=1 uj φj A Uh = (u1 , ..., uNh ) . RNh uh 0 ;$ < . ? J ⎞ ⎛ Nh uj φj , φi ⎠ = L(φi ) ∀ 1 ≤ i ≤ Nh , . Uh ∈ RNh a ⎝

ÉC

OLE

T Y POL

N H EC j=1

, 8 , # ;$<

Kh Uh = bh ,

.A 1 ≤ i, j ≤ Nh A (Kh )ij = a(φj , φi ),

(bh )i = L(φi ).

E U IQ

. 8 0 a(u, v) P ) 8 Kh A . 0 9 (A . Uh ∈ RNh A

N H EC

2 Nh 2 Kh Uh · Uh ≥ ν u φ . C > 0, j j ≥ C|Uh | j=1

T Y POL

. ) ;| · | RNh < 1 HA # a(u, v) Kh 1 Kh 9.10'+) &) 0'('&'1%

ÉC

E L O

/ , O , V 6 Vh * A I ( u − uh R u V ;$< uh Vh ;$ < * . J ν > 0 . M > 0 8 0 a(u, v) . ) a(u, u) ≥ νu2 ∀ u ∈ V, |a(u, v)| ≤ M u v ∀ u, v ∈ V

E U IQ

.A G ? N 'A u uh I 5*'=,09%9)*1 3.0 0.33,01 .5 :,5:):3.+) Vh u Vh

H C E T

N

$ ) #!)"#+ %,,, " u "$ %, " uh %,, $

)99) C C $&) %.6

Y L O P

u − uh ≤

LE O ÉC

M inf u − vh . ν vh ∈Vh

;$<

' !

"

N H EC

E U IQ

%9,*:10.1',*C * Vh ⊂ V A A 8 . ;$< ;$
T Y POL

a(u − uh , wh ) = 0 ∀ wh ∈ Vh .

9 wh = uh − vh 0

LE O ÉC

νu − uh 2 ≤ a(u − uh , u − uh ) = a(u − uh , u − vh ) ≤ M u − uh u − vh ,

,R , ;$<

>)0+'+) C C 6 0 2( ( a(u, v) : ( u − uh ≤

M inf u − vh . ν vh ∈Vh

E U IQ

L I uh 2 $ *

N H EC

2 A . . A + Vh 6 h > 0 , ) / A , h → 0 , ;$ < L. M . 8 . ;$<

ÉC

T Y POL

E L O$ ) #!)"#+ %,,, $ ))

)99) C C

&" $ ) V ⊂ V $ $ V " $ ))"$ )) ,3%0.1)50 &!'*1)03,-.1',* " lim v − rh (v) = 0

h→0

rh

V

$

Vh

;$"<

∀ v ∈ V.

"# ))&"$ -"$$ $"$ $-0' " ;$$<

lim u − uh = 0.

h→0

%9,*:10.1',*C > 0 * V A v ∈ V u − v ≤ * A h0 > 0 ; < A v ∈ V A v − rh (v) ≤ ∀ h ≤ h0 . 9 . $ A

Y L O P

H C E T

N

E U IQ

u − uh ≤ Cu − rh (v) ≤ C (u − v + v − rh (v)) ≤ 2C,

OLE

,R ,

ÉC

' $

N H EC

""

E U IQ

$A $ $ 6 ,50 ,/1)*'0 5*) .330,>'9.1',* *59%0';5) &) -. :,-51',* )>.+1) &5 30,/-?9) 2.0'.1',**)- $C 6H '- =.51 '*10,&5'0) 5* ):3.+) Vh &) &'9)*:',*

T Y POL

@*') 35': 0%:,5&0) 5* :'93-) :B:1?9) -'*%.'0) .::,+'% E -!.330,>'9.1',* 2.0'.1',**)--) '*1)0*) $C6 / A Vh , . 8

LE O ÉC

, J

. , rh V Vh 6 8 ;$"< ;R # V 8
) ? 8 L0M Vh 3. , A 4%

5 -

N H EC

E U IQ

4% ) E , , , H A , . ; , 0 6 < 9 , . 6 @ , = 0 V 0 ) A 6 PA * "A , 0 0 (ei )i≥1 V @ V 6 0 0 ; 0 ) < 0 G V 9 h = 1/nA ) Vh 6 ) (e1 , ..., en ) 2 6 A , , rh I Vh ; ) V V < 5 # $A $ $ 8 uh . . u + uh . # Kh Uh = bh R Uh . Rn uh 0 (e1 , ..., en ) 7 . 6 A 4% , . 9 (A Kh , 0 L MA ,6?6 : A L6 MA ,6?6 # 0 0 , 1 . A ) 0 8 80 ? 4%

ÉC

E L O

T Y POL

E U IQ

N H C E 5 ( 6 )7 T Y L O P E L O ÉC

) ,6 Vh 8 H 1 (Ω), H01 (Ω), H 2 (Ω), ...

' !

"$

N H EC

E U IQ

) 0 9.'--.() Ω . , . J A A ;. A A 2 $&< / ) , ) 1 h Vh ? 1.'--) 9.>'9.-) &): 9.'--): 5# 0 Vh 8 -,+.-':% ' J , A h → 0A , Vh L M , V A , A Kh # ;$< +0)5:)A ,6 ?6 : ; G < ) : 0 9 ? 0 , 00

LE O ÉC

.

T Y POL

N H EC

-

T Y POL

N =1

E U IQ

* ) , O 6 ) , Ω =]0, 1[ 9 , 6 (xj )0≤j≤n+1 ; ( ) A . ' < x0 = 0 < x1 < ... < xn < xn+1 = 1.

ÉC

E L O

5*'=,09) xj A ,6?6 xj = jh . h =

1 , 0 ≤ j ≤ n + 1. n+1

xj :,99)1: ; < * A , A 0 .

−u = f ]0, 1[ ;$&< u(0) = u(1) = 0, . , H01 (Ω) f ∈ L2 (Ω) ;. ' "< 1 Pk ,0 #X ? : , . 0 8 ? k

( P1

Y L O P

H C E T

N

E U IQ

) P1 , 8 06 : Vh = v ∈ C([0, 1]) v [xj ,xj+1 ] ∈ P1 0 ≤ j ≤ n , ;$-<

LE O ÉC

' ! N = 1 6

T Y POL

N H EC

"&

E U IQ

V0h = {v ∈ Vh v(0) = v(1) = 0} .

;$<

) P1 , . 6 $ Vh V0h ) ;$-< ;$<

LE O ÉC

vh

φj

x0 =0

x1

x2

E L O

T Y POL

N H EC

xj

xn

E U IQ xn+1=1

$ W 7 Ω =]0, 1[ 8 0 ) P1

ÉC

@ 8 Vh V0h A : A ? , 8 0 L8 M φ )

1 − |x| |x| ≤ 1, φ(x) = 0 |x| > 1. 8A 0 ≤ j ≤ n + 1 ) 8 0 ;. 2 $< x − xj φj (x) = φ ;$< . h

N

E U I Q

)99) CC ) Vh ' *$ ) %,' " $ ) H (0, 1) $$ n + 2' " " " $"$ vh ∈ Vh " *$ $+ $ ) - & " (xj )0≤j≤n+1

Y L O P

n+1

LE O ÉC vh (x) =

j=0

vh (xj )φj (x)

H C E T

∀x ∈ [0, 1].

1

' !

"-

N H EC

E U I Q $$

n' " 8 E' V0h ' *$ ) %,' " $ ) " " $"$ vh ∈ V0h " *$ $+ $ ) - & " H01 (0, 1)

(xj )1≤j≤n

T Y POL vh (x) =

LE O ÉC

n

∀x ∈ [0, 1].

vh (xj )φj (x)

j=1

%9,*:10.1',*C + , . A 8 C 1 ? H 1 (Ω) 1 Vh V0h 0 6 H 1 (0, 1) . φj (xi ) = δij A R δij #0 B% . i = j ;. 2 $< )9.0;5) CC 0 (φj )A ) ;$
E U IQ

, Vh 8 A . A . . @ &!%-%9)*1: @*': &) )09'1) A 8 P1 A , Vh A ) ;$-
ÉC

E L O

T Y POL

N H EC

1 . 0%:,-51',* 30.1';5) 0 1 ;$&< ) P1 8 . ;$ < , . J 1 1 uh (x)vh (x) dx = f (x)vh (x) dx ∀ vh ∈ V0h . ;$< . uh ∈ V0h 0

0

@ uh 0 (φj )1≤j≤n vh = φi n j=1

1

uh (xj ) 0

φj (x)φi (x) dx

9 Uh = (uh (xj ))1≤j≤n A bh =

9.10'+) &) 0'('&'1%

LE O ÉC Kh =

0

1

1 0

=

1

f (x)φi (x) dx. 0

H C E T

f (x)φi (x) dx

Y L O P

1≤i≤n

φj (x)φi (x) dx

1≤i,j≤n

,

N

E U IQ

A

' ! N = 1

N H EC

"

E U IQ

8 . V0h . ? Rn # Kh Uh = bh .

T Y POL

' 8 0 φj L M A , φj φi . . : Kh ⎧ −h−1 j = i − 1 ⎪ ⎪ 1 ⎨ 2h−1 j = i φj (x)φi (x) dx = −1 j = i + 1 −h ⎪ 0 ⎪ ⎩ 0

LE O ÉC

Kh ⎛

2 −1 0 ⎜ −1 2 −1 ⎜ ⎜ Kh = h−1 ⎜ ⎜ ⎝ −1 2 −1 0 −1 2

T Y POL

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠

N H EC

E U IQ

;$ <

* 0 0 bh 8 xi+1 (bh )i = f (x)φi (x) dx 1 ≤ i ≤ n.

ÉC

E L O

xi−1

,. 0 bh H : 0 86 f 9 ? =,095-): &) ;5.&0.150) ; 8 , < ) bh * A 8 L M xi+1 xi+1 + xi 1 ψ(x) dx ≈ ψ , xi+1 − xi xi 2 8 LFM 1 xi+1 − xi

0 1 xi+1 − xi

xi+1

xi+1

ψ(x) dx ≈

xi

ψ(x) dx ≈

xi

1 (ψ(xi+1 ) + ψ(xi )) , 2

H C E T

1 1 2 ψ(xi+1 ) + ψ(xi ) + ψ 6 6 3

Y L O P

N

xi+1 + xi 2

E U IQ

.

8 8 ψ :A #X ; bh

LE O ÉC

' !

$

N H EC

E U IQ

f ∈ Vh < 8 ψ A 8 . , O(h2 )A O(h2 )A O(h3 ) . # Kh Uh = bh G ', 8 + Kh . 0 $

LE O ÉC

T Y POL

)9.0;5) CC Kh ? I?

, ( ) 9 8 A hKh ; < ; 1/c< , . , ) / . ? ,9 $ , , , [ •

0,/-?9) &) )59.**C . ) P1 0 / .

−u + au = f ]0, 1[ u (0) = α, u (1) = β.

T Y POL

N H EC

E U IQ

;$<

+ ;$< H 1 (Ω) f ∈ L2 (Ω)A α, β ∈ RA a ∈ L∞ (Ω) a(x) ≥ a0 > 0 Ω ;. ' "< 8 . ;$ < , . J . uh ∈ Vh

1

ÉC 0

E L O

(uh (x)vh (x)

1

+ a(x)uh (x)vh (x)) dx =

f (x)vh (x) dx − αvh (0) + βvh (1),

0

vh ∈ Vh 9 uh 0 (φj )0≤j≤n+1 A 8 . Vh . ? Rn+2 # Kh Uh = bh ,

. Uh = (uh (xj ))0≤j≤n+1 A . Kh = 0

1

φj (x)φi (x) + a(x)φj (x)φi (x) dx

, 0≤i,j≤n+1

1 (bh )i = 0 f (x)φi (x) dx 1 ≤ i ≤ n,

1 (bh )0 = 0 f (x)φ0 (x) dx − α,

1 (bh )n+1 = 0 f (x)φn+1 (x) dx + β.

Y L O P

H C E T

N

E U IQ

a(x) , 8 A , 8 . : Kh ; ,. 8 , 0 bh <

LE O ÉC

' ! N = 1

N H EC

E U IQ

$

>)0+'+) CC G % P1

−u = f ]0, 1[ u(0) = α, u(1) = β,

T Y POL

6% %* :

LE O ÉC

>)0+'+) CC > a(x) = 0 Ω " : % P1 * " $ :

1

f (x) dx = α − β,

0

E U IQ

$ 0 <% !2.

N H EC

>)0+'+) CC G % E 0% 2 6% & $ % $ ( : + B Kh + M bh E "B >

ÉC

E L O

T Y POL

>)0+'+) CC (n + 2) *

xj = j/(n + 1) 0 ≤ j ≤ n + 1 B k > 0 + % * fj 6 $%: % * $ * : $

L %% $ : $

' $ #

* . ) P1 6 , . 6 $ / ) ,0 ,3%0.1)50 &!'*1)03,-.1',* rh ; $<

H C E T

N

$ )) )" $")"$ P1 ))"$ $ rh $ Vh *$' ) " " v ∈ H 1 (0, 1)' )

%@*'1',* CC H 1 (0, 1)

E U IQ

Y L O P

(rh v)(x) =

LE O ÉC

n+1 j=0

v(xj )φj (x).

' !

$

N H EC

E U IQ

' ) 0 A . A 8 H 1 (0, 1) . 0 ) , 6 rh v , 8 v 8 : [ . v xj ;. 2 $ < + , , , ) 8 H 1 (0, 1)A 8 H 1 (0, 1) ; <

T Y POL

LE O ÉC

rhv

v x0 =0

xn+1=1

x2

$

N H EC

E U IQ

T Y POL

W P1 , 8 H 1 (0, 1)

. ) P1 .

E L ' -* ÉCO

)99) CC $&!'*1)03,-.1',*6 v ∈ H (0, 1) 1

" rh )" $")"$ P1 , " "

lim v − rh vH 1 (0,1) = 0.

h→0

8 ) ' v ∈ H 2 (0, 1)' &" $ $"$" C $ )$ $" h " v − rh vH 1 (0,1) ≤ Chv L2 (0,1) .

/ 6 0 . ) P1 0 1

" u ∈ H01 (0, 1) " uh ∈ V0h "$ %,F " %,' )"-$", ' "# $" *$ P1 $-0' "

4%,0?9) CC

lim u − uh H 1 (0,1) = 0.

h→0

C E T Y L PO

HN

E U IQ ;$<

8 ) ' u ∈ H 2 (0, 1) " - f ∈ L2 (0, 1)' &" $ $"$" C $ )$ $" h "

LE O ÉC

u − uh H 1 (0,1) ≤ Chu L2 (0,1) = Chf L2(0,1) .

;$"<

' ! N = 1

N H EC

E U IQ

$

)9.0;5) CC ;$< 5 $ $ .

0 f ? H −1 (0, 1) , ;$"< . . ) P1 ' I6 ? hA ) P1 . + 5 $ $ . .0 Kh 0 bh . ' A 6 ) P1 . , 8 Kh bh ;. T"U< •

LE O ÉC

T Y POL

)9.0;5) CC / 5 $ $ , 6

8A ,6?6 xj [0, 1] ; xj+1 − xj = h< ? 6 8 ; 1) $
E U IQ

)9.0;5) CC @ 8 . ,

N H EC

, ) . , ( ) + A , 5 A . , ( ) 0 . ;86 < ) X ) I , $ "A X 0 . 8 0 ;0< •

ÉC

E L O

T Y POL

%9,*:10.1',*C $ " , .

$ P ;$< * 0 ;$"
Q $ "

/ $ " 8

E U Q" " &" $ $"$" $ )$ $" " ' I ) N ' H C E T Y L O P LE O ÉC

)99) CC

v ∈ H 2 (0, 1)

"

C

h

v − rh vL2 (0,1) ≤ Ch2 v L2 (0,1) ,

;$$<

v − (rh v) L2 (0,1) ≤ Chv L2 (0,1) .

;$&<

' !

$

N H EC

E U IQ

%9,*:10.1',*C v ∈ C ([0, 1]) * ) A , rh v 8 : A x ∈]xj , xj+1 [A v(xj+1 ) − v(xj ) v(x) − rh v(x) = v(x) − v(xj ) + (x − xj ) xj+1 − xj x xj+1 x − xj v (t) dt − v (t) dt = x − x j+1 j xj xj ;$-< = (x − xj )v (xj + θx ) − (x − xj )v (xj + θj ) ∞

LE O ÉC

T Y POL (x − xj )

=

xj +θx

v (t) dt,

xj +θj

8 ) . 0 ≤ θx ≤ x − xj 0 ≤ θj ≤ h @ , ' #6 CF 2 xj+1

2

|v(x) − rh v(x)| ≤ h2

|v (t)| dt

xj

xj+1

≤ h3

E U IQ

|v (t)|2 dt.

N H EC xj

;$<

9 ;$< ? x , . [xj , xj+1 ]A 0 xj+1 xj+1 |v(x) − rh v(x)|2 dx ≤ h4 |v (t)|2 dt, xj

E L O

T Y POL

xj

A j A ;$$< * . v ∈ H 2 (0, 1) ;$&< ? 8 J v ∈ C ∞ ([0, 1]) x ∈]xj , xj+1 [ 1 xj+1 v(xj+1 ) − v(xj ) v (x) − v (t) dt = v (x) − (rh v) (x) = v (x) − h h xj 1 xj+1 x = v (y) dy. h xj t

ÉC

. A ' #6 CF 8 j 0 ;$&
)99) CC v ∈ H 1 (0, 1)'

"

&" $ $"$"

C

$ )$ $"

rh vH 1 (0,1) ≤ CvH 1 (0,1) ,

Y L O P

lim v − (rh v) L2 (0,1) = 0.

LE O ÉC h→0

" ' ) " "

H C E T

v − rh vL2 (0,1) ≤ Chv L2 (0,1) .

8 ) ' ) " " v ∈ H 1 (0, 1)' $

h

N

E U IQ ;$ < ;$ < ;$

<

' ! N = 1

N H EC

$"

E U IQ

%9,*:10.1',*C . ;$ < ;$ < H v ∈ H 1 (0, 1) 5 ,0

T Y POL

rh vL2 (0,1) ≤ max |rh v(x)| ≤ max |v(x)| ≤ CvH 1 (0,1) , x∈[0,1]

LE O ÉC

x∈[0,1]

. 1, A rh v :A Q ? ;-< : v 0 . v A

xj+1

|(rh v) (x)| dx 2

=

xj

=

(v(xj+1 ) − v(xj ))2 h 2 xj+1 xj+1 1 2 v (x) dx ≤ |v (x)| dx, h xj xj

' #6 CFA A j A ? ;$ < * 0 ;$ < ;$-< ,R , |v(x) − rh v(x)| ≤ 2

N H EC

xj+1

xj

T Y POL

|v (t)| dt.

E U IQ

9 . A ' #6 CFA ? xA j A 0 0 ;$ < * ? ;$ < > 0 ' C ∞ ([0, 1]) H 1 (0, 1)A v ∈ H 1 (0, 1) φ ∈ C ∞ ([0, 1])

ÉC

E L O

v − φ L2 (0,1) ≤ .

@ rh . ) ;$
φ )A ;$&< ? φ A h :6 A φ − (rh φ) L2 (0,1) ≤ . * A 0

N

E U IQ

v − (rh v) L2 (0,1) ≤ v − φ L2 + φ − (rh φ) L2 + (rh v) − (rh φ) L2 ≤ C,

;$ <

Y L O P

H C E T

>)0+'+) CC 6 $ <% 2 *

LE O ÉC

>

' !

$$

( P2

N H EC

E U IQ

) P2 , Vh = v ∈ C([0, 1]) v [xj ,xj+1 ] ∈ P2 0 ≤ j ≤ n ,

LE O ÉC

T Y POL

;$ <

6

V0h = {v ∈ Vh v(0) = v(1) = 0} .

;$ <

) P2 , . 6 6 $ ? Vh V0h ' 6 8 A 0 , ? , 8 0 ψj+1/2

1 ψj

0

ÉC

E L O

T Y POL xj-1

xj-1/2 xj

N H EC

E U IQ

xj+1/2 xj+1

$ W 8 0 ) P2

,0 [xj , xj+1 ] ) xj+1/2 = xj + h/2 0 ≤ j ≤ n @ ) 8 LM ⎧ ⎨ (1 + x)(1 + 2x) (1 − x)(1 − 2x) φ(x) = ⎩ 0

ψ(x) =

1 − 4x2 0

− 1 ≤ x ≤ 0, 0 ≤ x ≤ 1, |x| > 1, |x| ≤ 1/2, |x| > 1/2.

H C E T

N

E U IQ

8A 0 ≤ j ≤ n + 1 ) 8 0 ;. 2 $< ψj (x) = φ

x − xj h

Y L O P

, 0 ≤ j ≤ n + 1, ψj+1/2 (x) = ψ

LE O ÉC

x − xj+1/2 h

, 0 ≤ j ≤ n.

' ! N = 1

$&

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n

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∀x ∈ [0, 1].

j=0

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H (0, 1) 0 . 8 ψj (xi ) = δij A ψj+1/2 (xi+1/2 ) = δij A ψj (xi+1/2 ) = 0A ψj+1/2 (xi ) = 0 ;. 2 $< 1

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;$ "<

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* # , (x1/2 , x1 , x3/2 , x2 , ..., xn+1/2 ) 8 (xk/2 )1≤k≤2n+1 A 0 (ψ1/2 , ψ1 , ψ3/2 , ψ2 , ..., ψn+1/2 ) V0h 8 (ψk/2 )1≤k≤2n+1 1 0 Uh ∈ R2n+1 . uh . ) uh (x) =

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1

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lim u − uh H 1 (0,1) = 0.

h→0

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−Uhj−1 + 2Uhj − Uhj+1 = hbjh 1 ≤ j ≤ n.

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;$ -<

Uhj0 = minj Uhj Uh J , # A j0 j0 = 0A Uhj ≥ Uh0 = 0 j A j0 ≥ 1A Uhj0 < Uh0 = 0A

1 Uhn+1 = 0 j0 ≤ n ' bjh = 0 f ψj dx ≥ 0 # f A ;$ -< j0 Uhj0 − Uhj0 −1 + Uhj0 − Uhj0 +1 ≥ 0,

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. ; < Uhj0 * ) P1 . ) / . 6 . / . 2%0'@)0 *59%0';5)9)*1 -): 2'1)::): &) +,*2)0()*+) . 0 1 ;$&< ) . ' , .

− ((1 + x)u ) + (1 + cos(πx)) u = f 0 < x < 1 ;$ < u(0) = u(1) = 0

H C E T

N

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. f (x) = −π cos(πx) + sin(πx)(1 + cos(πx) + π 2 (1 + x))A u(x) = sin(πx) , , u − uh H 1 (0,1) A 8 , A , u 9 ; , 8 < , I Vh A ,6?6 , rh (u − uh )H 1 (0,1) ;

LE O ÉC

Y L O P

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, , *,09) &':+0?1) Vh < , H , rh (u − uh ) = rh u − uh ∈ Vh ; . ? , H 1 (0, 1) Vh < @ rh (u − uh )H 1 (0,1) 8 h A 5 $ $ . . ; h< , ) P1 A 5 $ . . ; h2 < , ) P2 1 , ;$ < ( . h 2 $ ; < ? A 8 8 h2 h3 . + , , 6 0 . . .

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8 0 8 ) ;$< Vh 86 , ? C 1 ([0, 1])A ? C([0, 1]) ', 8 ( ) = A . 1 φj

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8A 0 ≤ j ≤ n + 1 ) 8 0 ;. 2 $$< x − xj x − xj φj (x) = φ 0 ≤ j ≤ n + 1, ψj (x) = ψ 0 ≤ j ≤ n + 1. h h

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Vh

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2.-)50: )1 +)--): &) :. &%0'2%) [0, 1]

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2

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n

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1 . 0 . , ;$< ) ,= P3 8 . , 6 1 1 . uh ∈ V0h uh (x)vh (x) dx = f (x)vh (x) dx ∀ vh ∈ V0h . ;$"<

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@ uh 0 (φj , ψj )1≤j≤n Uh = (uh (xj ), uh (xj ))1≤j≤n . 0 8 . ;$"< . ? R2n #

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5 ) , Ω N = 2 N = 3 @ 8 N 6 @ N 6 K RN ,. . (N +1) (aj )1≤j≤N +1 RN A K E G 26 36 ;. 2 $< @ N 6 K (aj )1≤j≤N +1 , ? H # RN ; LM< (ai,j )1≤i≤N . aj A

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K ⎛ a1,1 a1,2 ⎜ a2,1 a2,2 ⎜ ⎜ A = ⎜ ⎜ ⎝ aN,1 aN,2 1 1

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" Ω $ -" $$& )! RN , :$ 9.'--.() "$ 0 $ "$0 "$ Ω " $ $/ Th N )& $$ 0$ (Ki )1≤i≤n -*$" , Ki ⊂ Ω " Ω = ∪ni=1 Ki' , $""$ Ki ∩ Kj & N )& "$" " $ m)&' - 0 ≤ m ≤ N − 1' $" " " $" " Ki " Kj , $ $$ N = 2' $""$ & "$0 " " - ' " " $ " $' " $ E" $ )*1'?0) G $ $$ N = 3' $""$ & ""+ " " - ' " $ " $' " $ E" $ $"+' " $ $ $"+, :,99)1: *,)5&: 0 Th $" " N )& Ki )$", $-$"$' )+" h 0$ & +" N )& Ki ,

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1 N 6 K , 0#6 + A K N 6 (aj )1≤j≤N +1 A +,,0&,**%): /.0B+)*10';5): (λj )1≤j≤N +1 x ∈ RN ) N +1 j=1

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1 ≤ i ≤ N,

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* k = 1 , ,0 K A k = 2 H ;. 2 $< 1

A Σk 0 ) (σj )1≤j≤nk / ) ,0 Pk #X ? : RN R 8 ? k A ,6?6 p ∈ Pk , 8 p(x) = αi1 ,...,iN xi11 · · · xiNN . x = (x1 , ..., xN ). i1 ,...,iN ≥0 i1 +...+iN ≤k

E U Q I ' " N " " $ )&, $ $" ' *$ ) %,3' $" )$" $" $" , 'H " " )!$H C " "$ $+ $ ) - & )$" , "$" E "' &" $ / " Y T OL P LE O ÉC

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K

k≥1 (σj )1≤j≤nk

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1 ≤ i, j ≤ nk .

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card(Σk ) = dim(Pk ) =

(N + k)! N ! k!

; . ) , A k = 1, 2< ' , A ? #X Pk A 8 . Σk A : , I . , 0 I . #X p ∈ Pk , Σk 7A N A p RN * N = 1A , #X k , (k + 1) . ? , N −1 ' x 0# (λj (x))1≤j≤N +1 A ) #X q(λ) = p(x) k . 0 λ ∈ RN +1 , ) λj ,0 {0, 1/k, ..., (k − 1)/k, 1} , λ = (λ , λj )A 0 #X qj (λ ) = q(λ) N − 1 +1 . 0 ; N j=1 λj = 1< Σk ? . ) λj ' , k , (N − 1)6 ,# λj )A ,# qj = 0 3 A 8 λj (λj − 1/k) · · · (λj − (k − 1)/k)(λj − 1) . q A . 8 q(λ) 8 ? k A 8 q = 0A

LE O ÉC

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" K " K & N )& !$" $ $ Γ = ∂K ∩ ∂K , " $ $" k ≥ 1, ' " k ' Σk " Σk D$ $" "" Γ, 8 ) ' "$" $$ pK " pK & )!$H Pk ' $"$ v *$ )

pK (x) x ∈ K v(x) = pK (x) x ∈ K " $"$ K ∪ K ' " $" pK " pK $" - D$ $" & )$" " $ Γ,

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%9,*:10.1',*C ? 8 K ,

Σk , k ,# 8A 8 * A Σk Σk [ 8 Γ #X pK pK [ Σk ∩ ΓA $ ΓA . v

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1 , 2

λl (ajj ) = 0 l = j, j .

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Ki ∈ Th .

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dl

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k = 1 0 [ .

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Vh (φi )1≤i≤ndl

1 ≤ i, j ≤ ndl ,

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dimVh =

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Ω

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@ uh 0 (φj )1≤j≤ndl vh = φi ndl uh (ˆ aj ) ∇φj · ∇φi dx = f φi dx. Ω

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$ W 2 0 P1 N = 2 * : Kh A 8 , . @ (λi (x))1≤i≤N +1 0# x , N 6 K * α1 , ..., αN +1 ∈ NA α1 ! · · · αN +1 ! N ! . λ1 (x)α1 · · · λN +1 (x)αN +1 dx = Volume(K) ;$< (α1 + ... + αN +1 + N )! K

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* 0 bh ; H . Kh
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Kh .::)9/-.() &) -. 9.10'+) . 8 H F A G 8 0 ' ,

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K

3

Aire(K) ⎝ 3 ψ(ai ) + 8 60 i=1

⎞ ψ(aij ) + 27ψ(a0 )⎠

1≤i<j≤3

ψ ∈ P3

>)0+'+) CC ) (bi )1≤i≤I $ N K (ωi )1≤i≤I ) ψ(x) dx ≈ Volume(K) K

I

ωi ψ(bi )

i=1

ψ ∈ Pk " ( * ψ (

1 Volume(K)

4 h K

ψ(x) dx = K

LE O ÉC

I

Y L O P i=1

H C E T

ωi ψ(bi ) + O(hk+1 ),

N

E U IQ

' ! N ≥ 2 1

T Y L 8

O P LE O ÉC

4

N H EC 5

9

2

E U IQ

-

6

7

3

$ W 9

>)0+'+) CC Ω =] − 1, +1[2 7*

0 * Kh P1 > : *

>)0+'+) CC G % P1 6%

Ω =]0, 1[2 * * 7* 2 " * Kh B $ % E + h2 ( bh E

ÉC

E L O $

T Y POL

N H EC

E U IQ

W 7 8 ,

>)0+'+) CC $' ; n * B B % @* *A * * " * Kh P1 $ n2 * $ 2n n * " B % B : * Ω =]0, 1[3 + $ n3 * $ 2n2 4 n * $ B Ω

H C E T

N

E U IQ

)9.0;5) CC ' , ,9 $

Y L O P

0 ; 8O . 8 0< D Kh A ,6?6 , ' ;. A

LE O ÉC

' !

-

N H EC

E U IQ

A
LE O ÉC

T Y POL

)9.0;5) CC * ) ,# ; 0 .< 8 : N 6 K Th ? N 6 L8M K0 * . 0A ? K0 9 A . ! N K0 = x ∈ RN xi ≤ 1, xi ≥ 0 1 ≤ i ≤ N , ;$$< i=1

. ) N 6 K , 8 : K0 9 (A 0# A ) ;$-< H K K0 A λ = (λj )1≤j≤N +1 A x˜ = (x, 1) KA x ˜0 = (x0 , 1) K0 A Aλ = x ˜A A0 λ = x ˜0 A R A A0 ) ;$&< . 0 @ x˜ = AA−1 ˜0 A 0 x ,6?6 , B A . 0 , N A . b ∈ RN x = Bx0 + b •

ÉC

E L O

T Y POL

N H EC

E U IQ

$ W 7 ) 2 $&

E U IQ

, . ) P1 . )

H C E T

N

>)0+'+) CC $ B = (bij )1≤i,j≤n "

( i(

n

LE O ÉC bii > 0,

k=1

Y L O P

bik > 0,

bij ≤ 0 ∀ j = i.

' ! N ≥ 2

N H EC

-"

E U IQ

" " M

T Y POL

>)0+'+) CC N = 2 ) uh % 6% % P1 * * Ki ∈ Th * + π/2 " uh (x) ≥ 0 Ω f (x) ≥ 0 Ω L I ( > 0( Kh + Id " ( 4 Kh *

LE O ÉC

ÉC

E L O

T Y POL

$ W 5 f

N H EC

E U IQ

, ;$$<

,# . : ? ) Pk ? , 0 ;$$<

>)0+'+) CC G % Pk : $ !! " * Kh $ N ndl 4 N $ ndl *

E U IQ

>)0+'+) CC ' * Kh % Pk >

−∆u + au = f ∂u ∂n = g

LE O ÉC

H C E T

N

Ω ∂Ω,

Y L O P

f ∈ L (Ω)( g ∈ L (∂Ω)( a ∈ L (Ω) a(x) ≥ a0 > 0 Ω 2

2

∞

;$&<

' !

-$

N H EC

E U IQ

>)0+'+) CC " * Kh

% Pk E $' !22 :

T Y POL

>)0+'+) CC $ !&; % $ : O N = 2 - * * Th $

LE O ÉC

Vh = v ∈ C 1 (Ω) v |Ki ∈ P5 Ki ∈ Th .

" :9 p ∈ P5 * K 2 v(aj ), ∇v(aj ), ∇∇v(aj ),

∂p(bj ) ∂n

;$-<

j = 1, 2, 3,

E U IQ

4 (a1 , a2 , a3 ) K ( (b1 , b2 , b3 ) K ( ∂p(bj )/∂n * bj " Vh H 2 (Ω) v . % $ B * ' % $ $G*: !&;

T Y POL

N H EC

/ 6 , 0

) P1 0 1 ;$$< 0 2 $ @ 0 ( , , ; . L#M 0 < 0 1 J L M 2 $&A L)M 2 $ 2 $"A . @ . uh . )

; H< ', 8 8 h , F uh . . , I ( ( . ; . FA . ;"<
ÉC

E L O

H C E ' $ # T Y L O P LE O ÉC

N

E U IQ

/ . , ) Pk 0 1 ;$$< 8 , , , 0 A

' ! N ≥ 2

LE O ÉC

T Y POL

N H EC

E U IQ

-&

#

ÉC

E L O

T Y POL

N H EC

E U IQ

$" W

H C E T

N

E U IQ

uh , ( ;$$<

2 $& ; < ) 2 $ ;0<

LE O ÉC

Y L O P

' !

--

LE O ÉC

T Y POL

N H EC

E U IQ

$

$$ W

N H EC

E U IQ

uh , . 6 ( ;"<<

T Y POL

. , 0A / ;$&< / . 0 ,# * N 6 K J &'.9?10) diam(K) 0,*&)50 ρ(K)A ) 0 K A

ÉC

E L O

diam(K) = max x − y, x,y∈K

ρ(K) = max (2r). Br ⊂K

E GA I diam(K)/ρ(K) > 1 ' , K L M J ? K 9 A A 8 . , N 6 K

%@*'1',* CC " (Th )h>0 $ " 0 Ω, $ " 0" $ " 9.'--.(): 0%(5-')0:

, " h = maxK ∈T diam(Ki ) "$ - ' , &" $ $"$" C " ' ) " " h > 0 " " " K ∈ Th ' i

h

diam(K) ≤ C. ρ(K)

Y L O P

H C E T

N

E U IQ ;$<

)9.0;5) CC 9 N = 2 ;$< . ? . K J θ0 > 0 ; 8 h< K ∈ Th 8

LE O ÉC

' ! N ≥ 2

T Y POL

N H EC

-

E U IQ

diam(K)

LE O ÉC

ρ(Κ)

$& W 1 diam(K) ρ(K) , K ;$< ,# . . . •

E U IQ

/ . 6 : . ) Pk . .

N H C 0 , "

$ " T E 0 " ' "$ )/+ 8#" " ' $ ) Y %,3%' L )&"$ $"$ %,9 )O "# $" *$ , "# P " $" *$ $-0' E L O C É 8 ) ' " ' $ ""$ 4%,0?9) CC H01 (Ω)

(Th )h>0

Ω

uh ∈ V0h Pk

Pk

;$"<

lim u − uh H 1 (Ω) = 0.

h→0

u ∈ H k+1 (Ω)

u ∈

k + 1 > N/2

;$"<

u − uh H 1 (Ω) ≤ Chk uH k+1 (Ω) ,

. C " $ $"$" $ )$ $" h " u, )9.0;5) CC 5 $ , 8 ? ,

) # ; A ) 6 $< 9 (A , , 0 8 Vh . 0A I 0 ) # ;. + $$< + A # N = 2 N = 3A k + 1 > N/2 I 8 k ≥ 1 •

Y L O P

H C E T

N

E U IQ

)9.0;5) CC , , ;$"< 5 $ , .

LE O ÉC

u A , I u ,

' !

N H EC

E U IQ

A . ;. 2 $" ,< 1, A . ;$"
LE O ÉC

T Y POL

#

T Y POL

N H EC

E U IQ

$- W 7 uh ; )

? <

E L O

5 $ ) . , ,3% 0.1)50 &!'*1)03,-.1',* rh , * $$

ÉC

+ . (ˆ ai )1≤i≤ndl 8 0 (φi )1≤i≤ndl 0 V0h ) Pk ;. * $&< * 8 v A ) rh v(x) =

ndl

;$" <

v(ˆ ai )φi (x).

i=1

( . , 8 N = 1 A 8 H 1 (Ω) , N ≥ 2A , , rh , ) H 1 (Ω) ; . , 8 H 1 (Ω) , < / A , ,# k + 1 > N/2A rh ):1 /')* &%@*' :50 H k+1 (Ω) 8 H k+1 (Ω) ;H k+1 (Ω) ⊂ C(Ω) , 5 "<

H C E T

N

E U IQ

" (Th )h>0 $ " 0 0 Ω, $ )) k + 1 > N/2, ' ) " " v ∈ H k+1 (Ω) $") rh v " /$ *$' " &" $ $"$" C ' $ )$ $" h " v' " 0,3,:'1',* CC

LE O ÉC

Y L O P

v − rh vH 1 (Ω) ≤ Chk vH k+1 (Ω) .

;$"<

' ! N ≥ 2

N H EC

E U IQ

3 , * $$ . ? . ) Pk

T Y POL

%9,*:10.1',* &5 4%,0?9) CC C @ 0 6 $ * ;$"< $ . V = Cc∞ (Ω) 0 H01 (Ω) ' Cc∞ (Ω) ⊂ H k+1 (Ω)A , ;$"< * $$ . ) ,# ;$"< $ ; 8 , 0 k + 1 > N/2 * $$< * 0 , , ;$"< ' $ u − uh H 1 (Ω) ≤ C inf u − vh H 1 (Ω) ≤ Cu − rh uH 1 (Ω) ,

LE O ÉC

vh ∈V0h

rh u 0 ? H 1 (Ω) * * $$ ? u 0 ;$"<

E U N#! 5 I Q % ' N 0 " H % &

" 4 C E * " 8 '9 ( % T % FC/H! 0 * "

Y L N/! <

O ; %' : P %

" ! E L * " N##!

N#/! % % O C É # D( NCC %' " u 0 h

V0h Kh

bh

Pk

N

K

P2k−2

uh

P1

•

Ωh

Ω

N

E U IQ

$ W 3 # Ωh , . Ω

Y L O P

H C E T

#" 6" '% " " Ω

"

! $

( Ω " Ωh " Th % J NI! $ ( Ωh

LE O ÉC

' !

E U %

! I Q " " " ' ! % N H C E T Y $ L %

0 (

& O P 1 ' % " (( L E 4

! " " O % FC/H! " C ; É 4 h

C

Ω

dist(∂Ω, ∂Ωh ) ≤ Ch2 .

uh

Th

Vh Ωh ⊂ Ω Vh

Pk

V

N =2

u ∈ H 2 (Ω)

P1

u

N/#!

u − uh H 1 (Ωh ) ≤ ChuH 2 (Ω) ,

" P2 u ∈ H k+1(Ω) N//! " ( * P1 " % N/#! " N/! % Pk % k ≥ 2 $ & 8 " 9 K Ω ' & ' 8 '9 ' * N * * & A " NC ! N = 2 % * < " *

* 8 9 ' 5

( * ∂Ω ∂Ωh $ " N/! % F H FC/H! • 4 & NCN " 0 (" 1 K N k Σk $ rK * v K rK v = p ∈ Pk " p(x) = v(x) ∀ x ∈ Σk . N/N! .

NCC " < Pk " % Σk K " N/N! ' rK

=! u − uh H 1 (Ωh ) ≤ Ch3/2 uH k+1 (Ω) .

ÉC

E L O

T Y POL

N H EC

E U IQ

#! ( 1 2 , k + 1 > N/2 1

E U IQ

rK H k+1 (K) H k+1(K) C(K) v ∈ H k+1 (K) v − rK vH (K) ≤ C(K)|v|H N/G! (K) ,

% |v|H (K) ! 1 ! & k+1

k+1

|v|2H k+1 (K)

=

LE O ÉC |α|=k+1

K

Y L O P

H C E T

k+1

|∂ α v|2 dx = v2H k+1 (K) − v2H k (K) .

N

' ! N ≥ 2

E U D( #C/ " I" Q % * '

* N H < " C T E " % - $ " Y . OL N/)! P E

% ;& * M % L O #/! ! 6 " É C H k+1 (K) ⊂ C(K)

k +1 > N/2

H k+1 (K)

rK v v ∈ H k+1 (K)

rK

H m (K)

H

k+1

(K)

m∈N

vH k+1 (K) ≤ C(K) |v|H k+1 (K) + rK vH k+1 (K) ,

vn ∈ H k+1 (K)

N/I! ( N/I! " " vn ' H k+1(K) D( A( #C vn " % H k (K) N/I! " " % ∂ α vn |α| = k + 1 % % E L2 (K) " vn % H k+1 (K) % v " % & N/I!! |v|H (K) = 0, rK vH (K) = 0. NN ! NN ! " v ∈ Pk K #/! N/N! rK rK v = v v ∈ Pk NN ! " rK v = v = 0 " % ( N/I! ' N/G! " N/)! & (v − rK v) " " rK (v − rK v) = 0 " |v − rK v|H (K) = |v|H (K) " % k + 1 < Pk %

NCI @ 'B' " N/G! K $

% 1 = vn H k+1 (K) > n |vn |H k+1 (K) + rK vn H k+1 (K) .

k+1

ÉC

E L O

N H EC k+1

T Y POL

k+1

E U IQ

k+1

# $ k + 1 > N/2 diam(K) ≤ 1 : K v ∈ H (K) k+1

C

v − rK vH 1 (K) ≤ C

(diam(K))k+1 |v|H k+1 (K) . ρ(K)

NN!

$ A " NC " " N K * N * K0 N#N!

%' B % b K ! " x ∈ K x0 ∈ K0 % x = Bx0 + b. NN! ' NN! N/G! ' K0 " (

%' NN! 5 % % & K 4 "

' FC/H! 7 '

( %' det(B) % K ' & %

LE O ÉC

Y L O P

H C E T

N

E U IQ

' !

E U % I Q " * % N H C E T Y OL $ N/G! P LE O ÉC −1

K0

B v(x)

|v0 |H l (K0 ) |v|H l (K)

≤ ≤

C v0 (x0 ) = v(Bx0 + b)

K

CBl | det(B)|−1/2 |v|H l (K) CB −1 l | det(B)|1/2 |v0 |H l (K0 ) .

|v − rK v|H 1 (K) v − rK vL2 (K)

≤ ≤

CBk+1 B −1 |v|H k+1 (K) CBk+1 |v|H k+1 (K) .

% * " B ≤

diam(K) , ρ(K0 )

B −1 ≤

diam(K0 ) . ρ(K)

0 ' ' NN!

' # v ∈ H

E U IQ k+1

rh v N K rK v " v − rh v2H 1 (Ω) =

N H EC

v − rKi v2H 1 (Ki ) .

T Y POL Ki ∈Th

(Ω)

$ " ; NN! & (" Ki % C !

N#I! ; * diam(Ki )/ρ(Ki ) $

ÉC

E L O

v − rh v2H 1 (Ω) ≤ Ch2k

|v|2H k+1 (Ki ) ≤ Ch2k v2H k+1 (Ω)

Ki ∈Th

"

>)0+'+) CC " * *( P1 ( $ $ rh N = 2 3 v − rh vL2 (Ω) ≤ Ch2 vH 2 (Ω) .

(

E U IQ

Ω # ;,6?6 Ω . 6 # 8
LE O ÉC

Y L O P

H C E T

N

' ! N ≥ 2

LE O ÉC $

T Y POL

N H EC

"

E U IQ

W 9 N = 2

%@*'1',* CC " Ω $ -" $$& )! RN , :$ 9.'--.() 0)+ 1.*(5-.'0) Ω " $ $/ Th N "$0 $$ 0$ (Ki )1≤i≤n

E U IQ

-*$" , Ki ⊂ Ω " Ω = ∪ni=1 Ki' , $""$ Ki ∩ Kj & N "$0 "$" " $ m"$0' - 0 ≤ m ≤ N − 1' $" " " $" " Ki " Kj , $ $$ N = 2' $""$ & "$0 " " - ' " $ " $' " $ $ $"+, :,99)1: *,)5&: 0 Th $" " N "$0 Ki )$", $-$"$' )+" h 0$ & +" N "$0 Ki ,

ÉC

E L O

T Y POL

N H EC

/ ) ,0 Qk #X ? : RN R 8 ? k 3.0 0.33,01 E +4.;5) 2.0'./-)A ,6?6 p ∈ Qk , 8 p(x) = αi1 ,...,iN xi11 · · · xiNN . x = (x1 , ..., xN ). 0≤i1 ≤k,...,0≤iN ≤k

+ p H ? k A ( , Qk Pk * k ≥ 1 ) 10)'--': &!,0&0) k N 6 K ,0

1 k−1 xj − lj Σk = x ∈ K ∈ {0, , ..., ;$$< , 1} 1 ≤ j ≤ N . L j − lj k k

H C E T

N

E U IQ

* k = 1 , ,0 K A k = 2 N = 2 A H A 0# ;. 2 $ < Σk , N 6 K 5*':,-2.*1 Qk A ,6?6 , #X Qk

LE O ÉC

Y L O P

' !

$

Σ1

HN

Σ2

O P LE O ÉC

C E T LY

EΣ U IQ

3

$ W 5 , A A ; <

" K $ N "$0, " $ $" k ≥ 1, ' " " )!$H Qk " "$ $+ $ ) - & )$" " k ' Σk ' *$ ) %,%3,

)99) CC

N H EC

E U IQ

%9,*:10.1',*C @ . ) Σk Qk [

T Y POL

N

card(Σk ) = dim(Qk ) = (k + 1) .

' , A ? #X Qk A 8 . Σk A : , 0 0 Qk .

xµ Σk ) xµj − lj µj = . 0 ≤ µj ≤ k, ∀ j ∈ {1, ..., N }. L j − lj k

ÉC

E L O

@ ) #X p ∈ Qk ⎛ ⎞ N k # ⎜ # k(xj − lj ) − i(Lj − lj ) ⎟ p(x) = ⎝ ⎠ (µj − i)(Lj − lj ) i=0 j=1

. x = (x1 , ..., xN ).

i=µj

@ . ) 8 p(xµ ) = 1 p , Σk A

H C E T

N

E U IQ

' . . ? . 8 ; A ? 8 ? $A <

Y L O P

" K " K & N "$0 !$" $ $ Γ = ∂K ∩ ∂K , " $ $" k ≥ 1, ' " k Σk " Σk D$ $" ""

)99) CC

LE O ÉC

' ! N ≥ 2

&

E U & )!$H I Q' $"$ *$ , 8 ) ' "$" $$ " N ) H C E T Y " $"$ O' L" $" " $" - D$ $" & )$" " $ , P LE O ÉC Γ

pK

p

v(x) =

Qk

K

x∈K x ∈ K

pK (x) pK (x)

K ∪ K

v

pK Γ

pK

9 A Q1 Q2 2 $ 8 0 Q1 N = 2 ; # . ) 8 Q1 : P1 <

ÉC

E L O $

T Y POL

N H EC

E U IQ

W 2 0 Q1 N = 2

>)0+'+) CC

) K = [0, 1]2 N = 2 a = (0, 0)( a = (1, 0)( a3 = (1, 1)( a4 = (0, 1) x3 = 1 − x1 ( x4 = 1 − x2 ( i i ?F + ( % ai xi = xi+1 = 0 Q1 1

2

pi (x) = xi+2 xi+3

1 ≤ i ≤ 4,

Q2 Pi (x) = xi+2 (2xi+2 − 1)xi+3 (2xi+3 − 1) 1 ≤ i ≤ 4 Pi (x) = −4xi+2 (xi+2 − 1)xi+3 (2xi+3 − 1) 5 ≤ i ≤ 8 P9 (x) = 16x1 x2 x3 x4 .

H C E T

N

E U IQ

)9.0;5) CC 9 A 8 , ) Q2 ) A :A Q∗2 9 N = 2A , ) Q∗2 ) - 8 0 (pi )1≤i≤8 ,9 $ ; . p9 < @ . ) 0 Q∗2

LE O ÉC

Y L O P

' !

-

N H EC

E U IQ

H ; 0#< 9 N = 3A , ) Q∗2 ) 0 - H 0 ; ,# 0 ? , < •

T Y POL

I"$" $$ $ 0 "$0 Th $ -" Ω' "# $" *$ Qk " *$ ) ) " %@*'1',* CC

LE O ÉC

Vh = v ∈ C(Ω)

" v |K

i

) " " Ki ∈ Th

∈ Qk

;$$<

.

$ )) $ &)(0%: &) -'/)01% $/ )$" (ˆai)1≤i≤n " k # $ N "$0 Ki ∈ Th , dl

' A 1) $ " Q ? . ; , <

E U IQ

) Vh ' *$ ) %,%9' " $ ) H 1 (Ω) $" $$ " $/ 0 /" ndl , 8 ) ' &" $ / Vh (φi )1≤i≤n *$ ) 0,3,:'1',* CC dl

"

T Y POL

ÉC

E L O

v(x) =

N H EC

1 ≤ i, j ≤ ndl ,

φi (ˆ aj ) = δij

ndl

v(ˆ ai )φi (x).

i=1

' ) Qk ) # A H . ) Pk / . ) 5 $ , L M . ; 1) $ , <

4%,0?9) CC " (Th )h>0 $ " 0 "$0 0 Ω, " u ∈ H01 (Ω)' "$ &" )/+ 8#" %,3%' " uh ∈ V0h ' "$ ))# ) "# $" *$ Qk , "# $" *$ Qk $-0' " lim u − uh H 1 (Ω) = 0.

h→0

H C E T

N

8 ) ' u ∈ H k+1 (Ω) " k + 1 > N/2' $ ""$

Y L O " $ $"$" $ )$ $"P " , E L O ÉC

u − uh H 1 (Ω) ≤ Chk uH k+1 (Ω) ,

. C

h

u

E U IQ

' ! N ≥ 2

E U # " $ I Q

* N $

H 'C

! $ % " E * & " ' T ' $ Y L ( ' !$ O ' (" P E " * " ! K ( L % A " NC) & ;! O C

* % É " " " & * ! % & FC/H Qk

N

[0, 1]N 2N

F N

2

Ω F (Qk ) F (Qk ) = Qk

N

Qk

N

Q1

N =2

•

# ! $ Ω N

N ( Pk Qk • % & FC/H

E U IQ

##$ $ Pk Qk N = 3 8 " k9 1 " Ω = ω×]0, L[ % ω % R2 $ ω Ti ]0, L[ [zj , zj+1 ] $ R3

Ti × [zj , zj+1 ] % " Ω $ * ' Pk Qk % & FC/H •

T Y O L ( P E L ÉCO

N H EC

) ? # , . ; # , < 0 ' , # % ;"&< ? , 0 D ; . . < , 0 D 0 I ) 0 . ; ,# , I D . , ? # < + A 0 Ω ⊂ RN A 8 f (x)A , D A % , . ⎧ ⎨ ∇p − µ∆u = f Ω Ω divu = 0 ;$$"< ⎩ u=0 ∂Ω

H C E T

N

E U IQ

R µ > 0 . D 1 6 " . 8 . ;$$"< 5 . u ∈ V µ∇u · ∇v dx = f · v dx ∀ v ∈ V, ;$$$<

ÉC

OLE

Y L O P Ω

Ω

' !

N H EC

E U IQ

R V , = 0 ) V = v ∈ H01 (Ω)N divv = 0 Ω .

T Y POL

;$$&<

' V , 0 divv = 0A : 6 . ;$$$< ,. 8 I , * A : ) ; 6 < 6 Vh V ) , . Q 8 0 ) Pk Qk * A Th = (Ki )1≤i≤n , . # ΩA ) Vh = v ∈ C(Ω)N divv = 0 Ω, v |Ki ∈ PN k Ki ∈ Th ,

LE O ÉC

, Vh L M 0 9 A divv = 0 ) Vh v A : , 0 Vh @ , 8 . ;$$&< ) , ) 9 A 8 . % ? 8 , 0 ) , ?

8 . 9 ;$$"< 8 v ∈ H01 (Ω)N 8 q ∈ L2 (Ω)A 0 J . (u, p) ∈ H01 (Ω)N × L2 (Ω)/R ⎧ ⎪ µ∇u · ∇v dx − pdivv dx = f · v dx ⎪ ⎪ ⎨ Ω Ω Ω ;$$-< ⎪ ⎪ ⎪ ⎩ qdivu dx = 0,

ÉC

E L O

T Y POL

N H EC

E U IQ

Ω

(v, q) ∈ H01 (Ω)N × L2 (Ω)/R H ;$$-< ,# ;$$$< 0 . 0 ;# < . ;$$-< / . ) , .

E U IQ

)99) CC " (u, p) ∈ H01 (Ω)N × L2 (Ω)/R, ) (u, p) " "$ %,% " $" " "$ /' $ A#+ =,3, "$ "J %,%=,

H C E T

N

8 . ;$$-< @

V0h = v ∈ C(Ω)N v |Ki ∈ PN v = 0 ∂Ω , k Ki ∈ Th Qh = q ∈ C(Ω)/R q |Ki ∈ Pk Ki ∈ Th ,

LE O ÉC

Y L O P

' ! N ≥ 2

N H EC

E U IQ

. ) 0 V0h ×Qh 6 ) , . ;$$-< ⎧ ⎪ µ∇u · ∇v dx − p divv dx = f · vh dx ⎪ h h h h ⎪ ⎨ Ω Ω Ω ;$$< ⎪ ⎪ ⎪ ⎩ qh divuh dx = 0, H01 (Ω)N ×L2 (Ω)/R

LE O ÉC

T Y POL Ω

(vh , qh ) ∈ V0h × Qh 9 ;$$< 9 nV V0h nQ Qh A 0 (φj )1≤j≤nV V0h 0 (ψj )1≤j≤nQ Qh . 8 0 ) ;. * $&< @ uh ph 0 uh (x) =

nV

uh (ˆ aj )φj (x),

ph (x) =

j=1

nQ

uh (ˆ aj )ψj (x),

j=1

E U IQ

= ph (ˆ aj ) 1≤j≤n A 0 #

N H EC

9 Uh = (uh (ˆ aj ))1≤j≤nV Ph Q . Uh Ah Bh∗ bh ;$&< , = Bh 0 Ph 0

R Bh∗ I ; < Bh A bh = Ω f · φi dx 1≤i≤nV A

E L O

T Y POL

Ah = µ ∇φi · ∇φj dx

ÉC

Ω

, Bh = − ψi divφj dx

1≤i,j≤nV

Ω

. 1≤i≤nQ , 1≤j≤nV

, , . I #6 ;$&< + Ah # ) . , nV A Bh nQ × nV A A 0 ;$&< # , nV + nQ A , ) . / .

!"+ $ %,F " " > $ "$ (Uh , Ph ) $ -" Uh " $ ' "$ Ph " $ "$ )+ $ $" KerBh∗ ,

)99) CC RnV × RnQ ,

E U IQ

%9,*:10.1',*C ' ( KerBh )⊥ = ImBh∗ A 8 . ;$&<

. ?

H C E T

N

. Uh ∈ KerBh Ah Uh · Wh = bh · Wh Wh ∈ KerBh .

Y L O P

: , 5 67 0 , , Uh KerBh * A ;$&< (Uh , Ph ) RnV × RnQ ' Uh ? KerBh A

LE O ÉC

' !

N H EC

E U IQ

R * A . ) Ph ? , , KerBh∗ nV

T Y POL

? R A , # KerBh∗ , I

. , 8 H L M 5 8 k k ) .

E $! L O É C )$" $" 0

)99) CC

$"$" $ -" 1I Rn $" " "

"$" "' )$ +" ph " $

KerBh∗ 1,

Q

*$ $ $"$" )+,

%9,*:10.1',*C rh ∈ Qh wh ∈ V0h * ) Wh · Bh∗ Rh = Bh Wh · Rh =

rh divwh dx. Ω

@ rh = 1 I ? Qh A divwh dx = wh · n ds = 0 Ω

∂Ω

T Y L

N H EC

E U IQ

wh ∈ V0h A Rh = 1I = (1, ..., 1) ? KerBh∗

PO 0

ÉC

E L O

-1

+1

0

-1

+1

0

-1

+1

0

-1

+1

0

-1

+1

0

$ W 7 0 8 ;6 ) P1 . <

k = 2 " k = 1 $" *$ P2 ) -" " P1 ) )$' $! KerBh∗ " $$ $' $0$ ) -" 1I "$" "' )$ +" ph " $ $ $"$" )+, k = k = 1 $" *$ P1 ) -" " )$' $! KerBh∗ " $ 0$ $$ ""$" ) 0$ $ "$" "' )$ +" ph $" ) $ ' E $ $"$" )+,

)99) CC

Y L O P

H C E T

N

%9,*:10.1',*C rh ∈ Qh wh ∈ V0h * )

LE O ÉC

Wh · Bh∗ Rh = Bh Wh · Rh =

rh divwh dx = −

Ω

∇rh · wh dx. Ω

E U IQ

' ! N ≥ 2

N H EC

E U IQ

k = 2 k = 1A ∇rh Ki A

T Y L

∇rh · wh dx =

O P LE O ÉC Ω

n

∇rh (Ki ) ·

wh dx.

Ki

i=1

@ 8 ,9 $& A wh ∈ P2 A |K| wh dx = wh (aij ) N (N + 1)/2 j K

R (aij ) N (N + 1)/2 H ai aj K 9 . ; ?
N H EC

E U IQ

9 wh . aij A Wh · Bh∗ Rh = 0

T Y POL

;$&<

|Ki |∇rh (Ki ) + |Kj |∇rh (Kj ) = 0.

8 rh , , ? , ' rh ? P1 ΩA ? , 8 * A A 0 ;$&< rh A ,6?6 rh 8 9 A Wh · Bh∗ Rh = 0 Wh Rh = '1IA 1 6 , k = k = 1 @ 8 Ω =]0, 1[2 ;. 2 $ < @ ) 8 p0 ∈ Qh A . k = 1A . −1, 0, +1 Ki ;. 2 $ < 5 I ) A Bh Wh · Rh = rh divwh dx.

ÉC

E L O

Ω

E U IQ

7 wh : Ki A . Ki rh divwh dx = Ω

n

divwh (Ki )

Y L O P i=1

H C E T

rh dx

N

Ki

. F rh = p0 Ki p0 dx = |K3i | (0 + 1 − 1) = 0 * p0 . . KerBh∗ A 1I

LE O ÉC

' !

N H EC

E U IQ

):1 :10'+1)9)*1 3-5: 1 A &?: ;5) -. &'9)*:',* &) (0.*&) ;5) 5*H -. 9%14,&) &): %-%9)*1: @*': +,00):3,*&.*1) ):1 '*51'-' :./-)C 9 (A dim( KerBh∗ ) > 1A # KerBh∗

T Y POL

;$&< . ? J , Ph ( ? KerBh∗ @ '*:1./-) + I , p0 $ , ? , dim(KerBh∗ ) = 1A 8 , Ph . A 0 # Ω 1 A ,# , . Uh ) * , ) D A .# ? TU

LE O ÉC

/ ,. , # ;$&< * A ,FCA , 6 A . ' , ? , 0 , ,# , 9 0 . , ; # . ' < @ % . ? 0 , ;. ,9 "< / . A H A # ;$&< . ? .

E L O

J(Uh ) =

ÉC

min

T Y POL

Vh ∈ KerBh

N H EC

J(Vh ) . J(Vh ) =

E U IQ

1 Ah Vh · Vh − bh · Vh . 2

, ,FC 0 * G A , ) P2 . P1 @ 8 A P1 ^0 . P1 , ) P1 . , V0h . I 0 RN 8 /5--) ) λ1 (x)...λN +1 (x)A R λj (x) 0# x Ki ' 8 0 0 Ki . ? , A 0 0# ' 0 , .

H C E T

N

E U IQ

>)0+'+) CC " % P1 P P1 dim(KerBh∗ ) = 1

Y L O P

0 , , ) ( ) H # , .6 A , .

LE O ÉC

' ! N ≥ 2

N H EC

E U IQ

"

>)0+'+) CC ) = ! N = 1

$ B u = 0 p f ( - Ω = (0, 1)( * xj = jh h = 1/(n + 1) 0 ≤ j ≤ n + 1 % E $ 2

LE O ÉC

T Y POL

⎧ −uj+1 +2uj −uj−1 p −p + j+12h j−1 = f (xj ) 1 ≤ j ≤ n ⎨ µ h2 uj+1 −uj−1 = 0 1 ≤ j ≤ n 2h ⎩ u0 = un+1 = 0.

" : $ * (

(pj ) + $ $ $ $ (1, 0, 1, 0, ..., 1, 0)

)9.0;5) CC , 8 . ;$$-< , 6 0 ? *

−div (A∇u) = f Ω u=0 ∂Ω,

T Y POL

N H EC

E U IQ

σ = A∇uA 8 . ⎧ ⎪ ⎪ − divσv dx = f v dx ⎨ Ω Ω ⎪ ⎪ ⎩ A−1 σ · τ dx + udivτ dx = 0,

ÉC

E L O

Ω

Ω

(v, τ ) ∈ L2 (Ω) × H(div) , ) ( 9 • )

9

1 6 2':5.-':. 1',* 0 ) ) 6

E U IQ

? , ; 0< . 0 ;R , ? . R< F 9 N = 2 , . ^ , A . 2 $" $$ 8 8 . . 2 $" R ; H 0 < , 0 , 8 ( 9 N = 3 8 ; 8 R , <

0

LE O ÉC

Y L O P

H C E T

N

' !

$

LE O ÉC

$

T Y POL

N H EC

E U IQ

N H EC

E U IQ

W ' , . , 0 ( 61

ÉC

E L O

#

T Y POL

H C E T

N

E U IQ

$ " W 8 0 ( 61 ;.

? <

LE O ÉC

Y L O P

' ! N ≥ 2

N H EC

E U IQ

&

9 ,A 0 ( , , ; < ; 0 0< @ ) Q1 0 1 ;$$< . 0 1 8 0 0 3 0 1 L6 M u = 1A 0 / 2 $ . u A 2 $ " 8 u . 0 . ;R , ? . RN < ( *A A # , ;. 6 "< @ D . A # , : ? ? 0 L 8M , # ; N = 2 3< + . u(x)

x , 8 J A 8 x+u(x) / , , 0 . ; 1 < 0 0 ; / < ? ; 8 f . . < Y . 2 $ $ ,. ) 8 , # ,

LE O ÉC

ÉC

T Y POL

E L O

T Y POL

N H EC

E U IQ

$ $ W 1 ? A A ) 8A

; . . 8< ?

LE O ÉC

Y L O P

H C E T

N

E U IQ

' !

-

LE O ÉC

ÉC

T Y POL

E L O

N H EC

T Y POL

LE O ÉC

E U IQ

N H EC

Y L O P

E U IQ

H C E T

N

E U IQ

ÉC

T Y POL

N H EC

E U IQ

E L O

/

)%

,

T Y POL

N H EC

E U IQ

' ? . A ,6?6 ? , . 8 . 0 1, A . , A ; . 0 2.-)50: 30,30): . 1 Ω . 0 RN (λ, u) ∈ R × H01 (Ω)A . u = 0A

−∆u = λu Ω ;&< ∂Ω. u=0

ÉC

E L O

E U IQ

λ 2.-)50 30,30)A 8 u(x) 9,&) 30,30) ,5 =,*+ 1',* 30,30) ,0 . ;&< @

N

8 , ;&< 0 . . , A , nA Au = λu

Y L O P

H C E T

. (λ, u) ∈ R × Rn ,

;& <

: , −∆ L M ) , A ) ;&<

LE O ÉC

( $

N H EC

E U IQ

0 ,. A # 0 #0 A A ,6?6 , ;&"< , ;&&< / A ;&< # A . 0 . 3 . . 6 0 . ;&'9.1',* *59%0';5) . 8 , . 9 A 9.10'+) &) 9.::) M . KA . ;&< . # Ku = λMuA ) , ;&< . ;& <

LE O ÉC

T Y POL

N H EC

E U IQ

T Y O L

" P & E L ÉCO

3. . 0 A , 0 . 0 ,. * 8 . 6 # ( ) 1 A # A ) .A , n @ λk . rk . A 1 ≤ k ≤ nA Ark = λk rk @ # ( ⎧ ⎨ ∂u + Au = 0 t ≥ 0 ∂t ⎩ u(t = 0) = u0 ,

;&<

N

E U IQ

R u(t) 8 C 1 R+ Rn A u0 ∈ Rn 0

;&< 0 A * A 8 u0 = nk=1 u0k rk A n u(t) = u0k e−λk t rk .

ÉC

OLE

Y L O P k=1

H C E T

( $

N H EC

E U IQ

# ( ⎧ 2 ∂ u ⎪ ⎪ + Au = 0 t ≥ 0 ⎪ ⎪ ⎨ ∂t2 u(t = 0) = u0 , ⎪ ⎪ ⎪ ⎪ ⎩ ∂u (t = 0) = u1 , ∂t

LE O ÉC

T Y POL

;&<

+ R u(t) 8 C 2 R Rn A u0 , u1 ∈ RN 9 n 0 8 u0 = k=1 uk rk u1 = nk=1 u1k rk A ;&<

u(t) =

n u1 λk t + √ k sin λk t rk . u0k cos λk k=1

E U IQ

A 0 ,. ;&< ;&< 3 . 6 A ? 8 8 . / A , −∆A , Rn , = 0 L2 (Ω)A L M , , 3) . ;&< 0 L0M 8 0 . A L . 0M , , 6 . 8 9 ,0 A L 0 M ;. < A u , . 8

ÉC

E L O

T Y POL

N H EC

u(x, t) = φ(t)u(x),

,6?6 , . 0 , u , ∂u − ∆u = 0, ;&"< ∂t . ; 8< ∆u(x) φ (t) = = −λ φ(t) u(x)

H C E T

N

E U IQ

R λ ∈ R t x @ φ(t) = e−λt u H 0 .

Y L O P −∆u = λu

LE O ÉC

;&$<

( $

N H EC

E U IQ

1 H A u ,

O P LE O ÉC .

∂2u − ∆u = 0, ∂t2

T Y L

;&&<

∆u(x) φ (t) = = −λ φ(t) u(x)

R λ ∈ R ' 8 6 √ √ A λ > 0 ; ( . 0<

)9.0;5) C C / . . , # −iωt

N H EC

E U IQ

u(x, t) = e u(x), √ R ω = λ 8 u(x) , ? , . # #0 ' H # . , ,. #0 9 # . 0 ; A < ; A 6
ÉC

E L O

T Y POL

>)0+'+) C C ) Ω = RN " u(x) = exp(ik · x) & |k|2 = λ H

* A ? . -!%;5.1',* &) +40N&'*()0 6 ;. ' <

>)0+'+) C C ) * V (x) " ( u(x, t) = e−iωt u(x)

i

∂u + ∆u − V u = 0 RN × R+ ∗, ∂t

u(x) −∆u + V u = ωu RN .

T Y L PO

N H C E

;&-<

E U IQ ;&<

@ . H # 0 ;&$
, 8 # ;&-< . A A

LE O ÉC

(

N H EC

E U IQ

L 0M 6 V A . , 0 ) 00 ; . . # $"<

T Y POL

>)0+'+) C C ) V (x) = Ax · x A : " u(x) = exp(−A1/2 x · x/2) &/ ω = tr(A1/2 ) H

LE O ÉC /

&

1 0 6 = 0 ;. T $U< 0 . . ? ) 0 ) : # 0 0 6 9

-

T Y POL

N H EC

E U IQ

1 V = 0 , x, y

E L O

%@*'1',* CC " A $ ))"$ $ $"$ V $ V , $ )) - )) A $ λ ∈ R " &" $ $" $$ $ x ∈ V -* Ax = λx, :$ " -" x " )) -" )) - )) λ,

ÉC

4%,0?9) CC " A $ ))"$ $ $"$ V $ V , &" $

$ ))"$ $ $"$ A∗ V $ V ' " >$"' " Ax, y = x, A∗ y

∀x, y ∈ V.

%9,*:10.1',*C * y ∈ V )A L ∈ V 8 )

L(x) = Ax, y * 5 - + FA z ∈ V L(x) = z, x @ ) , A∗ V V A ? y z @ . ) 8 A∗ A 0 L(x) = Ax, y = x, A∗ y

H C E T

N

" A $ ))"$ $ $"$ V $ V , $ " " " >$" D$ - $ >$"' " A∗ = A,

%@*'1',* CC A

E U IQ

Y L O P

" A $ ))"$ $ $"$ V $ V , $ " A " *$ )"- Ax, x > 0 ) " " x ∈ V $$ $ ,

%@*'1',* CC

LE O ÉC

( $

N H EC

E U IQ

@ , ) 6 I 0 0 / . , ) 6 I +,93.+1): ) ,

LE O ÉC

T Y POL

:$ $/ K ⊂ V " " )" ' " " " (un )n≥1 $" K ' $ ) " &" $ " un $-0$" $ K , :$ $/ K ⊂ V " " "-$" )" ' " " " (un )n≥1 $" K ' $ ) " &" $ " un $-0$" $ V ,

%@*'1',* CC

0 A V ) A 60 V 8 0 7 A , . ) 9 (A 60 I 8 0 , . .

E U IQ

)99) CC 8$ $ ) @/" V $$ $*$' / $" $" > )",

N H EC

%9,*:10.1',*C ' , ) A

T Y POL

46 ) (en )n≥1 ' 6 0 ? 0 8 * A n = p

E L O

en − ep 2 = en 2 + ep 2 − 2en , ep = 2,

C " " & ) @/" " $ ))"$ $ É $"$ $ , $ " " )" 0 ) / $" . , 6 en , ' #

%@*'1',* CC V

V

W

W

A

A

V " "-$" )" $ W ,

A

1 .A A A 0 xn V A 6 Axn . W W V ) A ' , . W V ) A , .

E U IQ

>)0+'+) CC " $ Id O V $ 5 &2

H C E T

N

>)0+'+) CC ) $ O 2 x = (xi )i≥1

i≥1 |xi | < +∞( x, y = i≥1 xi yi ) (ai )i≥1 ( |ai | ≤ C < +∞ i ≥ 1 $ A Ax = (ai xi )i≥1 A " A limi→+∞ ai = 0 2

LE O ÉC

Y L O P

(

N H EC

"

E U IQ

>)0+'+) CC ) U ( V W O ( A

V W ( B U V " $ AB A B ' $ $ 5 $

T Y POL

*

# E L O

ÉC

6 .

4%,0?9) CC " V $ ) @/" $$ $*$ " A $ ))"$ $ $"$ ' *$ )"-' " >$"' )" V $ V , - )) A $" $ " (λk )k≥1 ""$" )" "$ - ' " &" $ / #/"$$ (uk )k≥1 V -" )) A' - Auk = λk uk ) k ≥ 1.

E U IQ

)9.0;5) CC ' 5 & -A . H 6 A 0 &%+,93,:'1',* :3)+10.-) v ∈ V v=

+∞ k=1

E L O

T Y POL

v, uk uk .

N H EC

v = 2

+∞

|v, uk |2 .

k=1

•

>)0+'+) CC %: % <% &2. " ( v ∈ V ( $ Au = v u ∈ V v

ÉC

+∞ |v, uk |2

k=1

λ2k

< +∞.

, A , 5 & - 0 8 , .

>)0+'+) CC ) V = L2 (0, 1) A $ V V

(Af )(x) = (x2 + 1) f (x) A ( ( 5 " A $ (A − λ Id) $ λ ∈ / [1, 2]

E U IQ

D( G) % '

$ " V * A 1/ ! V V &

Au, u

Au, u , M = sup . m = inf u∈V \{0} u, u

u, u

H C E T

OLE

Y L O P

2 A = max(|m|, |M |) m

ÉC

u∈V \{0}

M

A

N

( $

$

E U I Q

max(|m|, |M |) ≤ A $ % * " | Au, u| ≤ Au .

A ; ' u, v ∈ V

N H EC

2

T Y POL

4 Au, v

LE O ÉC

= A(u + v), (u + v) − A(u − v), (u − v) 2 ≤ M u + v2 − mu − v2 ≤ max(|m|, |M |) u − v2 + 2v + u ≤ 2 max(|m|, |M |) u + v2 .

$ A = supu=v=1 Au, v " Au = supv=1 Au, v $ " A ≤ max(|m|, |M |) :

m ≤ M % K A = M ≥ 0 A = −m % m ≤ 0 5 A = M ≥ 0 A = −m " 2 A −A! 1 (un )n≥1 %

V " M & " lim Aun , un = M un = 1. n→+∞

E U IQ

5

A " Aun % V % v .

N H EC

Aun , un ≤ Aun ≤ A = M,

: " limn→+∞ Aun = M & " v = M J

T Y P O L

Aun − M un 2 = Aun 2 + M 2 − 2M Aun , un ,

' " limn→+∞ Aun − M un = 0 n " " un % % v/M M = 0 M M = 0 % " " " A = 0! A " Aun % % Av/M % v! " Av/M = v & " v % v = M = 0! & % M

ÉC

E L O

" V * A ! V V . δ > 0 ! & ] − δ, +δ[ 1 ! & ? " % V

" % A % λ " |λ| ≥ δ > 0 $ 1 " (uk )k≥1 V % (λk )k≥1 " Auk = λk uk |λk | ≥ δ k ≥ 1. $ Ek % * (u1 , u2 , ..., uk ) 5

Ek−1 Ek % vk ∈ Ek " ( & Ek−1 5

|λk | ≥ δ vk /λk ' V " A " Avk /λk % V 5 j < k Avj vk Avj Avk − = vk + (A − λk Id) − . G ! λ λj λ λj

OLE k

ÉC

Y L O P

H C E T

k

N

E U IQ

(

&

N H EC

E U I Q

$ % * " AEk ⊂ Ek " (A − λk Id)Ek ⊂ Ek−1 & G ! & Ek−1 5

vk ( & Ek−1 G !

OLE

T Y POL

Avk Avj λk − λj ≥ vk = 1,

" % % Avk /λk

%& "

G " ' % É C % "

G " ' A

'' %

A % % % 4 (λk ) % A Vk = Ker(A − λk Id)

G " (" Vk ! $ " " Vk ( & K + vk ∈ Vk vj ∈ Vj % k = j

A ;

Avk , vj = λk vk , vj = vk , Avj = λj vk , vj ,

N H EC

E U IQ

: " vk , vj = 0 " λk = λj 1 W ( V Vk

W =

T Y P O L "

u ∈ V, ∃K ≥ 1

E L O

u=

K

!

uk , uk ∈ Vk .

i=1

$ * ' (' W ' ( (" Vk ( ( ! ? " * W = V " % " (λk ) " V ! $ ( W

ÉC

W ⊥ = {u ∈ V

" u, v = 0 ∀ v ∈ W } .

5

W ' A AW ⊂ W ! % " W ⊥ ' A

Au, v = u, Av = 0 u ∈ W ⊥ v ∈ W $ A & W ⊥ " ;

G W ⊥ = {0} % % u ∈ W ⊥ " % % A 5 ' - % * " W ;& % A " W ∩ W ⊥ = {0} " W ⊥ = {0}

W * " W = {0}⊥ = V

E U IQ

N H C D( G) E %' T % % K % % % %L Y ' O & % !P LE O ÉC A

KerA

•

( $

-

N H ! & $

C E T Y OL P LE O ÉC /

E U IQ

0 " +

/ . . ' , H F , , ? 0 ( 1 = 0 V 8 0 a(·, ·)A :B9%10';5)A .A ,6?6 a(w, v) = a(v, w)A M > 0 ν > 0 |a(w, v)| ≤ M wV vV w, v ∈ V

a(v, v) ≥ νv2V v ∈ V.

* . A . A ? . = 0 H / 8 ,# 8 .

V ⊂ H . I ;&< V H.

T Y POL

N H EC

E U IQ

, L I M . , , I ? v ∈ V Iv = v ∈ H ;. 1) & &< 3 A ,# ;&< 0 V 6 . H H V H A ·, ·H ·, ·V . 8 / 0 . . . ; 0 < J . λ ∈ R u ∈ V \ {0}

ÉC

E L O

a(u, v) = λu, vH

;& <

∀ v ∈ V.

@ λ . 0 . ;& < ; 8 0 a< u .

)9.0;5) CC

,# ;&< H V . I . H V H 7 0 , I V H , , ;. ,9 & < •

E U IQ

1 # , *

. 0 ΩA V = H01 (Ω)A H = L2 (Ω)A 8 0 # )

H C E T

N

∇u · ∇v dx.

Y L O P

a(u, v) =

Ω

' Cc∞ (Ω) ? 8 H01 (Ω) L2 (Ω)A Q 5 + A ,# ;&< . )A . ' "

LE O ÉC

( $ #

N H EC

E U IQ

8 0 a 0 . V * A . 8 ;& < . ?

−∆u = λu Ω u=0 ∂Ω,

LE O ÉC

T Y POL

,6?6 λ u . 8 ;& < .

4%,0?9) CC " V " H & ) @/" $$ $*$, $ )) V ⊂ H - $>"$ )" " V " $ $ H , " a(·, ·) $ /$ !" $"$ " - V , - )) F, $" $ " $" (λk )k≥1 )" "$ - $*$' " &" $ / #/"$$ H (uk )k≥1 -" )) ' " uk ∈ V, " a(uk , v) = λk uk , vH ∀ v ∈ V. √ 8 ) ' (uk / λk )k≥1 " $ / #/"$$ V ) ) " a(·, ·),

N H EC

E U IQ

%9,*:10.1',*C * f ∈ H A . 0 .

. u ∈ V a(u, v) = f, vH 8 v ∈ V.

T Y POL

;&<

8 . ) # 5 67 ;&< u ∈ V @ ) A H V ? f u = Af 3 A , A )

ÉC

E L O

Af ∈ V a(Af, v) = f, vH v ∈ V.

;&<

9 v = Af ;&
, , I I V H * A , A H V @ ) A = IA H H A 0 ' I A A ;. ,9 & < * A 6 I A v = Ag ;&< 0 A f, g ∈ H A

E U IQ

f, AgH = f, AgH = a(Af, Ag) = a(Ag, Af ) = g, Af H = g, Af H ,

H C E T

N

? # aA . A 6 I ) 8 H @ 5 & - ? , A . ) # (µk )k≥1 8 . A 0 0 (uk )k≥1 H 8 . AA .

OLE

Y L O P

Auk = µk uk k ≥ 1.

ÉC

( $

N H EC

E U IQ

+ A A . uk 6 ? H ? V +. 0 . ;& < ,

LE O ÉC

T Y POL

a(u, v) = λu, vH = λa(Au, v) ∀ v ∈ V,

? ) ;&
* A . (λk )k≥1 0 . ;& < 6 . . (µk )k≥1 AA . H @ 1 uk λk = vk = √ . µk λk

E U IQ

* A . uk 8 0 0 H @ . ) uk , uj H a(uk , uj ) a(vk , vj ) = = λk = δkj , λk λj λk λj

T Y POL

N H EC

, (vk )k≥1 V , (uk )k≥1 H ; .
E L O

)9.0;5) CC 8 , AA ) ;&
ÉC

>)0+'+) CC 6 <% &2 4 $ $%: % a(·, ·) $%: % $ η > 0 ν > 0 a(v, v) +

ηv2H

≥

νv2V

v ∈ V.

H C E T

N

E U IQ

6 (λk )k≥1 ( λk + η > 0

Y L O P

/ . 6 0 . ;& ,5 &) ,50.*1':4)0

LE O ÉC

( $ #

N H EC

E U IQ

* +# ) A 8 v ∈ V \{0}A a(v, v) R(v) = . v2H

T Y POL

" V " H & ) @/" $$ $*$, $ )) V ⊂ H - $>"$ )" " V " $ $ H , " a(·, ·) $ /$ !" $"$ " - V , k ≥ 0 $ $" Ek $/ ) -" $$ k V , $ $" (λk )k≥1 " +0,'::.*1) - )) )/+ -"$$ F,, ' ) " " k ≥ 1' k+ - )) " $$ )

LE O ÉC

0,3,:'1',* CC $,50.*1':4)06

λk = min

W ∈Ek

max R(v) = max W ∈Ek−1

v∈W \{0}

min

v∈W ⊥ \{0}

R(v) .

$ )" ' )+ - )) -* λ1 =

N H EC

min R(v),

v∈V \{0}

;&"<

E U IQ

;&$<

" " " )$" $ $ F,% " $ -" )) λ1 ,

T Y POL

%9,*:10.1',*C (uk )k≥1 0 0 H 8 . √

;& < 1, 5 & -A (uk / λk )k≥1 0 0 V @ H V ? ;. + & < ! +∞ +∞ 2 2 H= v= αk uk , vH = αk < +∞ ,

ÉC

E L O

k=1

V =

v=

+∞

!

k=1

αk uk , v2V =

k=1

+∞

λk α2k < +∞ .

k=1

@ A . λk λ1 > 0A 8 0 P V 6 H @ +# +∞

2 k=1 λk αk R(v) = +∞ 2 , k=1 αk

H C E T

N

E U IQ

. 6 6 Wk ∈ Ek (u1 , u2 , ..., uk ) @ k

2 j=1 λj αj k 2 j=1 αj

+∞

2 j=k λj αj ∀ v ∈ Wk R(v) = +∞ 2 j=k αj

LE O ÉC

R(v) =

Y L O P

⊥ ∀ v ∈ Wk−1 ,

( $ ,R , λk =

max

N H EC

R(v) =

T Y POL v∈Wk \{0}

min

⊥ \{0} v∈Wk−1

E U IQ

R(v).

W 6 Ek ' W k Wk−1 ⊥ k − 1A , W ∩ Wk−1 , ? {0} * A

ÉC

OLE

max R(v) ≥

v∈W \{0}

max

⊥ \{0} v∈W ∩Wk−1

R(v) ≥

min

⊥ \{0} v∈W ∩Wk−1

R(v) ≥

min

⊥ \{0} v∈Wk−1

R(v) = λk ,

. ;&"< 1 HA W 6 Ek−1 A W ⊥ ∩ Wk , ? {0}A min

v∈W ⊥ \{0}

R(v) ≤

min

v∈W ⊥ ∩Wk \{0}

R(v) ≤

max

v∈W ⊥ ∩Wk \{0}

R(v) ≤

max

v∈Wk \{0}

R(v) = λk ,

. ;&"< u ;&$< * v ∈ V A 8 f (t) = R(u+tv) , . 0 t ∈ R t = 0 * . , t = 0 9 f (0) = λ1 A

N H EC

E U IQ

a(u, v) − λ1 u, vH f (0) = 2 . u2H

T Y POL

' v V A f (0) = 0 , , 8 . ;&
ÉC

E L O

9

@ 5 & - ? 8 . 6 . 1 A .

" Ω $ -" /$ 0 C 1 RN , &" $ " $" (λk )k≥1 )" "$ - $*$' " &" $ / #/"$$ L2 (Ω) (uk )k≥1 ' " # uk ))"$" H01 (Ω) " -*

4%,0?9) CC

−∆uk = λk uk uk = 0

),), $ Ω ),), ∂Ω.

H C E T

N

E U IQ ;&&<

%9,*:10.1',*C * . 1 A 6

Y L O P

V = H01 (Ω)A H = L2 (Ω)A 8 0 # ) a(u, v) = ∇u · ∇v dx,

ÉC

OLE

Ω

( $ #

N H EC

L (Ω) 0 G u, vH = uv dx. 2

T Y POL

E U IQ

Ω

@ . ) # 5 & - 4Q 5 + A V 0 H ' Cc∞ (Ω) ? 8 H V A V 0 H 9)A . ' " 8 0 a 0 . V * A (λk )k≥1 8 . , ) A 0 0 L2 (Ω) (uk )k≥1 A uk ∈ H01 (Ω) ∇uk · ∇v dx = λk uk v dx ∀ v ∈ H01 (Ω).

LE O ÉC

Ω

Ω

* ; H # 5 " < 0 ;&&< + , Ω . 5

L M ? 1

N H EC

E U IQ

)9.0;5) CC ,# 0 , . Ω 0

T Y POL

8 5 &" , 8 A 5 + ; , I H 1 (Ω) L2 (Ω)< 8 A 5 &" , 9 8 A , ) ; 00< . L M R 8 , ? L2 (Ω) 3 ,9 & Ω = RN •

ÉC

E L O

>)0+'+) CC ' N = 1( Ω =]0, 1[ 0

6% && G $

C &2/( +∞

ak sin(kπx)

k=1

* L2 (0, 1) 2 2 +∞ k=1 k ak < +∞

+∞ k=1

a2k < +∞( H 1 (0, 1)

N

E U IQ

>)0+'+) CC Ω =]0, L1 [×]0, L2[× · · · ×]0, LN [( 4

H C E T

(Li > 0)1≤i≤N 0 6% &&

Y L O P

5 &" , / .

LE O ÉC

( $

N H EC

E U $" $"+ IQ

∂Ω " Ω $ -" /$ 0 R ) $ & )" >$" 0 + ∂ΩN " ∂ΩD - <0 9,, &" $ " $" (λk )k≥1 )" $ "$ - $*$' " &" $ / #/"$$ L2 (Ω) (uk )k≥1 ' " # uk ))"$" H 1 (Ω) " -* ⎧ ⎨ −∆uk = λk uk $ Ω ∂ΩD uk = 0 ⎩ ∂u ∂ΩN . ∂n = 0

,0,--.'0) CC

LE O ÉC

N

T Y POL k

)9.0;5) CC 1 , / A ,6?6 ∂ΩD = ∅A 8 0 , . H 1 (Ω) * ' && 8 ,9 & • >)0+'+) CC + Ω

$' & 0 > ∂Ω

E U IQ

. ' 62 6 . 8 A , .

T Y POL

N H EC

>)0+'+) CC %: % <% &! " C $ * -

- ; λ1 &&

E L O

@ 8 A . 1 / A

ÉC

" Ω $ -" /$ 0 C ∞ , $"$ )) "$ F,F ))"$$$" C ∞ (Ω),

0,3,:'1',* CC

%9,*:10.1',*C uk k6 8 H01 (Ω) ;&&< @ uk 0 .

−∆uk = fk Ω uk = 0 ∂Ω,

E U IQ

. fk = λk uk ' fk ? H 1 (Ω)A 5 " $ uk ? H 3 (Ω) 1 A 0 fk ,

uk *

8 uk ? H m (Ω) m ≥ 1 9 . 5 " 8 H m (Ω) ;. + $
Y L O P

H C E T

N

/ 8 ? .

LE O ÉC

( $ #

"

E U $ )$ I$""$ #!)"#+ Q ")+ N " $$&, - A#+ F,3,=, $ )) -" H )) " ) ,, ) )) )$ $" " $$ " C E)" ) )" " $ , ) -" )) ) " E"T # Y OL P LE O ÉC 4%,0?9) CC $&) L0)'*519.*6

Ω

λ1

Ω

)9.0;5) CC

5 & B 6+ ) L M ;,6?6 , u ? . R< ' 8 , u ? . . ;. ,

# , < ( . , ;. 5 " < • , .0

%9,*:10.1',*C u ∈ H01 (Ω) . ?

. λ1 1, " u+ = max(u, 0) ? H01 (Ω) ∇u+ = 1u>0 ∇u ; H u− = min(u, 0)< * A 8 |u| = u+ − u− ? H01 (Ω) ∇|u| = sign(u)∇u 9 . * & ' 62

|∇v|2 dx Ω

λ1 = min , R(v) ≡ v 2 dx v∈H01 (Ω)\{0} Ω

T Y POL

N H EC

E U IQ

. @ λ1 = R(u) = R(|u|)A |u| . ? λ1 ' u+ u− 0 u |u|A 8 ? λ1 9 8 A u , Ω Q

L8M : A w ∈ C 2 (Ω) . )

ÉC

E L O

−∆w ≥ 0 Ω, w = 0 ∂Ω,

w ≡ 0 ΩA w > 0 Ω @ ? u+ u− ; ? * &< . H A , 6 ? λ1 @ # . 8 u1 u2 A ,6?6 u1 u2 dx = 0, Ω

N H C E

E U IQ

0 , A

T Y L PO

>)0+'+) CC ) Ω * " Ω > $

LE O ÉC

( $

$

N H EC

E U IQ

)9.0;5) CC ,0 6 : A ,6?6 0 . .

−div(A∇u) = λu Ω u=0 ∂Ω

LE O ÉC

T Y POL

R A(x) # . ;. 6 " <

•

.

, 6 ? 6 . 6 0 . / . 0 . ) 8 J # , % ;""$< , . 8 # . ; 00 ? , <

∂2u ρ ∂t2 − div (2µe(u) + λ tr(e(u)) Id) = f Ω × R+ ∗ ;&-< u=0 ∂Ω × R+ ∗,

T Y POL

N H EC

E U IQ

R ρ > 0 . e(u) = ∇u + (∇u)t /2 + : . ) µ > 0 2µ + N λ > 0 9 ,0 8 f ; ,. < ;&-< ,. , 6 & ' ? (, u) 0 . .

−div (2µe(u) + λ tr(e(u)) Id) = u Ω ;&< u=0 ∂Ω,

ÉC

E L O

R = ω 2 8 . 0 ; . . . 8 . : λ< 9 A 8 u 9,&) 30,30) &) 2'/0.1',* 9 . 6 . ; , <

E U " $ -" /$ 0 , I Q N $&"

$ " $" )" "$ - $*$' "H &" / C #/"$$ ' " # ))"$" " -* E T Y ),), $ L O ),), P LE O ÉC 0,3,:'1',* CC

(k )k≥1 L2 (Ω)N (uk )k≥1

C1

Ω

uk

−div (2µe(uk ) + λ tr(e(uk )) Id) = k uk uk = 0

RN

H01 (Ω)N Ω ∂Ω.

( $ #

LE O ÉC

T Y POL

#

ÉC

E L O

N H EC

&

E U IQ

T Y POL

N H EC

E U IQ

%

E U IQ

& W , L M

LE O ÉC

Y L O P

H C E T

N

( $

-

N H EC

E U IQ

8 uk * & , 8 , 0 ;&< * 5 & . . 8 8 ; 8 < ' ) Q1 , L M 0 ) ; 1 < 0 ; / < A 0A ? H . ; # 90◦ . , z < ;. 2 & 50 &< . L 0M

LE O ÉC

T Y POL

+ >

"

"

--"

$

& W > 2 &

E U IQ

/ % ;"&< . , 0 ,. # 0 ;. ;- << * 0 ,. , . 8 (λ, u, p) 0 . ⎧ ⎨ ∇p − µ∆u = λu Ω divu = 0 Ω ;& < ⎩ ∂Ω, u=0

E L O

T Y POL

N H EC

R µ > 0 . A u . p D 9 . 6 A , .

ÉC

>)0+'+) CC ) Ω * C 1 RN "

$ (λk )k≥1 $( % (uk )k≥1 L2 (Ω)N + * ( % uk + H01 (Ω)N ( pk ∈ L2 (Ω) ⎧ ⎨ ∇pk − µ∆uk = λk uk Ω Ω divuk = 0 ⎩ uk = 0 ∂Ω. 8 * & , 8 % ;& < * 5 & . . 8 8 ; 8 <

H C E T

N

E U IQ

>)0+'+) CC $ )%1

Y L O P

* V (x) = Ax · x 4 A : $ %

OLE

−∆u + V u = λu RN .

ÉC

;& <

( H = L (R ) 2

N

N H EC

E U IQ

V = v ∈ H 1 (RN ) |x|v(x) ∈ L2 (RN ) .

T Y POL

" V O

LE O ÉC

u, vV =

RN

∇u(x) · ∇v(x) dx +

RN

|x|2 u(x)v(x) dx,

$5 V H ' $ (λk )k≥1 $ % L2 (RN ) (uk )k≥1 &2 0 %% uk pk (x) exp(−Ax · x/2) 4 pk :9 * k − 1 L %:

>)0+'+) CC ) Ω * RN $ $

∆ (∆u) = λu ∂u ∂n = u = 0

Ω ∂Ω.

N H EC

E U IQ

" $ (λk )k≥1 $ % L2 (Ω) (uk )k≥1 + H02 (Ω)

T Y POL

E L O C ( É *

/

)

@ . 8 . 6 ? 6 & 6 Vh , = 0 V A ) A (λh , uh ) ∈ R × Vh a(uh , vh ) = λh uh , vh H

;&

∀ vh ∈ Vh .

<

5# A Vh , ) 1) 6 $" $ "A H , L2 (Ω) , ;& < 8 .

E U IQ

)99) CC $ ) #!)"#+ A#+ F,3,, - )) F, $" $ " $" *$ 0 < λ1 ≤ · · · ≤ λn - ndl = dimVh , " &" $ / Vh ' "#$ $ H ' (uk,h )1≤k≤n -" )) ' " uk,h ∈ Vh , " a(uk,h , vh ) = λk uk,h , vh H ∀ vh ∈ Vh . dl

LE O ÉC

Y L O P

H C E T dl

N

( $

N H EC

E U IQ

%9,*:10.1',*C ' H . .

5 & ;? ( , ) 0 ) . < / (A 0 A ? . (φi )1≤i≤ndl 0 Vh ; A 8 0 , , ) A . * $&< @ uh ;& < 8

LE O ÉC

T Y POL

uh (x) =

ndl

Uih φi (x).

i=1

9.10'+) &) 9.::) Mh ) (Mh )ij = φi , φj H

1 ≤ i, j ≤ ndl ,

9.10'+) &) 0'('&'1% Kh ) (Kh )ij = a(φi , φj ) 1 ≤ i, j ≤ ndl ,

N H EC

E U IQ

0 ;& < . ? . (λh , Uh ) ∈ R × Rndl Kh Uh = λh Mh Uh .

T Y POL

;& <

L M . + A R Vh , ) A Kh H ' $ , ) 0 @ . ) Mh Kh # ) . # ;& < 0 . L M ;. $ T U< : , . 0 Ph

ÉC

E L O

Mh = Ph Ph∗ , Kh = Ph diag(λk )Ph∗ .

* A ;& < . (λk ) . (Uk,h )1≤k≤ndl . Ph ' . 8 0A Kh Mh ; 6 0 . ? + & 0< 2 A . Uk,h . 0 (φi )1≤i≤ndl 8 uk,h 8 0 Vh H

H C E T

N

E U IQ

)9.0;5) CC 1 & # 5 & J A 8 0 a(u, v) :B9%10';5) @ . 0

Y L O P

, # 9 (A , # A # ;& < 0A ,6?6 , • 0 . ;& <

LE O ÉC

(

N H EC

E U IQ

, & ? , . ) 0 1 ;&&< @ V = H01 (Ω)A H = L2 (Ω)A , V0h 1) $" ; V0h 1 <

T Y POL

>)0+'+) CC N = 1

ÉC

OLE

−uk = λk uk 0 < x < 1 uk (0) = uk (1) = 0.

% P1 ) 2 " Mh ⎞ ⎛ 2/3 1/6 0 ⎜ 1/6 2/3 1/6 ⎜ ⎜ Mh = h ⎜ ⎜ ⎝ 1/6 2/3 1/6 0 1/6 2/3

T Y POL

λk (Mh ) =

E L O

⎟ ⎟ ⎟ ⎟, ⎟ ⎠

N H EC

E U IQ

h (2 + cos(kπh)) 1 ≤ k ≤ n. 3

" ( !( Mh = h Id 6 (

ÉC

)9.0;5) CC * . . 0

;& < 8 A A 8 '%# Mh = Lh L∗h A ˜hU ˜h = λh U ˜h K

˜ h = L−1 Kh (L∗ )−1 U ˜h = L∗ Uh , . K h h h

, . . / .# ? J , , G ˜ h 8 , 8 @ . K '%# Mh 8 . : Mh &'.(,*.-) ' , +,*&)*:.1',* &) 9.::) ; L M < 8 * A 8 ;$"< ; . , 8
Y L O P

H C E T

N

E U IQ

/ . 6 . :)5-): -): 30)9'?0): 2.-)50: 30,30): λk,h ; < .

LE O ÉC

( $

N H EC

E U IQ

λk ;H . < 8 8 8 . ; < 0 ;& < , ) # V * A . 0 1 ;&&
/ 6 $ . 0 , 0 ; 0 ) 0 < @ 0 1 ;&&< # @

, ) P1 2 & 50 &< @ . L 0M

LE O ÉC

T Y POL

+ >

&

"&-

$

$

$

" $

$ "

E U IQ

W > 2 &

N H EC

' $ #

T Y POL

1 6 A , .6

) Pk . . 0 1 ;&&< 0

? , 0 ? , # , )

E L O

C " É " )

(Th )h>0 $ " 0 "$0 0 Ω, H01 (Ω)' *$ ) "# $" *$ Pk ' $$ ndl , " (λi , ui) ∈ R × H01 (Ω)' ) i ≥ 1' - " -" )) "#$ $ L2 (Ω) )/+ 8#" %,3%' $0 ) $"

4%,0?9) CC V0h

0 < λ1 ≤ λ2 ≤ ... ≤ λi ≤ λi+1 ...

" - )) 0 < λ1,h ≤ λ2,h ≤ ... ≤ λndl ,h ,

E U IQ

))&"$ -"$$ F, $ V0h , " " i ≥ 1 *&' $ lim |λi − λi,h | = 0.

C E T Y L PO

h→0

HN

;& <

&" $ -" )) (ui,h )1≤i≤n F, $ V0h " ' λi " $ - )) )' $ dl

OLE

lim ui − ui,h H 1 (Ω) = 0.

ÉC

h→0

;& "<

(

ÉC

OLE

T Y POL

#

N H EC

E U IQ

N H EC

%

ÉC

E L O

T Y POL

E U IQ

&

$

&

Y L O P

H C E T

N

W $ , 0

LE O ÉC

E U IQ

( $

E U " $ I Q $ N H

8 ) ' ) $0$ ) (u1 , ..., ui) k + 1 > N/2' $ ""$

O P LE O ÉC

C E T LY

H k+1 (Ω)

|λi − λi,h | ≤ Ci h2k ,

"

;& $<

. Ci $ )$ ) h' " λi " $ - )) )' $ ;& &<

ui − ui,h H 1 (Ω) ≤ Ci hk .

)9.0;5) CC Ci ;& $< ;& &< . +∞ i →

+∞A 8 , ,# . ; A λndl ,h < . λi , . . 0 . . , ? , , , 6 I , . ;. * < . . ,0 . 9 (A . λi A . ui,h . , ( 0 .

6 ? λi •

ÉC

E L O

T Y POL

LE O ÉC

N H EC

Y L O P

E U IQ

H C E T

N

E U IQ

ÉC

T Y POL

N H EC

E U IQ

E L O

! 1

:

)%

,

N H EC

E U IQ

' ? ,# 0 ,. ; . 0 . 0 < / # # ( , . J # 0 A # #0 , # , 0 , ; # , ? % < # , #0 , ; 6 8 # , ? , # , < * A , . , ? 0 , 0 ,. A 6 # 0 #0 A A A , ! . ? ? 1 - - 6 , , ,

? . =,095-.1',* 2.0'.1',**)--) ' ,. 6 & A /.:): 4'-/)0 1')**): &) =,*+1',*: 30,30): ' & / &!):1'9.1',*: &!%*)0(') 0 , # I ) 1 - -" 30,30'%1%: ;5.-'1.1'2): A , , 00 A -)50: 30,30'%1%: ;5.-'1.1'2): :,*1 3.0 +,*10) 10?: &'A%0)*1): / . A .

ÉC

E L O

T Y POL

LE O ÉC

Y L O P "

H C E T

N

E U IQ

) #$

$

N H EC

E U IQ

8 ? , # ; A . ,
LE O ÉC

T Y POL

: 5 ) # &

E U IQ

* 0 0 A # ,# -!%;5.1',* &) -. +4.-)50 , # I? ' Ω . 0 RN 8 ∂Ω * 1 , ⎧ ∂u Ω × R+ ⎨ ∂t − ∆u = f ∗ ;-< u=0 ∂Ω × R+ ∗ ⎩ u(x, 0) = u0 (x) x ∈ Ω.

ÉC

E L O

T Y POL

N H EC

0 ;-< ,. u(x, t)

Ω 0 A ? t = 0A 8 u0 0 ∂Ω A ? . A . 8 ;, 1 u(x, t) = 0 ∂Ω × R+ < 8 f = f (x, t) / . 0 x ∈ Ω t ∈ R+ I X ( ;-< , , , . t x ; . 0 < , , # # ;-< * A ;-< ( , u ΩA 0 ,. u , D , ;# 1#
LE O ÉC

Y L O P

H C E T

N

E U IQ

) $

N H EC

&

E U IQ

, ( . 8 ; . < # % . . 9 u . p , D . ? 8 f A # , ⎧ ∂u + ∇p − µ∆u = f Ω × R+ ⎪ ∗ ⎪ ⎨ ∂t divu = 0 Ω × R+ ∗ ;- < u=0 ∂Ω × R+ ⎪ ∗ ⎪ ⎩ u(x, t = 0) = u0 (x) Ω

LE O ÉC

T Y POL

R µ > 0 . D + 1 , D ? Ω ;. '
:

E U IQ

5 ) # &

N H EC

* #0 6 -!%;5.1',* &): ,*&): , # I? ' Ω . 0 RN 8 ∂Ω * 1 , ⎧ 2 ∂ u ⎪ − ∆u = f Ω × R+ ∗ ⎪ ⎨ ∂t2 u=0 ∂Ω × R+ ∗ ;-< ⎪ u(t = 0) = u0 (x) Ω ⎪ ⎩ ∂u Ω. ∂t (t = 0) = u1 (x)

ÉC

E L O

T Y POL

0 ;-< A A

. , 0 A 0 , , , u(t, x) 8 -!%-.:1,&B*.9';5) . ,. , ;. ' "< * 8 # A , . A 0 0 ,. , ;-< / A ( . ;-< , u(t, x) 8 ? . . RN * A f (t, x) ;. < 8 A u(t, x) ⎧ 2 ⎪ ρ ∂ u − div (2µe(u) + λ tr(e(u)) Id) = f Ω × R+ ∗ ⎪ ⎨ ∂t2 u=0 ∂Ω × R+ ∗ ;-< ⎪ u(t = 0) = u0 (x) Ω ⎪ ⎩ ∂u Ω, ∂t (t = 0) = u1 (x)

LE O ÉC

Y L O P

H C E T

N

E U IQ

) #$

-

N H EC

E U IQ

R u0 A u1 . A e(u) = (∇u + (∇u)t )/2 8 ΩA ρ > 0A H . ) µ > 0 2µ + N λ > 0

T Y POL

)9.0;5) C C @ I ;-< ;-< A

LE O ÉC

∂2u ∂u − ∆u = f Ω × R+ +η ∗. ∂t2 ∂t

: η 8A ? 8 8 ? . @ , , , ,6 @ &!%;5.1',* &): ,*&): .9,01'): •

)9.0;5) C C , # ? . #0 ' A #0 , 0 ,. , ', ,9 A 7C 6 A # , #0 , , ? 0A • ( , 1

E L O

T Y POL

N H EC

E U IQ

2 +

ÉC

/ . , ?

' " 0 , , , 0 ' J A 0 8 . ; 6 -
:

+ $

, , 8 . 0 ? %;5.1',* &'A%0)*1')--) ,0&'*.'0) A ? ;&< * 6

E U IQ

, ;-< 8 v(x)

t 3 ? v , 0 , . ΩA . ,( ; 0 < * , 8 / 0 ∂u (x, t)v(x) dx + ∇u(x, t) · ∇v(x) dx = f (x, t)v(x) dx. ;-"< Ω ∂t Ω Ω

LE O ÉC

Y L O P

H C E T

N

)

N H EC

E U IQ

' Ω v(x) . . tA 8 d u(x, t)v(x) dx + ∇u(x, t) · ∇v(x) dx = f (x, t)v(x) dx. dt Ω Ω Ω

LE O ÉC

T Y POL

9 8 . 0 x t I X (A . 0 u(t, x) 8 t E 2.-)50: &.*: 5* ):3.+) &) =,*+1',*: ) Ω ;H f (t, x)< * A , ) T > 0 ;. ? +∞
? u(x, t) . u(t) (x) , H01 (Ω) . 0 . , ? 9 , , . 8 . 0 1 HA f 8 t ? . L2 (Ω) @ L2 (Ω) 8 0 a(w, v) ) w, vL2 (Ω) =

E L O

Ω

T Y POL

N H EC

w(x)v(x) dx a(w, v) =

E U IQ

∇w(x) · ∇v(x) dx. Ω

9 8 , H01 (Ω)A ;-"< 8 , &!%;5.1',* &'A%0)*1')--) ,0&'*.'0) t @ 0 8 . . J . u(t) 8 ]0, T [ ? . H01 (Ω) ⎧ ⎨ d u(t), v 2 ∀ v ∈ H01 (Ω), 0 < t < T, L (Ω) + a (u(t), v) = f (t), vL2 (Ω) dt ;-$< ⎩ u(t = 0) = u0 .

ÉC

? 8 . ;-$< J f uA ? . K 9 A 8 0 u(t) t = 0 ? u0 * . 0 , 8 , 8 8 t ? . 8 x

E U IQ

N H C " $ ) @/"' ) E 0$$"' $ ) ' YT ' , " $ ") *$ K$# *$ "!) $"' L , $ $" ' $ $" ) $"$ O , $ $" $ $"$4$" -/ P $ $ ' E L O ÉC

%@*'1',* CC

X

Ω

0 < T ≤ +∞

X = L2 (Ω) H01 (Ω) C(Ω) k≥0 C k ([0, T ]; X) [0, T ] X vX

k

X

) #$

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k

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([0, T ]; X)

m d v (t) sup . m 0≤t≤T dt

k

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m=0

X

$ $" L2 (]0, T [; X) ]0, T [ $ X " $"$ t → v(t)X " / " $"0/' "

LE O ÉC

T

vL2 (]0,T [;X) = 0

v(t)2X dt < +∞.

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u, vL2 (]0,T [;X) =

T

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0

N H EC

E U IQ

)9.0;5) CC X , L2 (Ω)A L2 (]0, T [; L2(Ω)) , ) ? , L2 (]0, T [×Ω) A 2 0 A T T v2L2 (]0,T [;L2 (Ω)) = |v(t)|2 (x) dx dt = |v(x, t)|2 dx dt = v2L2 (]0,T [×Ω) . 0

Ω

E L O

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* 1 ≤ p < +∞A 1) - , E Lp (]0, T [; X) 8 ]0, T [ X 8 t → v(t)X 0 p6 0 •

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1 f , L2 (]0, T [; L2 (Ω))A u -!):3.+) &!%*)0(') L2 (]0, T [; H01 (Ω)) ∩ C([0, T ]; L2 (Ω)) ' P 0 A <5:1'@%A 6 , , 8 . ;-$
: 3

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* , , 8 . 6 ;-$
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)

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. L M , ? , 8 ( @ 6 = 0 V H V ⊂ H . I ;. ;&< , , < 5# V = H01 (Ω) H = L2 (Ω)

LE O ÉC

T Y POL

$" V " H & ) @/" " V ⊂ H - $ >"$ )" " V " $ $ H , " a(u, v) $ /$ !" $"$ " - $ V , " $ ") *$ T > 0' $ $$ $" u0 ∈ H ' " $ " f ∈ L2 (]0, T [; H), )/+ 4%,0?9) CC

⎧ ⎪ ⎨ d u(t), vH + a (u(t), v) = f (t), vH dt ⎪ ⎩ u(t = 0) = u ,

∀ v ∈ V,

0 < t < T,

;-&<

0

E U IQ

. "$ ,F $ / $ ]0, T [ $ $ "$ u ∈ L2 (]0, T [; V ) ∩ C([0, T ]; H), 8 ) ' &" $ $"$" C > 0 $ )$ Ω "

N H EC

uL2(]0,T [;V ) + uC([0,T ];H) ≤ C u0 H + f L2 (]0,T [;H) .

T Y POL

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E L O

)9.0;5) CC 1 5 - ( 0 ,# .

8 0 # a(u, v) ; , I? ,9 &< @ 0 H , . ν > 0 η > 0 a(v, v) + ηv2H ≥ νv2V v ∈ V.

9 (A , ( 8 u(t) = eηt w(t)A . ;-&< . ? ⎧ ⎨ d w(t), v + a (w(t), v) + ηw(t), v = f (t), v ∀ v ∈ V, 0 < t < T, H H H dt ⎩ w(t = 0) = u0 ,

H C E T

N

E U IQ

R 8 0 a(w, v) + ηw, vH 0 . V ' # ( 0 A A ,9 - •

Y L O P

%9,*:10.1',*C . 1

A , , uA 0 8 u 8 , 0 H

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) #$

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V 9 A 8 . , 1 A . L2 (]0, T [; V ) C([0, T ]; H)A 0 ;-&< 1.3) C u ∈ L2 (]0, T [; V ) ∩ C([0, T ]; H) ;-&< # , 5 & 0 . ? 8 0 # a(u, v) * A 0 0 (uk )k≥1 H . ;& <

LE O ÉC

T Y POL

uk ∈ V, a(uk , v) = λk uk , vH

∀ v ∈ V.

@ )

αk (t) = u(t), uk H ,

α0k = u0 , uk H ,

βk (t) = f (t), uk H .

* u ∈ L2 (]0, T [; V ) ∩ C([0, T ]; H) f ∈ L2 (]0, T [; H)A αk (t) ∈ C([0, T ]) βk (t) ∈ L2 (]0, T [) ' (uk )k≥1 0 0 H A u(t) =

+∞

N H EC

αk (t)uk ,

k=1

T Y POL

E U IQ

v = uk ;-&< 0 ⎧ ⎨ dαk + λ α = β ]0, T [ k k k dt ⎩ αk (t = 0) = α0k .

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;-<

@ . ) , ;-< t αk (t) = α0k e−λk t + βk (s)e−λk (t−s) ds t > 0, 0

8 u ; < 1.3) C / t +∞ βj (s)e−λj (t−s) ds uj α0j e−λj t +

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0

j=1

E U IQ

. L2 (]0, T [; V ) ∩ C([0, T ]; H) A u(t) ;-&< ' ? , k

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t k 0 −λj t −λj (t−s) w (t) = + βj (s)e ds uj . αj e k

j=1

Y L O P 0

N

;-<

' wk ? C([0, T ]; H) αj (t) 7 wk ' # C([0, T ]; H) * l > k A

LE O ÉC

)

N H EC

E U IQ

8 uj A t l l l k 0 −λj t −λj (t−s) w (t) − w (t)H ≤ αj e uj + βj (s)e ds uj j=k+1 j=k+1 0 H H ⎞1/2 ⎛ ⎞ ⎛ 2 1/2 t l l ≤⎝ |α0j |2 e−2λj t ⎠ + ⎝ βj (s)e−λj (t−s) ds ⎠

LE O ÉC

T Y POL ⎛

j=k+1

⎞1/2

0

j=k+1

⎞1/2 T 1 ≤⎝ |α0j |2 ⎠ + ⎝ |βj (s)|2 ds⎠ 2λj 0 j=k+1 j=k+1 ⎞1/2 ⎛ ⎛ ⎞1/2 T l l 1 ⎝ ≤⎝ |α0j |2 ⎠ + √ |βj (s)|2 ds⎠ , 2λ 1 j=k+1 j=k+1 0 l

⎛

l

E U IQ

. (λj ) . ' u0 ∈ H f ∈ L2 (]0, T [; H) u0 2H =

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j=1

N H EC

f 2L2(]0,T [;H) =

O P E L ÉCO

T Y L

+∞ j=1

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k

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≤2

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k

l

k

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j=k+1

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l j=k+1

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j=k+1 t

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H C E T

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2

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−λj (t−s)

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Y L O P 0

1 ≤ λj

LE O ÉC

0

t

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.

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* A . 2 0

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l

l

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j=k+1

j=k+1

2 λj

T

|βj (s)|2 ds,

0

wk . ) T lim wl (t) − wk (t)2V dt = 0, k,l→+∞

0

N H EC

E U IQ

,6?6 , ' # L2 (]0, T [; V ) ' C([0, T ]; H) L2 (]0, T [; V ) A ' # wk . ) u

T Y POL

lim wk = u C([0, T ]; H) ∩ L2 (]0, T [; V ).

E L O k→+∞

9 A wk (0) . . u0 H A . A u(0) = u0 ; 8 H < 1, A u(t)A ;-< . ) √ 8 . ;-&< 8 v = uk ' (uk / λk ) 0 0 V A u(t) . ) 8 . ;-&< v ∈ V A ,6?6 u(t) 0 ;-&< * 0 , , ;--
ÉC

wl (t) − wk (t)H ≤ u0 H + √

0

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1 f L2(]0,T [;H) 2λ1

wl (t) − wk (t)2V dt ≤ u0 2H +

2 f 2L2 (]0,T [;H) . λ1

H C E T

N

E U IQ

9 k = 0 8 l . , ) A 0 ,

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H −1 ,H01 (0,T )

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0

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∀ φ ∈ H01 (0, T ). ]0, T [

T

−

u(t), vH 0

dφ (t) dt + dt

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T

f (t), vH φ(t) dt

a(u(t), v)φ(t) dt = 0

0

v ∈ V φ ∈ Cc∞ (]0, T [) " Cc∞ (]0, T [) H01 (0, T ) K u )G! " )G! % dtd u(t), vH & L2 (0, T ) " )G!

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.

N H EC

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/ 0 5 - ? , A . . 0 , . ,

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" Ω $ -" /$ 0 RN , " $ ") *$ T > 0'

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ÉC

⎧ ∂u ⎪ ⎨ − ∆u = f ∂t u=0 ⎪ ⎩ u(x, 0) = u0 (x)

),), $ Ω×]0, T [ ),), ∂Ω×]0, T [ ),), $ Ω.

;- <

" $ $ "$ u ∈ L2 (]0, T [; H01(Ω)) ∩ C([0, T ]; L2(Ω)), 8 ) ' &"

$ $"$" C > 0 $ )$ Ω " ' ) " " t ∈ [0, T ]' t

2

u(x, t) dx + Ω

0

|∇u(x, s)| dxds ≤ C 2

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2

u0 (x) dx + Ω

t

2

E U IQ

f (x, s) dxds . 0

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H C E T

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LE O ÉC

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) #$

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1 . 5 ? u(t) ∈ H01 (Ω) t ∈]0, T [A I ) u(t) t = 0 ; 8 ? . L2 (Ω)< u : ; A ∂u ∂t ∆u 2 ? L (]0, T [×Ω)A . , * -$
T Y POL

LE O ÉC

8 v(x) ∈ H01 (Ω) t ∈]0, T [ * A ;-< ∂u − ∆u − f = 0 ∂t

]0, T [×Ω.

E U IQ

1 u " u ∈ L2 (]0, T [; H01 (Ω)) ∩ C([0, T ]; L2 (Ω)) ' ; 5 * & A " )N )N!

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N H EC

T

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f vφ dx dt

0

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(" % )9.0;5) CC !):1'9.1',* &!%*)0(') ;-< 6 , , X ? I ? L. M # ;

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' u .2 # * ]0, T [×Ω( ( t ∈ [0, T ]( $ * $ * 1 2

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u(x, t)2 dx +

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t 0

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2 6 ( @ ?Q A I z [0, T ] R+

t

z(t) ≤ a + b

∀ t ∈ [0, T ],

z(s) ds 0

4 a, b ( z(t) ≤ aebt

∀ t ∈ [0, T ].

' ?Q z(t) = 12 Ω u(x, t)2 dx( . ( t ∈ [0, T ]( t 1 et u(x, t)2 dx + |∇u(x, s)|2 dx ds ≤ u0 (x)2 dx 2 Ω 2 0 Ω Ω ;-&< T 2 f (x, s) dx ds . +

ÉC

T Y POL

E L O

N H EC 0

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Ω

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z(t) ≤ a + 2

t

g(s) z(s)ds

∀ t ∈ [0, T ],

0

z(t) ≤

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∀ t ∈ [0, T ].

0

2 6 . ( t ∈ [0, T ]( 2

t

u(x, t) dx + 2 Ω

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Ω

LE O ÉC

Y L O P

+

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|∇u(x, t)|2 dx +

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E L O ⎪ ⎩

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∂u ∂n

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− div (A(x)∇u) = f u=0 ⎩ u(x, 0) = u0 (x) ∂u ∂t

Ω×]0, T [ ∂Ω×]0, T [ Ω,

H C E T

u ∈ L2 (]0, T [; H 1 (Ω)) ∩ C([0, T ]; L2 (Ω))

Y L O P

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R divu0 = 0

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N

Ω

Ω×]0, T [ Ω×]0, T [ ∂Ω×]0, T [ Ω

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T Y POL

N H EC

E U IQ

⎧ ⎨ d u(t) · v dx + µ ∇u(t) · ∇v dx = f (t) · v dx ∀ v ∈ V, 0 < t < T, dt Ω Ω Ω ⎩

E L O

u(t = 0) = u0 ,

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)!

: " )! *' ]0, T [ $ " D( )C & * % )! ( ( * %! ' u ∈ L2 (]0, T [; H01 (Ω))N ∩ C([0, T ]; L2 (Ω))N D % " )! ' )! .( % D( #CC & u(t) ∈ H01 (Ω)N " t ∈]0, T [ ; u(t) t = 0 u0 ∈ H % "

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∂u − µ∆u − f · v dx = 0 ∂t

E U IQ

" t ∈]0, T [ " " v(x) ∈ Cc1 (Ω)N " divv = 0 Ω 5

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E L O

, L M 8 v H01 (Ω) @ L2 (Ω) 8 0 a(w, v) ) w, vL2 (Ω) = w(x)v(x) dx a(w, v) = ∇w(x) · ∇v(x) dx. Ω

Ω

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Y L O P

H C E T

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N H EC

⎧ 2 d ⎪ ⎨ u(t), vH + a (u(t), v) = f (t), vH dt2 ⎪ ⎩ u(t = 0) = u0 , du (t = 0) = u1 , dt

O P E L ÉCO

T Y L

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∀ v ∈ V, 0 < t < T,

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a(v, v) + ηv2H ≥ νv2V v ∈ V.

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, u(t) = e

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R 8 0 a(w, v) + ηw, vH 0 . V , ;- -<

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N H EC

8 u ; < 1.3) C * t +∞ α1j 1 0 sin(ωj t) + βj (s) sin(ωj (t − s))ds uj αj cos(ωj t) + ωj ωj 0 j=1

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. C([0, T ]; V ) ∩ C 1 ([0, T ]; H)A . wk = k j=1 αj (t)uj ' # 1 V 6 a(u, v) 8 (uj ) * H V (uj ) ;. 5 & k A tA l dαj 2 l d l 2 2 k l k k (t) . a w − w ,w − w + λj |αj (t)| + w −w = dt dt H

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j=k+1

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@A ;- < dαk /dt A 0 t dαj 2 dαj + λj |αj (t)|2 = α1 2 + λj α0 2 + 2 (t) (s)ds. βj (s) j j dt dt 0

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1 8 ;-< 8

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t dαj 0 1 |βj (s)|ds. dt (t) ≤ ωj αj + αj + 0

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H01 (Ω)

RN ' " $ ") *$ T > 0, × L2 (Ω) " $ " f ∈

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2 ∂u 2 2 (x, t) + |∇u(x, t)| dx ≤ C |u1 (x)|2 + |∇u0 (x)| dx ∂t Ω Ω t + |f (x, s)|2 dxds .

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E L O

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H C E T

N

E U IQ

v(x) ∈ Cc1 (Ω) t ∈]0, T [ @ , ;-< u , 5 -A 0 H , L MA 5 - & ; < @ N +1 A σ = ( ∂u ∂t , −∇u) 8 ? . . R

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%9,*:10.1',*C 9 5 - . f = 0A ,6?6 βk = 0A ;- < , , .A ,6?6 dαj 2 0 2 1 2 2 dt (t) + λj |αj (t)| = αj + λj αj ,

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LE O ÉC

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u1 (x)2 dx + |∇u0 (x)|2 dx Ω Ω t ∂u +2 f (x, s) (x, s)dxds. ∂t 0 Ω

) #$

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2 ' $ C(T ) T

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2 ∂u 2 2 2 (x, t) dx + |∇u(x, t)| dx ≤ C(T ) u (x) dx + |∇u0 (x)| dx 1 ∂t Ω Ω Ω t Ω 2 + f (x, s) dxds .

LE O ÉC

0

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" $ C : T 2 ∂u 2 2 2 (x, t) dx + |∇u(x, t)| dx ≤ C u1 (x) dx + |∇u0 (x)| dx Ω ∂t Ω Ω Ω ⎞ 1/2 2 t + f (x, s)2 dx ds ⎠ . 0

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⎧ ∂2u ⎪ ⎪ ρ 2 − div (2µe(u) + λ tr(e(u)) Id) = f ⎪ ⎨ ∂t u=0 ⎪ ⎪ u(t ⎪ ⎩ ∂u = 0) = u0 (x) ∂t (t = 0) = u1 (x)

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u(t) − α01 e−λ1 t u1 L2 (Ω) ≤ C e−λ2 t ∀ t > 1,

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α01 =

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%9,*:10.1',*C u− = min(u, 0) 0 ? L2 (]0, T [; H01 (Ω))

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/ , ;-< . H ∂u ∂t , ? L2 (]0, T [; L2(Ω)) * A v = u− 8 . 6 ;-$< , 0 1 d |u− |2 dx + |∇u− |2 dx = f u− dx, 2 dt Ω Ω Ω

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t t 1 1 − 2 − 2 − |u (t)| dx + |∇u | dx ds = f u dx ds + |u− (0)|2 dx. 2 Ω 2 Ω 0 0 Ω Ω ' u− (0) = (u0 )− = 0 t 1 |u− (t)|2 dx + |∇u− |2 dx ds ≤ 0, 2 Ω 0 Ω ,6?6 u− = 0 ]0, T [×Ω

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0 . , 8 . @ A A ;- < J u0 ≤ u ˜0 Ω f ≤ f˜ ]0, T [×ΩA u u ˜ . ;- < (u0 , f ) (˜ u0 , f˜) .A u ≤ u ˜ ]0, T [×Ω . 6

LE O ÉC

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>)0+'+) CC ) Ω * RN u1

Ω 6% ( λ1 $ % u1 > 0 Ω <% NC &; $ ∂u1 /∂n > 0 ∂Ω ) f = 0( u0 ∈ L2 (Ω) u $ * .2 ) > 0 " $ K −Ku1(x) ≤ u(x, ) ≤ Ku1 (x)

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LE O ÉC

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k

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u ˆ(k, t) =

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u(t)L∞ (RN ) ≤ u0 L∞ (RN ) ∀ t > 0.

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N H EC

4 %

T Y POL

"$ &

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≥ 0

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" Ω $ -" /$ 0 RN ' " $ ") *$ T > 0, " (v0 , v1 ) ∈ × L2 (Ω)' " $ " f ∈ L2 (]0, T [; L2 (Ω)), "$ $ 0%10,(0.&) )* 1)93: $"0 $ $"$" ") )" T ⎧ ∂ v ⎪ − ∆v = f ),), $ Ω×]0, T [ ⎪ ⎨ ∂t ),), ∂Ω×]0, T [ v=0 ;-" < ⎪ v(x, T ) = v0 (x) ),), $ Ω ⎪ ⎩ ∂v ),), $ Ω ∂t (x, T ) = v1 (x) 1 " $ $ "$ v ∈ C([0, T ]; H0 (Ω)) ∩ C 1 ([0, T ]; L2(Ω)), 8 ) ' u(t, x) " "$ "$ $ ,3 " v0 (x) = u(x, T ) $ H01 (Ω) " 2 v1 (x) = ∂u ∂t (x, T ) $ L (Ω)' $ v(t, x) = u(t, x), 0,3,:'1',* CC

H01 (Ω)

2

2

H C E T

N

E U IQ

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LE O ÉC

) #$

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T Y POL

. 0 , 0 6 , ,# )A)1 0%(5-.0':.*1 , , 9 (A , A * -"A 0 ( L M ; ) T A , 0 ,( , < * A '- *!B . *' (.'* *' 3)01) &) 0%(5-.0'1% , @ :A A , ?

LE O ÉC

' " " Ω R " ! & T > 0 N

E U IQ

u0 ∈ H (Ω) u1 ∈ ! f ∈ L2 (]0, T [; L2 (Ω)) 1 2 u ∈ C([0, T ]; H0 (Ω)) ∩ C ([0, T ]; L (Ω)) <$$ 2 u C([0, T ]; H01 (Ω) ∩ H 2 (Ω)) ∩ C 1 ([0, T ]; H01 (Ω)) ∩ C 2 ([0, T ]; L2 (Ω)) 4 )/ " & )#N " % $ ' - 8 9 & ' " u " )CC! " % % " u0 u1 f % % ' % A " )#G! H01 (Ω) ∩ 1

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LE O ÉC

Y L O P

u(x, 0) = w(x) ,

H C E T

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$&

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LE O ÉC

T Y POL

>)0+'+) CC ) η > 0 $ ⎧ ⎪ ⎪ ⎨

∂2u ∂t2

Ω × R+ ∗ ∂Ω × R+ ∗ Ω Ω.

+ η ∂u ∂t − ∆u = f u=0 ⎪ u(x, 0) = u0 (x) ⎪ ⎩ ∂u ∂t (x, 0) = u1 (x)

N H EC

;-"<

E U IQ

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E L O

>)0+'+) CC ) u(t, x) ( M *( $

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t 2 t ∂u 1 dx = lim 1 |∇u|2 dx = E0 , ∂t t→+∞ t 0 2 0 Ω Ω

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2

2

|u1 (x)| dx +

E0 = Ω

|∇u0 (x)| dx. Ω

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:

9 (

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, A 30, 3.(.1',* E 2'1)::) @*') / . I? . ' ; N = 1

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, Ω = R< , X ; 6 < 0 , 8 , ;.

LE O ÉC

) #$

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2 < * A u (t, x) . u0 u1 [x − t, x + t] @ ? K = [kinf , ksup ] ⊂ RA t ? [kinf − t, ksup + t] 9 # . 0 ? . ) 0 ' 8 0 ( , ; . * -< , . N = 2, 3

LE O ÉC

T Y POL

>)0+'+) CC $ $ RN ⎧ ⎨

∂2u ∂t2

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;-"<

E U IQ

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T Y POL

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E L O

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1 N = 2, (M v)(x, t) = 2π

N = 3, (M v)(x, t) =

N H EC

∂(M u0 ) ∂t

+t

−t

(x, t),

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1 4πt

|ξ|
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E U IQ

>)0+'+) CC $ .! Ω ⊂ RN

N

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Y L O P

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, . 8 ( , N = 2 3

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)'

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OLE

1 .

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) +

1 A ) ;6 ' $< , ? , J ) A ( )

:

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, )* ):3.+) :)5-)9)*1 8 . ;-$< , ;-< * A 0 6 A . 6 V0h H01 (Ω)A ) 5# A V0h 6 , ) Pk ; Qk < ; < 1) $" $ " 6 ;-$< , . . J . uh (t) 8 ]0, T [ ? . V0h ⎧ ⎨ d u (t), v 2 h h L (Ω) + a (uh (t), vh ) = f (t), vh L2 (Ω) ∀ vh ∈ V0h , 0 < t < T, dt ;-""< ⎩ uh (t = 0) = u0,h

ÉC

E L O

T Y POL

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R u0,h ∈ V0h u0 ' ,6 L M @ 0 5 - ;-""< A 0 . ) ;-""< 8 # &!%;5.1',*: &'A%0)*1')--): ,0&'*.'0): ? : 8 , 1 A ;-""< 0 (φi )1≤i≤ndl V0h ;# A 0 )
ndl i=1

H C E T

E U IQ ;-"$<

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. U h = (Uih )1≤i≤ndl . uh ;-"$< 8 0 φi Uih (t) 8 t 1 HA

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) #$

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;-""< . A 1 ≤ i ≤ ndl A

E U IQ

⎧ ndl ndl dU h (t) ⎪ ⎨ + φj , φi L2 (Ω) i a(φj , φi )Uih (t) = f (t), φi L2 (Ω) dt j=1 j=1 ⎪ ⎩ h Ui (t = 0) = Ui0,h

T Y POL

LE O ÉC

; &< 9.10'+) &) 9.::) Mh ) (Mh )ij = φi , φj L2 (Ω)

1 ≤ i, j ≤ ndl ,

9.10'+) &) 0'('&'1% Kh ) (Kh )ij = a(φi , φj ) 1 ≤ i, j ≤ ndl ,

E U IQ

, . ;-""< . # &!%;5.1',*:

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⎧ h ⎨ M dU (t) + K U h (t) = bh (t), h h dt ⎩ h U (t = 0) = U 0,h

E L O

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' " " (T

h )h>0 ! Ω " V0h 1 H01 (Ω) & ! ! & Pk ! ndl " f (t) ∈ L2 (]0, T [; L2 (Ω)) u0 ∈ H01 (Ω) u ∈ L2 (]0, T [; H01 (Ω)) ∩ C([0, T ]; L2 (Ω)) <# " uh ! <55 V0h " limh→0 u0,h − u0 L (Ω) = 0 2

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N H C )N & * & E % T % 0 Y % FC/H $ L ' % %

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3 . , ,6 ) A 0 &'A%0)*+): @*'): )* 1)93: 'A ( ) # , ( ;-"&< 6 / . 0 I? ' 0 , 6 * ) A . # ;-"&< h ⎧ ⎨ M dU (t) + KU (t) = b(t) dt ;-"-< ⎩ U (t = 0) = U 0

* ) ,# . b(t) [0, T ] @ , . [0, T ] n0 . ∆t = T /n0 tn = n∆t

0 ≤ n ≤ n0 .

N H EC

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@ U n , U (tn ) * 6 ;-"-< θ:+4%9. ; I? . A . ; "<< M

T Y POL

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ÉC

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;-"<

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;-$<

+ , M , A A H A # U n+1 8 U n 0 b ; 8 8 , M A . + &< . A 8 , ' ' I ? . A :+4%9. &) #).0A M

3U n+1 − 4U n + U n−1 + KU n+1 = b(tn+1 ). 2∆t

H C E T

N

E U IQ ;-$<

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Y L O P

>)0+'+) CC " % 0 =>% ? $

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0 ≤ n ≤ n0 = T /∆t,

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8 MU · U = Ω |u|2 dx . u ∈ V0h 8 U 0 V0h ;M 0 ) .< + A 1) - 0 ( ) A ∆x U . , . u L2 (Ω) •

N H EC

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1/2 ≤ θ ≤ 1' θ# ,= " $$ "$$$" "/' "$ ' 0 ≤ θ < 1/2' " "/ $ "$ ;<

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. ) 8 N = 1 ,9 -$ 6 9 , θ6 θ < 1/2 0 ;-$ < 0 . J 0 , 0 G •

E U IQ

%9,*:10.1',*C @ ;-$< 0 M K ;. &< ˜ n+1

( Id + θ∆t diag(λi ))U

HN

= ( Id − (1 − θ)∆t diag(λi ))U + ∆t˜bn , ˜n

C E T Y L PO

;-$<

. M = P P ∗ A K = P diag(λi )P ∗ A U˜ n = P ∗ U n A ˜bn = P −1 (θb(tn+1 )+(1−θ)b(tn )) ˜ n U ˜ n . ) @ ;-$< U i ˜in = (ρi )n U ˜i0 + U

LE O ÉC

n ∆t (ρi )k−1˜bn−k . i 1 + θ∆tλi k=1

;-$<

)' . ρi =

N H EC

1 − (1 − θ)∆tλi . 1 + θ∆tλi

T Y POL

&

E U IQ

˜ n ≤ C * A 1 0 0 U n M = U

: 0 |ρi | ≤ 1 iA , , ;-$ < 0 ≤ θ < 1/2A I 8 θ ≥ 1/2

LE O ÉC

)9.0;5) CC , ;-$< θ A

: . ˜bn−k 9 8 A ? i 0 * A H . 1/2 < θ ≤ 1 θ6 0A 0 θ = 1 ; , < ', 0 . 0 L M 0 , '%6 / •

E U IQ

)9.0;5) CC # , ( ;-"-< L 6

N H EC

M A 8 . # exp(−λi t)A . ( ; mini λi = O(1) maxi λi = O(h−2 )< , , ( : θ6 ' + 6B ;. A A __ T&U< / A # ? . , , , 6 0 A A # ? ; 0 < •

ÉC

E L O

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@ . ) ? ;-"-< 0 θ6

>)0+'+) CC θ% .!/ 1/2 ≤ θ ≤ 1 U M = √ MU · U 6 $ $ * $ * .& T n0 n0 2 n n 0 2 2 ˆ ·U ˆ ≤ C U M + U M + ∆t KU f (t) 2 dt + O(1) . 0

n=0

L (Ω)

E U IQ

- ( .!/ Uˆ n = θU n+1 + (1 − θ)U n

>)0+'+) CC " % ? .

H C E T

N

>)0+'+) CC P1 % $

% .2 N = 1 M * C & $' &

K 2 $ $' " 07 . 2 : ∆t ≤ Ch2

LE O ÉC

Y L O P

) #$

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Pk

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θ

L2 (Ω)

h

∆t

lim

max u(tn ) − un h L2 (Ω) = 0.

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LE O ÉC

T Y POL

R u0,h ∈ V0h u1,h ∈ V0h u0 u1 * ;-$"< A 0 (φi )1≤i≤ndl V0h ;
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N H EC

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E U IQ

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ÉC

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i=1

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bhi (t) = f (t), φi L2 (Ω) , 1 ≤ i ≤ ndl ,

i=1

, . ;-$"< . # &!%;5.1',*: &'A%0)*1')--): ,0&'*.'0): , ? : ⎧ d2 U h ⎪ ⎪ (t) + Kh U h (t) = bh (t), 0 < t < T, ⎨ Mh dt2 ;-$$< h ⎪ ⎪ ⎩ U h (t = 0) = U 0,h , dU (t = 0) = U 1,h , dt R . H 9.10'+): &) 9.::) Mh )1 &) 0'('&'1% Kh , (Mh )ij = φi , φj L2 (Ω) , (Kh )ij = a(φi , φj ) 1 ≤ i, j ≤ ndl .

E U IQ

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N H EC

E U IQ

θ = 0A ;-$-< :+4%9. )>3-'+'1) ; , 8 . M < * 8 P U 0 U 1 A , 0 Q

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U = U0

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d2 U dU (t) + KU (t) = b(t) (t) + C 2 dt dt ¨n U (t), dU/dt(t), d2 U/dt2 (t) U n , U˙ n , U ⎧ ¨ n+1 + C U˙ n+1 + KU n+1 = b(tn+1 ) MU ⎪ ⎪ ⎪ ⎪ ⎨ ˙ n+1 ¨ n+1 + (1 − δ)U ¨ n) U = U˙ n + ∆t(δ U ⎪ ⎪ ⎪ 2 ⎪ ⎩ U n+1 = U n + ∆tU˙ n + (∆t) 2θU ¨ n+1 + (1 − 2θ)U ¨n 2 M

;-$<

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LE O ÉC

Y L O P

H C E T

N

)( *

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&&

+ δ = 1/2 /C% θ6 9 6 A δ A 8 0 ; , .
T Y POL

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LE O ÉC

δ = 1/2( $ 2 δ = 1/2 θ = 1/12( $ δ = 1/2 θ = 1/12 + $

@ 0 1) -$ * . A , 0 > / ;. + < . H $

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max λi (∆t)2 <

N H EC

2 , δ − 2θ

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R 0 # Ch ;. C > 0 3-'+'1)A ,6?6 , ,# # ? ? 9 (A , ? G 0 /C% ;-&< ,H θ = 0 M Q ? 8 , ;. + &< ' . . δ = 1/2 •

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) #$

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A ;-&< .A A ?

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T Y POL

LE O ÉC

. . ; λi . # KVi = λi MVi < ' ;-& < ? . ;. 6 "
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= Ai

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Uin Uin−1

(∆t)2 + 1 + θλi (∆t)2

2 − λi (∆t)2 ( 12 + δ − 2θ) , 1 + θλi (∆t)2

bni 0

a12 = −

. Ai =

Uin+1 Uin

= Ani

Ui1 Ui0

a12 0

,

1 + λi (∆t)2 ( 12 − δ + θ) . 1 + θλi (∆t)2

@

a11 1

E U IQ

n−1 p bn−p (∆t)2 0 i A Ui + . 0 1 + θλi (∆t)2 p=0 i

C E T LY

HN

;-&<

0 > / ρ(Ai ) ≤ 1 @ . Ai #X µ .

O P E L ÉCO

µ2 − a11 µ − a12 = 0

−4λi (∆t)2 + λ2i (∆t)4 ( 12 + δ)2 − 4θ ∆= . (1 + θλi (∆t)2 )2

@ . ) #X 8 ? , . J ∆ ≤ 0 a12 ≥ −1A ∆ > 0 1−a12 ≥ |a11 | 8 ? ;-&< ;. $A A __ T&U<

E U IQ

>)0+'+) CC .&( $ + δ = 1/2 λi (∆t)2 =

4 1−4θ

N

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Ai =

−2 −1 1 0

, Ani = (−1)n

Y L O P

H C E T n+1 n −n 1−n

.

C # $ $ * $ @ A Ani

LE O ÉC

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&

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LE O ÉC

T Y POL

i∈I

R I , 6 A xi (t) , ( d2 xi (t) + λi xi (t) = bi (t) dt2

E U IQ

, 8 A A ' A :53)03,:'1',*: &) 9,&):A . 0 9 ,. ,H .A ,6?6 , E GA A . . # •

N H EC

T Y J %

% ( O'L P " E )G# ! " L % % % & FC/H O C É ' " " +>!! ;, u

<$$ " (Th )h>0 ! Ω " V0h 1 ! & Pk " (∆t) H01 (Ω) & ! ! ? " unh ∈ V0h

U n ! & V0h ! @A!B " limh→0 u0h = u0 L2 (Ω) limh→0 u1h = u1 L2 (Ω) h ∆t <C lim

max u(tn ) − un h L2 (Ω) = 0.

h→0, ∆t→0 0≤n≤n0

E U IQ

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xi pi ≤ P

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R

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inf J(v) ≤ J(w) ≤ θJ(u1 ) + (1 − θ)J(u2 ) = inf J(v) .

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J (x0 ) ≥ 0 x0 = a , J (x0 ) = 0 x0 ∈]a, b[ , J (x0 ) ≤ 0 x0 = b .

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J(x) ≥ J(x0 )A ,R J(x0 ) + hJ (x0 ) + o(h) > J(x0 )A J (x0 ) ≥ 0 . 6 h 8 h . 0 1 H 0 6 J (x0 ) ≤ 0 x0 ∈]a, b] x = x0 − h + ;, , < x0 ∈]a, b[ J .0 x0 A J (x0 ) ≥ 0 ; (A h2 J(x0 ) + J (x0 ) + o(h2 ) ≥ J(x0 ) h F < 2 ,0 J ;x ∈ [a, b] , 6 < x0 ;x0 + h . h > 0 x0 ∈ [a, b[, x0 − h . h > 0 x0 ∈]a, b]< J &'0)+1',*: .&9'::'/-): @ ) . ; 8 5# ? , < ', 8 V . ? ? .0 8 , J ,0 . 0 '*%;5.1',* &!5-)0 Y , , A 0 8 . 95-1'3-'+.1)50: &) .(0.*() 14%,0?9) &) L54* )1 5+D)0 : A # . A , : @ # 08 O &5.-'1% 6 , ? # ( . 2 A " .-(, 0'149): *59%0';5): &!,31'9':.1',* @ (0.&')*1

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$ $ $"$ V ' L ∈ V ' "

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J(u + w) = J(u) + L(w) + o(w)

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L = J (u)

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1 A : . 8 L = J (u) ∈ V , 0 , p = J (u) ∈ V V ) ? V 9 A 8 . , L pA .

T Y POL

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>)0+'+) C C " ; J u " ( L1 , L2

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J(u + w) ≥ J(u) + L1 (w) + o(w) , J(u + w) ≤ J(u) + L2 (w) + o(w) ,

;<

J L1 = L2 = J (u)

>)0+'+) C C $)::)*1')- P6 ) a : V ×

V ) L V J(u) = 12 a(u, u) − L(u) " J V J (u), w = a(u, w) − L(w) u, w ∈ V

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RN $ u, v = (∇u · ∇v + uv) dx.

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Ω

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a(u, v) = Ω ∇u · ∇v dx( L(u) = Ω f u dx f ∈ L2 (Ω) " J (u) = −∆u − f V = H −1 (Ω) " ( V V ( J (u) = u0 4 u0 $ H01 (Ω)

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Ω ∂Ω

−∆u0 + u0 = −∆u − f u0 = 0

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N = 1 Ω =]0, 1[ ) L = L(p, t, x) RN ×R×Ω( ∂L % # + p t ( ∂L ∂p ∂t V = H01 (Ω) J(v) = L(∇v(x), v(x), x)dx Ω

" J H01 (Ω) J (u), w =

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2 ) N = 1 Ω =]0, 1[( ( u ∈ H01 (0, 1) J (u) = 0( u

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∂L ∂L ;< u (x), u(x), x − u (x), u(x), x = 0 , ∂p ∂t $ ]0, 1[ ) L x L = L(p, t) u ∈ C 2 (]0, 1[) d dx

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J(v) ≥ J(u) + J (u), v − u J (u) − J (v), u − v ≥ 0

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∀ u, v ∈ V .

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α v − u2 2

J (u) − J (v), u − v ≥ αu − v2

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∀ u, v ∈ V ,

∀ u, v ∈ V .

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9 ) 8 . J(v) I u ; 8 : v < ;< ;-< # . 9 A J(u) = 12 a(u, u) − L(u) . a 8 0 # V L 8 V A ,9 ;< ) . a •

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%9,*:10.1',*C : * " 0. α = 0 * 7 ;< ;< ' J α6.A . ) ; < A k ≥ 1A 1 1 1 1 1 α J 1 − k u + k v ≤ 1 − k J(u) + k J(v) − k+1 1 − k u − v2 , 2 2 2 2 2 2

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' & 1 α 1 1 − k u − v2 . 2 J u + k (v − u) − J(u) ≤ J(v) − J(u) − 2 2 2 k

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9 8 k . +∞A . ;< * 0 ;< : , 6 ;< . 6H u v 7 ;< ;< * u, v ∈ V t ∈ RA ϕ(t) = J(u+ t(v−u)) 3 ϕ .0 RA ϕ (t) = J (u+t(v−u)), v−uA A , ;<

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ϕ (t) − ϕ (s) ≥ α(t − s)v − u2

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; <

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θ ∈]0, 1[ 9 , ; < t = θ ? t = 1 s = 0 ? s = θA 0 αθ(1 − θ) v − u2 , θϕ(1) + (1 − θ)ϕ(0) − ϕ(θ) ≥ 2 ,6?6 ;<

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>)0+'+) C C " $ J V

J(v) > J(u) + J (u), v − u ∀ u, v ∈ V

J (u) − J (v), u − v > 0 ∀ u, v ∈ V

u = v ,

u = v .

5 6 ) &%0'2%) :)+,*&) J + ,0 , 8 1) ( 0 , 8 f ) V ? . = 0 W ; R< @ f ( 0 ; 2< u , L V W

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f (u + w) = f (u) + L(w) + o(w)

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o(w) =0. w→0 w

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J " $ $"$ & -/ V $ R' -*

1 J(u+w) = J(u)+J (u)w + J (u)(w, w)+o(w2 ), 2

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9 , J (u)(w, w) ,

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&

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V J(u) = 12 a(u, u) − L(u) " J

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J 8 .0 . . J . .A 8 .

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x − xK = min x − y.

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inf Ax − b2 .

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Ax − b = B ∗ p p ∈ Rm .

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1 2

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f v dx

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a(u, v − u) ≥ L(v − u) ∀ v ∈ K .

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K ⊂ V J1 " u ∈ K J1 + J2

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J1 (u), v − u + J2 (v) − J2 (u) ≥ 0

∀v ∈ K .

.A 5 A . , L M . ? 6 .

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ai , v = 0

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1 ≤ i ≤ M } ,

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J (u), w ≥ 0 ∀ w ∈ K .

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∃ λ1 , . . . , λM ≥ 0 ,

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n

v

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LE O ÉC

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lim wn = w ,

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E L O

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* 0 5 -A , ,9 $

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H C E T

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A . 0 B mA . x J x = A−1 (b + B ∗ p)A Bx = 0 0 −1 −1 p = − BA−1 B ∗ BA−1 b x = A−1 Id − B ∗ BA−1 B ∗ BA−1 b.

LE O ÉC

Y L O P

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9 A . p B m , A , p BA−1 B ∗ p = −BA−1 b , ,? , , . # B ∗ @ . ,9

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K = {v ∈

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v dx = 1}

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|∇v|2 dx . min J(v) = v∈K

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Ax·x≤1

sup b · x

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H C E T

N

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$ H <% ;2. $ 2 $ : $ n A ≥ B Ax · x ≥ Bx · x x ∈ Rn 6 ( A ≥ B ( B −1 ≥ A−1

LE O ÉC

Y L O P

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>)0+'+) CC ' % * # * # $ f (v) v ∈ RN ( * # ρ(

u( T ( *F f (v)

LE O ÉC

T Y POL

f (v) dv ,

ρ=

RN

ρu =

v f (v) dv ,

RN

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RN

|v|2 f (v) dv .

; -<

V # $ H(f )

f (v) log f (v) dv .

H(f ) = RN

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ρ (2πT )N/2

|v − u|2 exp − . 2T

T Y POL

N H EC

E U IQ

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L(v, µ) = J(v) +

M

µi Fi (v) = J(v) + µ · F (v)

i=1

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,

∂L (u, λ) = 0 , ∂µ

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/ *%+)::.'0) ,

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0,3,:'1',* CC $ ) #!)"#+ A#+ ,, " $ ) ) $"$ J " F1 , ..., FM $" & $"$4$" -/ " -" Fi (u) 1≤i≤M $" $$" $ )$ $", " λ ∈ RM ")"

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; <

i=1

%9,*:10.1',*C , 0 C 2 A ,6?6

8 t → u(t) [0, 1] V u(0) = u F (u(t)) = 0 t ∈ [0, 1] * ) A . u (0) X 0 K(u) @ j(t) = J u(t) fi (t) = Fi u(t) 1 ≤ i ≤ M. 9 . 0 j (t) = J u(t) , u (t) j (t)

=

fi (t) =

fi (t) = Fi u(t) , u (t) 1 ≤ i ≤ M,

J u(t) u (t), u (t) + J u(t) , u (t)

N H EC

E U IQ

Fi u(t) u (t), u (t) + Fi u(t) , u (t) 1 ≤ i ≤ M.

T Y POL

' fi (t) = 0 t 0 j(t)A j (0) = 0A j (0) ≥ 0A fi (0) = fi (0) = 0 fi (0) = 0 u (0) 6 Fi (u) 1≤i≤M ; ? K(u) 8 0
ÉC

E L O

J (u) +

M

λi Fi (u) = 0.

;<

i=1

fi (0) = 0 0=

M

λi Fi (u) u (0), u (0) + Fi (u), u (0) ,

i=1

j (0) ≥ 0

J (u) u (0), u (0) + J (u), u (0) ≥ 0.

H C E T

N

E U IQ

4Q ? ;< . u (0) 0 ; < M λi Fi (u) + J (u) u (0), u (0) ≥ 0,

LE O ÉC i=1

Y L O P

$

- # %

N H EC

E U IQ

, , ; < u (0) K(u) , 0 u(t) 8 ,0 u (0) X 0 K(u) , Q ? ,# 8 8 Fi (u) 1≤i≤M 0 ; <

LE O ÉC

T Y POL

>)0+'+) CC 0 $ ' / /& ,*10.'*1): &!'*%(.-'1% 1 K K = {v ∈ V

,

Fi (v) ≤ 0

1 ≤ i ≤ M } ,

;<

E U IQ

R F1 , . . . , FM I 8 V R , . X 0 K(v)A ;< I H X v R , K(v) 9 (A Fi (v) < 0A A : A Fi (v+w) ≤ 0 ; i . v < Fi (v) = 0 iA , , .

. w ∈ V A > 0 : A (v + w) 8 ;< . 8 A +,*&'1',*: &) ;5.-'@+.1',* 4 A . , 8 L. M , v ) ( # ) ; < / ) 0 -'*%.0':% , 0 8 . ' L . M . ) .

ÉC

E L O

T Y POL

N H EC

" u ∈ K , $/ I(u) = {i ∈ {1, . . . , M } , " )) $/ $"$" .+1'2): $ u,

%@*'1',* CC

Fi (u) = 0}

%@*'1',* CC $ " $"$" ,3 $" ;5.-'@%): $ u ∈ K " $" &" $ "$ w ∈ V " $ " ) " " i ∈ I(u) /$ Fi (u), w < 0 , ; < /$ Fi (u), w = 0 " Fi " 6$ .

H C E T

N

E U IQ

)9.0;5) CC w L M ; < u + w ∈ K ≥ 0 : E GA 8 Fi :A w = 0 ) 8 :

LE O ÉC

Y L O P

- #

N H EC

&

E U IQ

1) I ) 6 ) A , • : ; ' <

T Y POL

/ . *%+)::.'0): , ,0 ;<

$ )) " $$ ) ,3' $"$ " L E$" -/ O $"$" $" * $ , ' É C " $ $ $ " ' &" ' )) ")" 4%,0?9) CC

K

F1 , . . . , FM u

J

0$0' "

J (u) +

J

u

M

λi Fi (u) = 0,

u

λ1 , . . . , λM ≥ 0

K

λi ≥ 0,

λi = 0

Fi (u) < 0

∀ i ∈ {1, . . . , M }.

;<

i=1

E U IQ

)9.0;5) CC @ ;< 8 . J (u) +

M

λi Fi (u) = 0 ,

N H EC

λ≥0 ,

T Y POL

i=1

λ · F (u) = 0 ,

R λ ≥ 0 ) . λ = (λ1 , ..., λM ) .A A i ∈ {1, . . . , M }A Fi (u) = 0A λi = 0 8 λ · F (u) = 0 •

ÉC

E L O

%9,*:10.1',*C ' ,0 ,0 ˜ K(u) = {w ∈ V

,

Fi (u), w ≤ 0 ∀ i ∈ I(u)} .

;<

˜ , X K(u) 0< ;@ K(u) ˜ A δ > 0 / w 0 8 ; 0 F 8 )

i∈ / I(u)A Fi (u) < 0 Fi (u + ε(w + δw)) < 0 ε F i ∈ I(u) Fi (u), w < 0A Fi (u + ε(w + δw))

= Fi (u) + εFi (u), w + δw + o(ε)

H C E T

≤ εδFi (u), w + o(ε) ,

,R Fi (u + ε(w + δw)) < 0 ε > 0 F

Y L O P

N

E U IQ ;"<

9)A i ∈ I(u) Fi (u), w = 0A Fi :

OLE

Fi (u + ε(w + δw)) = Fi (u) + εFi (u), w + δw = εFi (u), w ≤ 0 .

ÉC

;$<

-

- # %

N H EC

E U IQ

2 A u J K A ˜ J (u), w + δw ≥ 0 ∀ w ∈ K(u) ,

T Y POL

∀ δ ∈ R∗+ .

˜ ' J (u), w ≥ 0 ∀ w ∈ K(u) Q 2% & 6

E L O É C + $/

)99) CC $&) .0D.:6

"

$" a1 , . . . , aM $" *& V , $ $

) * K = v ∈ V , ai , v ≤ 0 ) 1 ≤ i ≤ M ,

M * ) ˆ = v ∈ V , ∃ λ1 , . . . , λM ≥ 0 , v = − K λi ai .

) " " p ∈ V ' $ )"$

i=1

N H C ) "$" - $"' 0" $ " $ -$, E T Y L O P E L O ÉC ˆ. p, w ≥ 0 ∀ w ∈ K =⇒ p ∈ K

E U IQ

%9,*:10.1',*C 'O Kˆ 8 * . M = 1A , . 0 . ai 8 ? M

. (ai )1≤i≤M ,0 M n ˆ ; . λn ≥ 0 ∀ i ∀ n)A (v ) = − i=1 λni ai , K i . . v ∈ V 3 (λni ) . R+ . λi ≥ 0 ; 1 ≤ i ≤ M < . (ai )1≤i≤M ˆ 8 0 , , @ v = − M i=1 λi ai ∈ KA 8 . (ai )1≤i≤M A , : µi 8 M i=1 µi ai = 0A M ˆ * t ≤ 0A 8 v = − i=1 λi ai K M (λi + tµi )ai A t ≤ 0 v = − i=1

H C E T

N

E U IQ

λi + tµi ≥ 0 ∀ i ∈ {1, . . . , M } ∃ i0 ∈ {1, . . . , M } , λi0 + tµi0 = 0 .

' ˆ= K

M ) +

LE O ÉC i0 =1

Y L O P

v ∈ V , ∃ λ1 , . . . , λM ≥ 0 , v = −

i=i0

* λi ai .

;&<

- #

N H EC

E U IQ

* # A 0 0 ˆ ;&< 8A H K + ,0 J p, w ≥ 0 ∀ w ∈ K ˆ @ 5 , , p∈ / K ˆ 8 A ? ,. A . . . p K w = 0 V α ∈ R

LE O ÉC

T Y POL

;-<

ˆ. p, w < α < w, v ∀ v ∈ K

ˆ Y , A i ∈ {1, . . . , M } 7 A . α < 0 0 ∈ K . ;-< v = −λai . λ 0 A w, ai ≤ 0 @ 0 w ∈ K p, w < α < 0A 0

>)0+'+) CC ) A : $ n( B

m×n m ≤ n * m

1 J(x) = Ax · x − b · x , Bx≤c 2

min n

x∈R ,

N H EC

E U IQ

G <% ;2! $ $ * * p ∈ Rm $ x

T Y POL

Ax − b + B ∗ p = 0 ,

E L O

p≥0,

p · (Bx − c) = 0.

>)0+'+) CC ) f ∈ L2 (Ω) Ω -

ÉC

> 0 * min |∇u|2 dx. 1 u∈H0 (Ω) , u−f L2 (Ω) ≤

Ω

" u " ( u = f ( u = 0( λ > 0 u

−∆u + λ(u − f ) = 0 u = 0

Ω, ∂Ω.

,*10.'*1): &!%(.-'1% )1 &!'*%(.-'1%

E U IQ

@ 0 G # @ K K = {v ∈ V , G(v) = 0 , F (v) ≤ 0} , ;<

H C E T

N

R G(v) = (G1 (v), ..., GN (v)) F (v) = (F1 (v), ..., FM (v)) V RN RM 1 . A 8 ) ) @ I I(u) = {i ∈ {1, . . . , M } , Fi (u) = 0} ,0 , . u ∈ K

LE O ÉC

Y L O P

- # %

E U $ " $"$" ,3 $" $ IQ N " $" -" $" $$" $ )$ $" " &" H C

$ "$ " $ " ) " " E T Y L O P LE O ÉC . " $$ ) ,3, $ )) " %@*'1',* CC

;5.-'@%):

Gi (u) 1≤i≤N ,N - .⊥ w ∈ i=1 Gi (u)

u∈K

i ∈ I(u)

Fi (u), w < 0 .

;<

/ . *%+)::.'0): , ,0 ;<

4%,0?9) CC

u ∈ K

K

J

λi Fi (u) = 0 ,

λ ≥ 0 , F (u) ≤ 0 , λ · F (u) = 0 . ;<

" F $" -/ $ u' G " -/ $ $ -$0 u' " $"$" $" * $ u $ 8*$"$ ,,, ' u " $ $ J K ' &" ")" 0$0 µ1 , . . . , µN ' " λ1 , . . . , λM ≥ 0' " J (u) +

N

µi Gi (u) +

i=1

M i=1

N H EC

E U IQ

5 5 - "A ,

T Y POL

510): =,09): &): +,*&'1',*: &) ;5.-'@+.1',*

E L O

) :5J:.*1): # L 6 M 8 . ? ,0 K ? , u ∈ K ) 1) F ; ,H
ÉC

)9.0;5) CC 1 , A , ; ? + , < ) P ) 1) 9 (A u ∈ K . LI M ? . i ∈ I(u) I , V @ . ) . ; u , Fi (u) = 0 i ∈ I(u)< (Fi (u))i∈I(u) 8 0 ; <

H C E T

N

E U IQ

P ;
LE O ÉC

Y L O P

- &. /

/ .

N H EC

E U IQ

)9.0;5) CC 1 0 , , A

, ) 1) - ' L8M ;,6?6 . . )< ) + Gi (u) 1≤i≤N ;< (Fi (u))i∈I(u) 8 0

LE O ÉC

T Y POL

@ . ) 8 ;< P ;
)9.0;5) CC +. , A .A

) 0 . @ , v ∈ V , A i ∈ {1, . . . , M }A 8 Fi . A 0 Fi (v) < 0 , 0 Fi (v) = 0 Fi

;<

:

N H EC

E U IQ

,# ;< P ) u ∈ K 1) 9 (A i ∈ I(u) Fi (v) < 0A A , . ;&<

T Y POL

Fi (u), v − u = Fi (u) + Fi (u), v − u ≤ Fi (v) < 0 .

E L O

1, A i ∈ I(u) Fi (v) = 0A Fi :

ÉC

Fi (u), v − u = Fi (v) − Fi (u) = 0 ,

1) ) 8 . w = v − u ,. ,# ;< P u . 8 F1 , . . . , FM •

6

4(7 8 &97 (

/ . . + 0 , (u, λ) ; A < 3,'*1 :1. 1',**.'0) &5 .(0.*(')* L / 3,'*1:)--) 8 ; . A #A . *%+)::.'0): :5J:.*1):< / 0 . 14%,0') &) -. &5.-'1% @ , H A H . " ) 6 I X 8 14%,0') &): <)5>

LE O ÉC

Y L O P

H C E T

N

E U IQ

- # %

@

! =

N H EC

E U IQ

1 0 A V Q = 0 A L V × Q ; , U × P V × Q< R 1 5 - ; ,
T Y POL

LE O ÉC

%@*'1',* CC

$ " (u, p) ∈ U × P " $ )$" L U × P

∀q ∈ P

L(u, q) ≤ L(u, p) ≤ L(v, p)

;"<

∀v ∈ U .

. 6 0 . , ; < , ;< * )A . . A 8 q ≥ 0 q ∈ (R+ )M

N H EC

E U IQ

$ )) $"$ J, F1 , . . . , FM $" $"$ " *$ ) ,9 ,3, $ $" P = RM $ $"$" 0" ,9 " P = (R+ )M $ $"$" $0" ,3, " U $ -" V $"$$" K , (v, q) ∈ U × P ' $ ) L(v, q) = J(v) + q · F (v), " (u, p) $ )$" L U × P , u ∈ K " u " $ $ 0/ J K , 8 ) ' J " F1 , . . . , FM $" -/ $ u' $ 0,3,:'1',* CC

V ' " $/ K

ÉC

E L O

T Y POL J (u) +

M

pi Fi (u) = 0 .

;$<

i=1

%9,*:10.1',*C . 6 ∀q ∈ P

J(u) + q · F (u) ≤ J(u) + p · F (u) ≤ J(v) + p · F (v) ∀ v ∈ U .

;&<

9 ,0 , * P = R A ;&< F (u) = 0A u ∈ K J(u) ≤ J(v) + p · F (v) ∀ v ∈ U A 0 ; v ∈ K < u 0 J K 1 , A P = (R+ )M ;&< F (u) ≤ 0 p · F (u) = 0 ' . u ∈ K A 8 u 0 J K 9)A J F1 , . . . , FM .0 uA ;&< u J + p · F , . U A . , uA J (u) + p · F (u) = 0 ;8 + < M

LE O ÉC

Y L O P

H C E T

N

E U IQ

- &. / @

/ .

N H EC

2 A 2

E U IQ

/ . 0 , 6 ,0 K ;
LE O ÉC

T Y POL K = {v ∈ V ,

Fi (v) ≤ 0

1 ≤ i ≤ m} .

;-<

5 " , 1 6 . :5J:.*1) 8 G +,*2)>): 9 (A * : A (u, p) 6 A u J K * 0 . . , .A 0 A ,6?6 A u J K A p ∈ (R+ )M (u, p) 6 @ J, F1 , . . . , FM . V

E U IQ

)9.0;5) CC ' J, F1 , . . . , FM . A K . 8 , , 0 J K 5 & K . L ) ? , ) M ;< . ) •

N H EC

B 5 % ; 8 B A B 5 %< : A .A ,

5 " 8 *%+)::.'0) )1 :5J:.*1)

T Y P O L $ )) $"$

J' $" $-& $"$ V " -/ $/ K ,9, $ $" " 0$0$ L

E L O

4%,0?9) CC $&) L54* )1 5+D)06 F1 , . . . , FM

ÉC

L(v, q) = J(v) + q · F (v)

∀ (v, q) ∈ V × (R+ )M .

" u ∈ K $ )$" K . $"$" $" * $ 8*$"$ ,,3, u " $ $ 0/ J K " $" &" p ∈ (R+ )M " (u, p) " $ )$" 0$0$ L V ×(R+ )M ' $+ -$"' " F (u) ≤ 0 , p ≥ 0 , p · F (u) = 0 , J (u) +

M

pi Fi (u) = 0 .

;<

i=1

E U IQ

%9,*:10.1',*C u J K A 56

N

"A , ;
Y L O P

H C E T

)9.0;5) CC 5 B 5 % , ,

LE O ÉC

, A , A ' A

- # %

N H EC

E U IQ

0 &!%(.-'1% .J*): Av = b . , 8 , ;: .< Av − b ≤ 0 b − Av ≤ 0 ', . , 5 B 5 % ? 0 . , : •

T Y POL

, . , pi 0 . J . Fi J 6 A : G A 8 ? A

LE O ÉC

>)0+'+) CC $ ( inf

Fi (v)≤ui , 1≤i≤m

;"<

J(v),

u1 , . . . , um ∈ R %: % <% ; N% <= m∗ (u) ;!;

E U IQ

" p * * $ + ;!; u = 0( ∗

T Y L ∗

N H EC

;"<

m (u) ≥ m (0) − pu .

O P E L ÉCO

2 6 ;! u → m∗ (u) ( pi = −

∂m∗ (0). ∂ui

L $' /.

@ * . '* 2 ( ' 4 " ' % % 4 % & ' ' (( (u, p) L(v, q) = J(v)+q ·F (v) ? % " & (u, p) % 0 ' ' ' ! " 0 4 " " & ' & ! A% . C

E U IQ

N H C $# " * E & T Y 1 L O /! P E L O ÉC U ×P

V V ×Q

∀q ∈ P

Q

L

(u, p)

L(u, q) ≤ L(u, p) ≤ L(v, p)

∀v ∈ U .

L

U ×P

- &. / . v ∈ U q ∈ P

N H EC

J (v) = sup L(v, q)

T Y POL q∈P

/ .

E U IQ

"

/C!

G(q) = inf L(v, q) . v∈U

;! ! ;! ! !

OLE

v∈U

/#!

sup G(q) .

//!

inf J (v) ,

;! ;! ! ! C É q∈P

$# @ - ( ( % " J (v) = % v " G(q) = −∞ % q ? (u, p) . CN " J G ' {v ∈ U , J (v) < +∞} {q ∈ P , G(q) > −∞}

+∞

E U IQ

" * ' ! % " /! " J (u) = G(p) = L(u, p) ' ' • % " ' (u, p)

N H %& $#" ( , C 1 E ! T Y L O P $#!LE . /C! /N! " % & ÉCO L

(u, p)

U ×P

J (u) = min J (v) = max G(q) = G(p) . v∈U

q∈P

J

J (u) = min v∈U

sup L(v, q)

q∈P

q∈P

/N!

G

= max

inf L(v, q) = G(p) .

v∈U

/G!

1 * /G! & " % ! % " /G! ' ( " L 5 * " * L ! " " % & • 1 (u, p) L U × P 4 L∗∗ = L(u, p) v ∈ U /C! " J (v) ≥ L(v, p) : J (v) ≥ L /! 5

J (u) = L∗ " J (u) = inf v∈U J (v) = L∗ $ *2 " G(p) = supq∈P G(q) = L∗ A " " /N! L∗ = J (u) /C! J " L(u, q) ≤ J (u) = L∗ ∀ q ∈ P . /)! . K L(v, p) ≥ G(p) = L∗ ∀ v ∈ U , /I! * /)! /I! " L(u, p) = L∗ " " (u, p)

LE O ÉC

Y L O P

H C E T

N

E U IQ

- # %

$

E U $#$ ? I Q % . N H C E N ! T Y L 0 + P O E " % L " N ! + %! ' O É C N ! L

inf

v∈U

supq∈P supq∈P

v ∈ U inf v ∈U L(v , q) inf v ∈U L(v , q)

U ×P

sup L(v, q)

≥ sup

q∈P

q∈P

inf L(v, q) .

v∈U

q ∈ P L(v, q) ≥ inf v ∈U L(v , q) supq∈P L(v, q) ≥ v ∈ V inf v∈V supq∈P L(v, q) ≥ •

/0 $#

3"" " 4,, ! &! .

/0 $## 1 U !& P !& ! ! & V !&% Q (

3"" v → L(v, q) !& U q ∈ P q → L(v, q) !!& P v ∈ U 2 !

L U × P

E U IQ

N H C % E 4 " ' T % % Y L O N! P E L % $ % C O É 33-'+.1',*

inf

v∈V , F (v)≤0

J

F = (F1 , . . . , FM )

J(v)

V

L(v, q) = J(v) + q · F (v)

∀ (v, q) ∈ V × (R+ )M .

. % * " v ∈ V J (v) =

sup q∈(R+ )M

L(v, q) =

J(v) +∞

F (v) ≤ 0 ,

N!

" " ' inf v∈V J (v) ' N! S . * G(q) ' ' /C! /C! ' % . G(q) * %

* % 0 IC! " '

E U IQ

N H C ' % " T ' N! E S 5

( Y L

" * ( ,EV! ,O ' D( W ( D 3 C# C) P % E L O ÉC sup

q∈(R+ )M

G(q) ,

- &. /

/ .

E U IQ

5 $# J, F , . . . , F

HN 1

&

M

V " u ∈ V F (u) ≤ 0 & u ' & #$ 2 u ! !! J V p ∈ (R+)M

p ! !! G (R+)M # (u, p) 1 L V × (R+ )M $ (u, p) ∈ V × (R+)M & ! > F (u) ≤ 0 , p ≥ 0 , p · F (u) = 0 , J (u) + p · F (u) = 0 . NC! 5 C % 1 " ' * & " ' ! u ' p ' . " (u, p) & " u '

O P LE O ÉC

C E T LY

E U Q " % ( ( * I N ' " ' H

5 C ; ( ( * %

& C E T Y C L $# 5 O P " " % ' E

L O N#! ÉC min L(v, p) . v∈V

u

J

RN

)

min

v∈RN , F (v)=Bv−c≤0

J(v) =

1 Av · v − b · v 2

*

,

: A N × N " % b ∈ RN B M × N c ∈ RM L(v, q) =

1 Av · v − b · v + q · (Bv − c) 2

N/!

∀ (v, q) ∈ RN × (R+ )M .

4 % ;& * N! J " ' N#! 0 ' q ∈ (R+ )M ' min L(v, q)

v∈RN

E U IQ

" " v → L(v, q) * * % 5 % ∂L (v, q) = Av − b + B ∗ q = 0 v = A−1 (b − B ∗ q) $ ' ∂v

H C E T

G(q) = L A−1 (b − B ∗ q), q ,

'

Y L O P

1 1 sup − q · BA−1 B ∗ q + (BA−1 b − c) · q − A−1 b · b 2 2 q≥0

LE O ÉC

.

N

NN!

-

- # %

E U 5 * & NN! I Q (" 6 ' % * " " N H 5 5 C " C $ % " * " *

E T %! ? % ' NN! Y L * % * " O ' ' " P M % & 1 /C " E %

(

L ' O ÉC B

BA−1 B ∗

M

(q ≥ 0)

•

D * " ' ( ;

/0 $# 1 ! !" ⎛ ⎜ ⎜ ⎝

A=⎜

1 −3 −4 −2 −1

0 2 2 4 2

4 −1 −2 −1 −6

2 2 0 6 3

3 −5 −1 −2 −1

5 2 2 2 1

⎞ ⎟ ⎟ ⎟. ⎠

N H EC

E U IQ

(

5 ! " i ! j ! ! 6 & ! " " 5 !7! aij ! A 5 !&

−aij 2 " ! ' % 3 5 % &! ! ! A 8

E L O

T Y POL

C É . " "

6

#

1 "

&

( ( (( @

5( / % % " ' %

−∆u = f Ω NG! ∂Ω, u=0

E U IQ

: Ω % ' RN f ∈ L2 (Ω) " % &

min

v∈H01 (Ω)

J(v) =

1 2

Ω

|∇v|2 dx −

H C E T

Y L O P

f v dx

Ω

N

N)!

% /G! 4 % % & 0 I) " N)! "

& 0 # " " 0 * % NG! (" N)! % NG!

LE O ÉC

-

E U

* ' " I Q * ! " N

" H * * $ " C ( & NG!

0& E ; Y T " ! (" " N)! O L 4 & N)! ' " P E * L N)! " %

O %' C & É u

f

e ∈ L2 (Ω)N

e = ∇v

1 J˜(v, e) = 2

min

v∈H 1 (Ω) , e∈L2 (Ω)N 0 e=∇v

|e|2 dx −

Ω

f v dx .

Ω

$ '

˜ e) + M(e, v, τ ) = J(v,

τ · (∇v − e) dx , Ω

% τ ((

∈ L2 (Ω)N

L(v, τ ) =

N H EC

M(e, v, τ ).

min

e∈L2 (Ω)N

E U IQ

$ e '

5

e → M(e, v, τ ) * % "

* "

T Y POL

1 L(v, τ ) = − 2

E L O

2

|τ | dx −

Ω

Ω

NI!

τ · ∇v dx.

f v dx + Ω

$ % " ' NI! ' N)!

ÉC

max

L(v, τ )

⎧ ⎨

τ ∈L2 (Ω)N

" '

min

v∈H01 (Ω)

L(v, τ )

1 = G(τ ) = 2 ⎩ −∞ −

= J(v) ,

− divτ = f Ω

|τ |2 dx

Ω

G !

$

%& $ : (u, σ) L(v, τ ) H (Ω) ×

E U IQ 1 0

L2 (Ω)N

L(u, σ) =

min L(v, τ ) =

max

τ ∈L2 (Ω)N v∈H01 (Ω)

min

max

v∈H01 (Ω) τ ∈L2 (Ω)N

L(v, τ ).

H C E T

N

2! u 2 N ! !! J(v) H01 (Ω) σ ! !! G(τ ) L (Ω) J(u) =

Y L O P

min J(v) =

v∈H01 (Ω)

LE O ÉC

σ = ∇u

max

τ ∈L2 (Ω)N

G(τ ) = G(σ),

- # %

E U $ ' G ! 5

Q(" I ! N H ' ( " C . & % " ' * E T * =

Y " ' O L , " D( # " ; P " ' ' E A " " L O ÉC max G(τ ) = − min(−G(τ )) 1 2

Ω

|τ |2 dx

−divτ = f

Ω

v

−divτ = f

G(τ ) ≤ J(v)

J(u) = G(σ) =

1 2

f u dx

Ω

" " % * • % "

5 C " "

! ( ! & ' $ ' G ! " −divτ = f " x ∈ Ω 4 % * % " 0 +

N H EC

G(τ ) ≤ L(v, τ ) ≤ J(v),

T Y POL

E U IQ

" ' "

u σ % $

u NG!

"

ÉC

E L O

J(u) =

1 2

|∇u|2 dx −

Ω

f u dx = − Ω

1 2

|∇u|2 dx = − Ω

1 2

f u dx . Ω

1 σ = ∇u NG! " −divσ = f ' G(τ ) ≤ J(u) = G(σ),

& " σ

G (u, σ) L(v, τ ) @ ' $ 0 II '

" " $ +

! y(t) & % RN dy = Ay + Bv + f dt y(0) = y0

0 ≤ t ≤ T

Y L O P

H C E T

N

E U IQ G!

: y0 ∈ RN f (t) ∈ RN v(t) ∈ RM

" A B % N × N N × M

LE O ÉC

-

E U $ % (

& I Q " " N H C E T Y L 8'9 :

; O 8'9 " % % A " " * P E %' & % G! L % " ( ( O C ( B' * É ' 8 9 * ( v

J(v) =

1 2

T

Rv(t) · v(t)dt +

0

z(t)

1 2

T

0

1 Q(y − z)(t) · (y − z)(t)dt + D (y(T ) − zT ) · (y(T ) − zT ) , 2

zT

R, Q, D

R

y(t)

v

L2 (]0, T [; RM )

v

]0, T [

RM

B'! %

% * % K RM " '

' ' inf J(v). G! v(t)∈L (]0,T [;K) 2

5

2 % " G! '

N H EC

E U I Q 2 C

$ # f (t) ∈ L (]0, T [; R ) v(t) ∈ L (]0, T [; K) 2

2

N

! y(t) ∈ H (]0, T [; R ) [0, T ] 5 ' f v 6 L2 $ * t 1

ÉC

E L O

N

T Y POL

exp (t − s)A (Bv + f )(s) ds

y(t) = exp(tA)y0 +

0

" % y H 1 (]0, T [; RN )

#CC " y [0, T ] $

' $ : u ∈ L (]0, T [; K) ! ! C# - !1 2

! ! u

T

Q(yu − z) · (yv − yu )dt 0

T

Ru · (v − u)dt

+ 0

+D(yu (T ) − zT ) · (yv (T ) − yu (T )) ≥ 0 ,

GC!

E U IQ

v ∈ L2 (]0, T [; K) % yv C !! v $

" " v → y * 0 + G! yv = y˜v + yˆ : y˜v Y L O P

d˜ yv = A˜ yv + Bv dt y˜v (0) = 0

LE O ÉC

H C E T

0≤t≤T

N

G#!

- # %

yˆ

N H EC

dˆ y = Aˆ y+f dt yˆ(0) = y0

T Y POL

0 ≤ t ≤ T

E U IQ

6 " yˆ v v → y˜v L2 (]0, T [; K) H 1 (]0, T [; RN ) " v → J(v) * " " % v

* " " * ! J % * % R % 5

L2 (]0, T [; K)

% * % D( IN & &

u G! . D( J (u), v − u ≥ 0 ( -

LE O ÉC

lim

→0

J(u + w) − J(u) = J (u), w.

5

J(v) " " " yu+w = yu + ˜yw $ ' GC! " " yu − yv = y˜u − y˜v

E U Q! " I

N

(

" % 1 /! $ '

H C E T Y G/! L O P E : * L " " O C É GC! * ' S 0 $ 0 G! * J (w)

w ∈ L2 ]0, T [

T

J (w)v dt

0

u

T

T

Rw · v dt +

=

0

+D(yw (T ) − zT ) ·

Q(yw − z) · y˜v dt

0 y˜v (T )

,

L2 ]0, T [

v

•

+ u (" * v yv , *2 % ' '

J (u) & G/! & 6 " * (

< ?

A " #) ! ' G! ; p

" dp = −A∗ p − Q(y − z) dt p(T ) = D(y(T ) − zT )

0 ≤ t ≤ T

E U : G!

; % " IQ N ; " ; " = GN! H ' C E T %& $ Y L O GG! P E L O ÉC y

u

A∗

J (u)

J

u

J (u) = B ∗ p + Ru .

GN!

-

E U D ! > ;! C# IQ N H C G)! E T LY $ PO * GG! " & GN!

&

E D( #N L O ; GN! 5 + * % * C * G! % É G/! " ("

T

(B ∗ p + Ru) · (v − u) dt ≥ 0

∀ v ∈ L2 (]0, T [; K).

0

w ∈ L2 ]0, T [

∗

B p + Rw w

p

J (w) =

y

v

v

•

1 p GN! y˜v G#! GN! y˜v G#! p " % + D ' y˜v (0) = 0 p(T ) = D(y(T ) − zT ) T d˜ yv dp dt = D(y(T ) − zT ) · y˜v (T ). GI! · y˜v + p · dt dt 0

. " '

N H EC

E U IQ

) ! T Y L O $ PGI! ) ! G/! % E L O C É " GG! G)! 0

T

d˜ yv dp · y˜v + p · dt dt

T

T

dt = −

0

J (u)v dt =

T

Q(y − z) · y˜v dt +

0

Bv · p dt.

0

T

T

Ru · vdt +

Bv · pdt,

0

0

$ " 5

' * ' GN! " ; J (v) X 0 * ' G! $ " G!

%' v y

J(v) " p &

L(v, y, p)

T

Rv(t) · v(t)dt +

= 0

T

Q(y − z)(t) · (y − z)(t)dt 0

+D (y(T ) − zT ) · (y(T ) − zT ) + −p(0) · (y(0) − y0 ) ,

T

p· − 0

E U IQ

dy + Ay + Bv + f dt dt

H C E T

N

: p G! v y J

G! ' " & "

LE O ÉC

Y L O P

∂L ∂L ∂L = = = 0. ∂v ∂y ∂p

- # %

E U % GG! I Q " % ; " % 6 * N H " * " 8' 9 " C ; E T Y D( #N L ' " % O % 0 #! (

P

" E

G! ( 0 ' L

& O % 8

9 " ;

C É % G)!! p

y

•

y

u

K = RM

p

u = −R−1 B ∗ p

' $ ! K [0, T ] !

= RM f = 0 z = 0 zT = 0 N

dP = −A∗ P − P A + P BR−1 B ∗ P − Q dt P (T ) = D

"

0 ≤ t ≤ T

P (t)

)!

E U IQ

2 P (t) ! t ∈ [0, T ] p(t) = P (t)y(t) t ∈ [0, T ] " & y0 ∈ RN * (y, p, u)(t) : u = −R−1 B ∗ p

! 0 ;% % 0 # ';% RN RN " y(t) → p(t) RN RN P (t) N " p(t) = P (t)y(t) 0 % " G! GN! '

ÉC

E L O

T Y POL

N H EC

dP y = −A∗ P y − P Ay + P BR−1 B ∗ P y − Qy dt

y(t) " " " " y0 ! $ " )! $ ' P (T ) = D p(T ) = Dy(T ) y(T ) " " RN $ )!

/0 $ (

K = R f = 0 z = 0 z M

t ∈ [0, T ]

T

p(t) · y(t) = Dy(t) · y(t) +

Qy(s) · y(s) ds + t

T

T

= 0

2

R−1 B ∗ p(s) · B ∗ p(s) ds .

t

E U IQ

/ t0 ∈ [0, T ] y(t0 ) = 0 y(t) = p(t) = 0 t ∈ [0, T ] 9 !

H C E T

N

/0 $ ( & 4,$$ ) 4,$ ⎧ ∂y ⎪ − ∆y = v + f ]0, T [×Ω ⎨ ∂t ⎪ ⎩ y = 0 ]0, T [×∂Ω y(0) = y0 Ω

LE O ÉC

Y L O P

- : y0 ∈ L

2

(Ω) f ∈ L (]0, T [×Ω) v ∈ L (]0, T [×Ω) 2

inf

: z ∈ L

J(v) =

(]0, T [×Ω)

LE O ÉC

T

0

TEC

v 2 dt dx +

Y O L P

v∈L2 (]0,T [×Ω) 2

2

Ω

2

zT ∈ L (Ω)

T

0

E U ! I Q N H |y − z|2 dt dx +

Ω

"

|y(T ) − zT |2 dx,

Ω

/0 $ # ; !! ! '

@ ? & $ 0 I & < ' " * * f '

−∆u = f + v Ω )! ∂Ω, u=0

E U IQ

: u % ' v * < " %' $ % ω ⊂ Ω " < * % vmin ≤ vmax L2 (ω) 6 " " * L2 (ω) E Ω\ω $ ' < ' K = v ∈ L2 (ω) " vmin (x) ≤ v(x) ≤ vmax (x) ω v = 0 Ω \ ω . )C!

N H EC

T Y L O 1 D( / " " P (( & < ' "

* - L E ÉCO

u ∈ H01 (Ω) $ u0 ∈ L2 (Ω) $

f ∈ L2 (Ω)

1 J(v) = 2

)#!

|u − u0 |2 + c|v|2 dx ,

Ω

: u )! v! c > 0 ' inf J(v) . )/! v∈K

' $ $ : E ! v ∈ K ;! <5 $ " " * v → u L (Ω) H (Ω) 2

1 0

" J(v) * " " % v % 0

* % " J(v) ≥ cv2L (Ω) . K % * % L2 (Ω) " D( IN & &

)/! 2

H C E T

N

E U IQ

' " '

1 # ; p

" H01 (Ω)

−∆p = u − u0 Ω )N! ∂Ω. p=0

LE O ÉC

Y L O P

- # %

$

E U I Q ' $ 0 N H C E .

% T <( > ! L E

! Y O )G! P E L ))! O )I! ÉC J(v)

K

J (v) = p + cv ,

p

v

v

−∆u = f + v

u ∈ H01 (Ω),

Ω,

p ∈ H01 (Ω), p v = 1Iω P[vmin (x),vmax (x)] − , c ω 1 1 ω −∆p = u − u0

Ω,

% 1Iω Ω \ ω P[v (x),v (x)] / ! [vmin (x), vmax (x)] & P[v (x),v (x)] w = min vmax (x), max vmin (x), w(x) 5

## ( - min

max

min

max

lim

→0

J(v + w) − J(v) =

Ω

J (v)w dx .

N H EC

5

J(v) " " '

T Y POL

J (v)w dx =

Ω

: u˜w

((u − u0 )˜ uw + cvw) dx,

Ω

E L O

E U IQ

Ω I ! ∂Ω. ; K I ! p )N! u˜w

ÉC

−∆˜ uw = w u ˜w = 0

∇p · ∇˜ uw dx =

Ω

Ω

∇˜ uw · ∇p dx =

(u − u0 )˜ uw dx wp dx

Ω

Ω

"

J (v)w dx =

Ω

(p + cv)w dx, Ω

E U IQ

: D(

N I! H C 0 & * ' T E * ' % E 8 9L Y I! " ! O P E L O ÉC

(p + cv) (w − v) dx ≥ 0

∀w ∈ K .

Ω

w

v

p(x) + cv(x)

ω

w(x) − v(x) ≥ 0

x

∀ w(x) ∈ [vmin (x), vmax (x)] .

ω

-" %

&

E U

5 " I Q ; ( % D( ! J ' N H )I! " " * & C E $ 5

L T A " #) "

% Y * )N! " ; $ ' O P )/! " " )! " E ! * %'L O ÉC v(x)

−p(x)/c

[vmin (x), vmax (x)]

K

v

1 L(v, u, p) = 2

ω

u

2

2

|u − u0 | + c|v|

dx +

Ω

p(∆u + f + v) dx,

Ω

: p )! v u J ' " & " ∂L ∂L ∂L = = = 0. ∂v ∂u ∂p

E U IQ

% )I! " % ; p " % u • 6

@

T Y POL

N H EC

#

,

E L O

,0I # 6 A ,.330,+4)0 0 , 6 5 ( .

0 , 6 ' . J ? , 6 u0 A (un )n∈N , . A #A . u 0 , 3 . +,*2)0()*+) &) +): .-(,0'149): ;,6?6 A . (un ) . u u0
ÉC

LE O ÉC

Y L O P

H C E T

N

E U IQ

-

- # %

N H EC

E U IQ

)9.0;5) CC / # ; A 6 A < @ 8 # 8 ? 00 ; ,0 ; < 8 • . ;? 8 F . <

LE O ÉC

T Y POL

@ . 6 7

'O 0 , ,0 J 8 α6. ( 0 ) ,6 = 0 V A 0 ; <

inf J(v) .

v∈V

E U IQ

1, 5 $ uA , + , ,9 J (u) = 0 .

T Y POL

-(,0'149) &) (0.&')*1 E 3.: ,31'9.-

E L O

N H EC

, ? L M , un . ? 8 G J(v) ? un J (un ) 9 (A , un+1 8

ÉC

;<

un+1 = un − µn wn ,

. µn > 0 wn V A , . wn = J (un ) , . . J(un+1 ) ; ,0 J (un ) , 8 . < ' A # ;< L MA ? , (0.&')*1 E 3.: ,31'9.-A 0 ? . 0 ;H V , ) < 3 u0 V A (un ) ) un+1 = un − µn J (un ) ,

C E T Y L PO

R µn ∈ R ? J(un+1 ) = inf J un − µ J (un ) .

LE O ÉC

µ∈R

' . , .

HN

E U IQ ;<

;"<

-" %

E U $ )) " $-& 5$"/ I Q " " ) #"7$ " " /$ ' " H N C E T Y 0"# 0 $" ) )" $-0 O " ' " L *$ ) ,9P"O ,= $-0 - "$ ,, LE O ÉC 4%,0?9) CC

J

J

α

V

∀ M > 0, ∃ CM > 0, v + w ≤ M ⇒ J (v) − J (w) ≤ CM v − w . u0

;$< (un )

u n %9,*:10.1',*C 8 f (µ) = J u − µ J (un ) 8 . .0 R ; J (un ) = 0 Y A I? . A un = u V< 0 ;"< 0 A f (µ) = 0A , J (un+1 ), J (un ) = 0 .

;&<

' L M . * ;&< J (un+1 ), un+1 −un = 0A ,α6. J α J(un ) − J(un+1 ) ≥ un − un+1 2 . ;-< 2 * J(un ) ; J(u)
E L O

T Y POL

N H EC

E U IQ

. ;$< v = un w = un+1 ;&
ÉC

αun − u2 ≤ J (un ) − J (u), un − u = J (un ), un − u ≤ J (un ) un − u ,

αun − u ≤ J (un )A ,R , . ,

)9.0;5) CC , H

J . . A I • 0 , un − u

-(,0'149) &) (0.&')*1 E 3.: @>)

E U IQ

, ? ) , un ) un+1 = un − µ J (un ) , ;<

H C E T

N

R µ 8 ) ' , ? A , 8 ? , ;"< . # µ .

LE O ÉC

Y L O P

- # %

E U " " ) $ )) " $-& 5$"/ IQ N #"7$ ' " &" $ $"$" " H C E T Y ' 0"# 0 $" ) *& $-0 O " ' L O ' " *$ ) ,= $-0 - "$ ,, P LE O ÉC 4%,0?9) CC

J

J

α

V

C>0

J (v) − J (w) ≤ Cv − w

u0

;<

∀ v, w ∈ V .

0 < µ < 2α/C 2 (un )

u

%9,*:10.1',*C * v n = un −u ' J (u) = 0A v n+1 = v n −µ J (un )− J (u) A ,R .

2 = v n 2 − 2µJ (un ) − J (u), un − u + µ2 J (un ) − J (u)

v n+1 2

≤ 1 − 2αµ + C 2 µ2 v n 2 ,

;<

, ;< ,α6. 0 < µ < 2α/C 2 A 8 . 1 − 2αµ + C 2 µ2 ∈]0, 1[A . ;<

E U IQ

)9.0;5) CC A

N H EC

, A . O ;< ,# 8 0 ;$< 8 A , 6 ? )A ? ( ? A J(un ) , •

E L O

T Y POL

)9.0;5) CC 0 # ;< @ , I

wn J (un )

/ 6 "A , 8 # 12 Ax · x − b · x •

ÉC

)9.0;5) CC ' . . A ( K G * A ,. A , ? ) , ? . A , H un A ? P 8 G , ? ) @ .

A . A 2'1)::) &) +,*2)0()*+) , A ) 0 , , un − u 8 ? ) * A , ;< . ,

? ) A un − u ≤ γ n u0 − u . γ = 1 − 2αµ + µ2 C 2 .

LE O ÉC

Y L O P

H C E T

N

E U IQ

-" %

N H EC

E U IQ

' , A . , # . A - ? 8 µ . α/C 2 , . 0, 2α/C 2 A . γ 9 8 A . 6 ( . ; ) un − u1/n ) A 0 1A n . +∞< •

LE O ÉC

T Y POL

>)0+'+) CC - V = R2 J(x, y) = ax2 + by 2 a, b > 0(

$ * % * + * a = b x0 y 0 = 0( * * 8

* $ * % * + I µ * ( K

@

. 6 $ 7

E U IQ

@ 0 , . inf J(v) ,

v∈K

N H EC

; <

R J 8 α6. ( 0 ) K A 60 . 8 . , = 0 V 5 $ , , u ;
E L O

T Y POL

6 A 8 ? # ,0 K

ÉC

-(,0'149) &) (0.&')*1 E 3.: @>) .2)+ 30,<)+1',* , ? ) , 0 ; < . ? . * µ > 0A ;< , u − u − µJ (u) , v − u ≥ 0 ∀ v ∈ K . ;< / PK , I ,0 . K A ) 5 I . ;. + < 3A , A ;< , , u I u − µJ (u) K 3 A u = PK u − µJ (u) ∀ µ > 0 . ;"<

H C E T

N

E U IQ

8 . ;"< 8 . ? ;) .2)+ 30,<)+1',* ;

I< ) , un+1 = PK un − µJ (un ) , ;$<

LE O ÉC

R µ 8 )

Y L O P

- # %

E U " ) $ )) " $-& 5$"/ I Q " ' 0"# #"7$ $"$" ' - ,, H 'N 0 $" ) *& - )>"$ $-0 " ' " CO ,, E *$ ) ,% $-0 - T "$ Y L O P LE O ÉC 4%,0?9) CC

J

V

J

α

C

0 < µ < 2α/C u0 ∈ K

2

(un )

u

%9,*:10.1',*C 5 " 0. . ;< , v → v − µJ (v) 0 < µ < 2α/C 2 A ,6?6 ∃ γ ∈]0, 1[ , v − µJ (v) − w − µJ (w) ≤ γv − w .

* I PK 8 0 , ;
>)0+'+) CC ) V = RN K = {x ∈ RN

N

i=1 xi = 1} ' $ 5 %* PK ;; * *

-(,0'149) &!Q.I.

T Y POL

N H EC

E U IQ

? ) . I 0 ? 0 , . . 7 . A , I PK , J I , v ∈ V . 8 V H : ? V A ) ; V = RM
ÉC

E L O

i=1

;. . ai = −∞ bi = +∞ i< 9 (A 8 . A x = (x1 , x2 , . . . , xM ) ∈ RM A y = PK (x) yi = min (max (ai , xi ), bi )

1 ≤ i ≤ M ,

;-<

A : I L M x ' A I A . ? . 9 (A H 0 8 .

0 K 0 I PK H A 0 8 0 8 ;&
6

LE O ÉC

Y L O P

H C E T

N

E U IQ

-" %

N H EC

' 0 .

T Y POL

E U IQ

;<

inf J(v) ,

F (v)≤0

R J 8 . ) V F 8 . V RM # 5 B 5 % A ;< . ? . 6 (u, p)

LE O ÉC

;<

L(v, q) = J(v) + q · F (v) ,

V × (R+ )M 3 1) 6 ∀ q ∈ (R+ )M

L(u, q) ≤ L(u, p) ≤ L(v, p)

;<

∀v ∈ V ,

(p − q) · F (u) ≥ 0 q ∈ (R+ )M A ,R A µ > 0A (p − q) · p − p + µF (u) ≤ 0 ∀ q ∈ (R+ )M , A , ;
N H EC

E U IQ

; <

p = PRM (p + µF (u)) ∀ µ > 0 , +

T Y POL

PRM I RM (R+ )M + 3 . ;
ÉC

E L O

L(un , pn ) = inf L(v, pn ) , v∈V

;<

pn+1 = PRM (pn + µF (un )) , + µ 8 )

4%,0?9) CC $ )) J " α$-& 5$"/' F " $-& " )#"7$$ V $ RM ' " &" $ $"$" C " F (v) − F (w) ≤ Cv − w

E U IQ ;<

∀ v, w ∈ V ,

" &" $ )$" (u, p) 0$0$ , V × (R+ )M , ' 0 < µ < 2α/C 2 ' 0"# :7N $-0 O " $" $" p0 ' " (un ) *$ ) ,3 $-0 - "$ u )/+ ,,

Y L O P

H C E T

N

%9,*:10.1',*C + ,0 , , u ;< 6 (u, p) ;. *
LE O ÉC

- # %

N H EC

E U IQ

;< 0 u 1, ,9 $A ,9 8 u un , . J (u), v − u + p · F (v) − F (u) ≥ 0 ∀ v ∈ V , ;"< n

T Y POL

J (un ), v − un + pn · F (v) − F (un ) ≥ 0

LE O ÉC

;$<

∀v ∈ V .

* . v = un ;"< v = u ;$< A 0 J (u) − J (un ), un − u + (p − pn ) · F (un ) − F (u) ≥ 0 , ,R ,α6. J rn = pn − p rn · F (un ) − F (u) ≤ −αun − u2 .

;&<

E U IQ

1, A I PRM 8 0 , ;
N H EC

rn+1 ≤ rn + µ(F (un ) − F (u)) ,

T Y POL

rn+1 2 ≤ rn 2 + 2µrn · F (un ) − F (u) + µ2 F (un ) − F (u)2 .

E L O

;< ;&
ÉC

rn+1 2 ≤ rn 2 + C 2 µ2 − 2µα un − u2 .

0 < µ < 2α/C 2 A . β > 0 C 2 µ2 − 2µα < −β A ,R ;-<

βun − u2 ≤ rn 2 − rn+1 2 .

' rn 2 J 0 ;-< . 0A P un . u 3 A , ,FC , ;< 6 O 0 . 0 ;< 3 A pn A ,

H C E T

pn+1 = max (pni + µFi (un ), 0) 1 ≤ i ≤ M , i

N

E U IQ

, ;-< 8 5 " .6

(pn ) 9 8 A . , # A , , , , p ∈ (R+ )M (u, p) 6 ;. + ,9 " 6 <

LE O ÉC

Y L O P

-" %

N H EC

E U IQ

"

? 8 , ,FC A ,. I? + ,0 0 ;< ,

T Y POL

;<

sup G(q) , q≥0

LE O ÉC

RA )

; <

G(q) = inf L(v, q) , v∈V

p 0 ;< 9 8 A # F A G ( 0 G (q) ? F (uq )A R uq ,

0 ; < 9 (A 8 G(q) = J(uq ) + q · F (uq ),

. ? q

N H EC

E U IQ

G (q) = F (uq ) + (J (uq ) + q · F (uq )) uq = F (uq ),

T Y POL

? , uq @ . -!.-(,0'149) &!Q.I. *!):1 .510) ;5) -. 9%14,&) &5 (0.&')*1 E 3.: @>) .2)+ 30,<)+1',* .33-';5%) .5 ;< , 30,/-?9) &5.

E L O

pn + µG (pn ) ; ? ;$< .

pn+1 = PRM + 8 0 ;< 0 6 < . ) 8 ? , .

ÉC

>)0+'+) CC G $ * % $H# Q C ;2 M ) B * M ( $ p $ C ;2( pn * p %*.-':.1',* &): +,*10.'*1):

E U IQ

/ 6 . 0 . # ,6 0 . 0 Y , 3%*.-':.1',* / . L M L , M , ? ( . 0 H ? , , 6 " ' , 6 . : A 0 L M ; < . L M ;. 6 <

LE O ÉC

Y L O P

H C E T

N

$

- # %

N H EC

E U IQ

/ O ) R V = R A . 0 .

T Y POL

N

; <

inf J(v) ,

F (v)≤0

LE O ÉC

R J 8 . RN R F 8 . RN RM * ε > 0A 0 M 1 2 inf J(v) + [max (Fi (v), 0)] ; < , ε i=1 v∈RN

Fi (v) ≤ 0 L M @ 6 .A A ε A 0 ; < L 0 M 0 ; <

E U IQ

$ )) J " $"$ ' ""$" $-&' " $*$ $*$' $"$ Fi $" $-& " $"$ ) 1 ≤ i ≤ M ' " $/ 0,3,:'1',* CC

N H EC

T Y " $$ - , $ $"$" $ "$ , "' ) L "$ ,' $ O P E L O ÉC K = v ∈ RN

,

Fi (v) ≤ 0

∀ i ∈ {1, . . . , M }

u

ε > 0' uε

$

lim uε = u .

ε→0

%9,*:10.1',*C ,0 K . 8A , , u

5 . J 1 A 8 M G(v) = i=1 [max (Fi (v), 0)]2 . 8 R R ? x max (x, 0)2 . @ 8 Jε (v) = J(v) + ε−1 G(v) .A A ) ? , ) G(v) ≥ 0A , , uε ' G(u) = 0A G(uε ) ≤ Jε (u) = J(u) . Jε (uε ) = J(uε ) + ; < ε ' J(uε ) ≤ Jε (uε ) ≤ J(u) , ; <

H C E T

N

E U IQ

uε 0 , L ) ? , ) M @ 8 (uε ) (uεk ) . . u∗ εk . 0 @ 0 ≤ G(uεk ) ≤ εk (J(u) − J(uεk )) , ; < * ? A 0 G(u∗ ) = 0A u∗ ∈ K ' ; < J(u∗ ) ≤ J(u)A u∗ = uA A (uεk ) . . H u

LE O ÉC

Y L O P

-" %

N H EC

E U IQ

&

>)0+'+) CC ' %: % - ;!;(

J F1 , . . . , FM , E I(u) $ u( u 6 ;2 '( Fi (u) i∈I(u) ( M $ * * λ1 , . . . , λM J (u) + i=1 λi Fi (u) = 0( λi = 0 i ∈ / I(u) " ( i ∈ {1, . . . , M }

LE O ÉC

T Y POL &

lim

ε→0

' 2 max (Fi (uε ), 0) = λi . ε

)9.0;5) CC

/ . ? 6 8 L0 M •

@

5 % B

E U IQ

@ ) V = RN 9 /C F 8 C 2 RN RN u F F ,6?6

N H EC

F (u) = 0 F (u) . 0

T Y POL

8 5# . v

F (u) = F (v) + F (v)(u − v) + O u − v2 ,

ÉC

,6?6

E L O

u = v − (F (v))

−1

F (v) + O v − u2 .

/C ? 8O . 6

* u0 ∈ RN A un+1 = un − (F (un ))

−1

F (un ) n ≥ 0.

; "<

+ , , . F (un ) ; "< , # , ? 1 . , A /C , . J 8 C 3 RN RA u J F = J A , J (u) = 0 ' A . /C 3 . 5#

H C E T

N

E U IQ

1 J(w) = J(v) + J (v) · (w − v) + J (v)(w − v) · (w − v) + O w − v3 , ; $< 2

Y L O P

J(w) . v 8 /C ? ?

LE O ÉC

- # %

-

N H EC

E U IQ

; $< −1 w = v − (J (v)) J (v) J (v) ) . @ . 8 . ; "< ,. /C . 0

T Y POL

$ ' " $ L E ,, " $"$"$$-/, 7O 0 &" $ " ' C " 7 )# $ . ' "# N"$ É ) ,= $-0' " " $-0 - ' " &" *$

$ 0,3,:'1',* CC F

u0

F F (u) = 0 u

$"$" C > 0 "

C2

F (u) u − u0 ≤ (un )

RN

RN >0

u

u

; &<

un+1 − u ≤ Cun − u2 .

%9,*:10.1',*C * F > 0 F . 0

E U IQ

0 u # un uA R u − un ≤ A F (un ) . 0 ' F (u) = 0A ; "< −1 un+1 − u = un − u − (F (un )) (F (un ) − F (u))

T Y POL

N H EC

A . 5# u A . n

un+1 − u = (F (un ))

E L O

−1

O un − u2 .

' u − un ≤ A , C > 0 ; n F F 0 u # < un+1 − u ≤ Cun − u2 . ; -<

ÉC

: ? C ≤ 1A ; -< un+1 0 u # ' . ) ,# u − un ≤ n ≥ 0A ; -< 0

)9.0;5) CC E GA 8 . ? , /C ; "< , # A G 1 A . ; L M< ; &< , F C 2 A u0 F uA # 0 . . I ,? 9( .A H RA /C . u0 Y 8 . ; &< , . , F A , /C ? 8 F (x) = x2 RN A . , * A /C , 8 J 6 A

LE O ÉC

Y L O P

H C E T

N

E U IQ

-" %

N H EC

E U IQ

. . J A . A 8 F J /C , A . ; &< •

T Y L O $

, % ; ( 4V P = E B % ! " ' L ' * * %' ( O ( " 4V % C É 4V *2 % J (v)

F (v)

J

J (v) = F (v)

Sn

$ * /!

(F (un ))

−1

n ≥ 0. 0 S n * un+1 = un − S n F (un )

S n+1 = S n + C n

E U IQ

: C n " un , un+1 , F (un ), F (un+1 ) ( & " S n − (F (un ))−1 % % 0 ( " 4V % & F/H F#H • $ ( 4V & * J % 1 J * C 3 RN R G = (G1 , ..., GM ) * C 3 RN RM % M ≤ N ! u

min J(v) . I! v∈R , G(v)=0

ÉC

E L O

T Y POL

N H EC

N

1 % (G1 (u), ..., GM (u)) D( ) J (u) +

M

λi Gi (u) = 0 ,

Gi (u) = 0

C !

1≤i≤M .

i=1

: λ1 , . . . , λM ∈ R $ C ! (N + M ) " & (N + M ) (u, λ) ∈ RN+M ( 4V $ F (u, λ) =

%

F (u, λ) =

J (u) + λ · G (u) G(u)

J (u) + λ · G (u) G (u)

Y L O P

,

H C E T (G (u))∗ 0

.

N

E U IQ

$ " ( 4V /! & * F (u, λ) F (u, λ) %' 4 % " 8 9 : & % *

LE O ÉC

- # %

"

E U %' IQ ;% 1

N H C E T Y L $ " P O LE O C É

F (u, λ)

w ∈ KerG (u) =

(w, µ)

J (u)w + λ · G (u)w + (G (u))∗ µ = 0 1≤i≤M Gi (u) · w = 0

M (

KerGi (u)

(J (u) + λ · G (u))w ∈ Im(G (u))∗ .

i=1

$

Im(G (u)) = [ KerG (u)]⊥

∗

" "

C! F (u, λ) %' $ " " C! 6 * ( ( C! " ( 4V $ % ( % F/H! 6 (

( $ L(v, µ) = J(v) + µ · G(v) % & v L L % " " J (u) + λ · G (u) (w, w) > 0

∀ w ∈ KerG (u), w = 0 ,

T Y L

N H EC

(un+1 , λn+1 ) = (un , λn ) − F (un , λn )

O P E L O % É C

−1

E U IQ

F (un , λn )

" un+1

' " " & min Qn (w) , C! w∈RN G(un )+G (un )·(w−un )=0

Qn (w) = L(un , λn ) + L (un , λn ) · (w − un ) +

1 n n L (u , λ )(w − un ) · (w − un ) 2

,

λn+1

C! $ " " C! + % D & w L(w, λn ) G(w) un

$ . C! " " * J $ ( % " " & %

min

w∈RN G(v)+G (v)·(w−v)=0

J(v) + J (v) · (w − v) +

1 J (v)(w − v) · (w − v) 2

.

H C E T

N

E U IQ CC!

?( ( ' CC! % S 0 % " B J (v) * B " *

• !

LE O ÉC

Y L O P

T Y POL

N H EC

E U IQ

L E O ÉC

E U IQ

6"

& $ -& 7

T Y POL

N H EC

E L O

1 A . +@ ; 6 < 1 +@A L M , ,0 J +@ A A 0 ) 3 4 1F A , . ,

A , , A ) E - 0 ;. TU < , 0 . . 0 +@ 0 , . J 0 , ; A # , ,A
ÉC

LE O ÉC

Y L O P "

H C E T

N

E U IQ

"

N H EC

E U IQ

I ; 0 # < , . +@ . , ; A , A < / , +@A X , 6 ? A , 0 A , 8 ? , J : 0 0 ,6 . .A A 0 0I ;0 DA 0 ,( A # A A A
/ . A ? , 3, -B?&0) )*1')0A 8

A 0 D 6 A 30,(0.99.1',* &B*.9';5)A 0 A 6 0 0 : " ,.-(,0'149) (-,51,* J 0 A . / .6 ) $A , A . 0 ,0 :%3.0.1',* )1 %2.-5.1',* ;L0 0 M
LE O ÉC

ÉC

T Y POL

E L O

T Y POL

N H EC

E U IQ

4

/ ,. ,9 # , 0A 3 , 0 .

*(

E U IQ

@ . 0 .A 30,(0.99) -'*%.'0) :,5: =,09)

:1.*&.0&A

inf

x∈Rn

" Ax=b, x≥0

c · x,

C E T Y L PO

HN

;<

R A m × nA b ∈ Rm A c ∈ Rn A x ≥ 0 ) x . 1 m ≤ n A m 9 (A rg(A) < mA A 0 J

LE O ÉC

%

N H EC

E U IQ

"

; ? < 0A 0 ;< 0 H , # x ≥ 0 , A #

ÉC

OLE

T Y POL x∈Rn

inf

" Ax≥b, A x=b

c · x.

8 ;< ? 9 (A ,0 , A x = b . 6 . , A x ≤ b A x ≥ b @ . ; , < inf

x∈Rn

" Ax≥b

; <

c · x.

E U IQ

1 ; < , . . 0A &!%+.01:A λ ∈ Rm , Ax ≥ b .6 ? Ax = b + λ . λ ≥ 0 3 ; < . ?

T Y POL

(x,λ)∈R(n+m)

inf

N H EC

" Ax=b+λ, λ≥0

;<

c · x.

2 A x . .A ,6?6 x = x+ − x− . x+ = max(0, x) x− = − min(0, x)A 0 ; < . ?

ÉC

E L O

(x+ ,x− ,λ)∈R(2n+m)

inf

" Ax

+ −Ax− =b+λ,

x+ ≥0,x− ≥0,λ≥0

c · (x+ − x− ).

;<

0 8 ; . . 0< ,# ? ;< / . I? .

0 ' ;. ,9 < ' , . , 6

min x1 + 4x2 + 2x3 . ;"< x1 ≥0,x2 ≥0,x3 ≥0 2x1 +x2 +3x3 =6

E U IQ

2 . ,0 (x1 , x2 , x3 ) . ) J T ', 8 R3 A 8 x1 + 4x2 + 2x3 # , M * 8 x1 + 4x2 + 2x3 = c 6 c 9

. c ? −∞A L0 M , R3 I ,? T A M 0 L M 3 A ;"< 0 T 8O . 8 x1 + 4x2 + 2x3

LE O ÉC

Y L O P

H C E T

N

"

x3

T Y POL

LE O ÉC

N H EC

E U IQ

x2

x1

N H EC

E U IQ

W 90 0 , ;"<

T Y POL

T . 0 T * , ;"< ; < (0, 3, 0) T / . , , , 8 J ;, < I . ,0 . x . ) L :M , ) . J , 6 8 ; < , . 6 * 0 6 ;
ÉC

E L O

$/ Xad -" Rn "$" $"$" ,' " Xad = {x ∈ Rn " Ax = b, x ≥ 0} , " )) $/ :,-51',*: .&9'::'/-):, $ )) " )$" &" Xad " " )$" x ∈ Xad $ ) " ) ) $ $ /$$ $-& $$ "- & " )$" Xad' " ' &" y, z ∈ Xad " θ ∈]0, 1[ " x = θy + (1 − θ)z ' y = z = x,

%@*'1',*

CC

H C E T

N

E U IQ

)9.0;5) CC .0 , # @ ; 0< . 8 * A

. ;< :,-51',* ,31'9.-) ; • <

LE O ÉC

Y L O P

%

N H EC

""

E U IQ

@ . ) 8 ,0 Xad 3,-B?&0) ;. . < ;+ , # ) 6 Rn < # Xad . A .

T Y P O L "

c · x = +∞.

L E &" $ $ "$ )" )$" $ O É C)0 $ "$ , " $" - $ " *$ inf

x∈Rn

)99)

Ax=b, x≥0

CC

−∞ <

inf

x∈Rn

" Ax=b, x≥0

c · x < +∞.

%9,*:10.1',*C (xk )k≥1 ;< @ 6

A )

A=

c∗ A

.

Axk X . n

T Y POL C=

N H EC !

E U IQ

xi Ai . xi ≥ 0 ,

i=1

R Ai A 1, 2% & X C 8A z0 lim Axk = ∈ C, b k→+∞

ÉC

E L O

x ≥ 0

z0 b

=

c·x Ax

,

x

$ )) /.:) , $ / Rm m $$ A, $ $" B "" / " $ " A' m $-/, )+ ) ""$ $$ $ ) " A (B, N ) . N " $ " " m × (n − m), 8 E P$ $ ) " ) x $ (xB , xN ) " $ %@*'1',*

CC

E U IQ

N H C )$" -" $" )) -/ E / " -/ T # /, :$ / "

$ -" Y$ " "$" / ", $ ) $ )$" O "L $ ' , P E L O ÉC Ax = BxB + N xN .

xB

:,-51',* /.:';5)

&%(%*%0%)

x ∈ Xad

xB

xN

xN = 0

"$

E U IQ

N H " )!+ C$" &"$" "$ / , E Y

T '" ' 1 O L ' " " 4 1 " P

& E ' O L C "

É A "

' 0 ? Xad

)99)

CC

Xad

x ∈ Xad Rn m m x = (x1 , ..., xm , 0, ..., 0) (b1 , ..., bm ) R xb = b i=1 i i 0<θ<1 y, z ∈ Xad x = θy + (1 − θ)z n− y z y z Xad m m m yb =b zb =b i=1 i i i=1 i i x=y=z x Xad x Xad k % b = ki=1 xi ai : (ai ) A " x '" % " * (a1 , ..., ak ) ' Rm ' ' B * ! 1 " K y = 0 " ki=1 yi ai = 0 (yk+1 , ..., yn ) = 0 5

(x1 , ..., xk ) % > 0 " (x + y) ∈ Xad (x − y) ∈ Xad * " x = (x + y)/2 + (x − y)/2 x x '"

N H EC

E U IQ

8 . , : # Xad

T Y POL

&" $ "$ )" )0 $ "$ ,' &" $ "$ )" / , 0,3,:'1',*

CC

E L O

%9,*:10.1',*C ? "

ÉC

x ∈ Xad ;< @ k 0 A . b=

k

xi ai ,

i=1

R (ai ) A 8 (a1 , ..., ak ) 0 Rm A x 0 (a1 , ..., ak ) A y = 0 k

yi ai = 0 (yk+1 , ..., yn ) = 0.

i=1

N

E U IQ

' (x1 , ..., xk ) .A > 0 (x ± y) ∈ Xad ' x A

Y L O P

c · x ≤ c · (x ± y),

H C E T

,6?6 c · y = 0 @ ) 8 z = x + y 9 . = 0A

LE O ÉC

%

N H EC

E U IQ

"&

,0 Xad I ,? . 0 ? z ≥ 0 . 3 A z0 ∈ Xad (k −1) @ , . x = z0 8 (k − 1) (ai ) 3 8 8 A 0 ) 8 0 0

LE O ÉC

T Y POL

)9.0;5) CC 9 * $ c = 0 ; 0
; 6 R2 < • >)0+'+)

CC C * max

x1 ≥0,x2 ≥0

x1 + 2x2

⎧ ⎨ −3x1 + 2x2 −x1 + 2x2 ⎩ x1 + x2

≤ ≤ ≤

T Y POL

2 4 5

N H EC

E U IQ

9 0 # Xad H 0 . 0 @ . ) , .

>)0+'+)

E L O

CC " $ % A m× n b ∈ Rm R Xad [0, 1]n−m M n − m Ax = b ' Xad

2n−m

ÉC

. )

, G ? 4 1F ? # 0 I ,? , .

; ( .

< , , A P . 8 c·x , . @ ;< + , ;

0 < ,0 0 Xad 0 B ;m 0 A< 3 A

H C E T

A = (B, N ) x = (xB , xN ),

Y L O P

N

E U IQ

, Ax = BxB + N xN 5 0 , xB = B −1 (b − N xN ) ≥ 0 xN ≥ 0 ? B ) ;, < xN = 0 xB = B −1 b ≥ 0 c = (cB , cN ) 0A

LE O ÉC

"-

N H EC

E U IQ

G , 0 x . 0 x @ . 8 c · x ≤ c · x

T Y POL

cB · B −1 b ≤ cB · B −1 (b − N xN ) + cN · xN .

LE O ÉC

;$<

@ , .

0,3,:'1',* CC ))$ "$ / B " $$ 0$ ' " B −1 b > 0, :$ $ "$ $ " 6$" ) "" "$ / B " )" " c˜N = cN − N ∗ (B −1 )∗ cB ≥ 0.

-" c˜N " )) 2)+1)50 &): +,M1: 0%&5'1:,

;&<

E U IQ

%9,*:10.1',*C x 0 ? B c˜N ≥ 0A 0 x

N H EC

c · x − c · x = c˜N · xN ≥ 0 ,

T Y POL

xN ≥ 0 1 ;&< : x + A , i c˜N cN · ei ) < 0 * > 0 ) . x() xN () = ei .A (˜ xB () = B −1 (b − N xN ()) * Ax() = b A B −1 b > 0A . : x() ≥ 0A x() ∈ Xad 1, A x(0) = x A > 0A

ÉC

E L O

c · x() = c · x(0) + (˜ cN · ei ) < c · x,

x , 1 ;&<

)9.0;5) CC 1 * -A 0 A ;&< : , •

E U IQ

@ * - P . 8 G c · x ? , 0 x ; < ' x 6 A . G c˜N c˜N · ei < 0 @ ) x() 6 * G P . A H ? . 0 , Xad ', ,

LE O ÉC

Y L O P

H C E T

N

% -(,0'149) &5 :'93-)>)

N H EC

"

E U IQ

W ; < J 0 B 0 0 x0 0 (B 0 )−1 b x0 = ≥ 0. 0

LE O ÉC

T Y POL

W ; < J ? , k ≥ 0A , 0 B k , −1 0 0 xk @ G c˜kN = ckN − (N k )∗ (B k )∗ ckB k k c˜N ≥ 0A x , ) A . 0 60 , i (˜ cN · ei ) < 0A ai A @ xk () = (xkB (), xkN ()) . xkN () = ei , xkB () = (B k )−1 (b − ai ).

W

> 0 , . . xk () ∈ Xad 1 A −∞ W . k j j 6 6 xk (k ) , @ 0 . 0 B k+1 B k O j 6 ai . 0 0 xk+1 G 8 ? xk 0 ? , 6 / .

E L O

T Y POL

N H EC

E U IQ

%(%*%0):+)*+) )1 +B+-.()

ÉC

@ I c · xk+1 ≤ c · xk A # . 0 0 xk A . k = 0 ; , A * - < @ 0 G J , # H , . # , A # ,P I 9 ,0 # A , 60 Xad ; 8O < G ' # 0 ) A , . G @ .

E U IQ

" " "$ / / xk ) " ) 0 "# )& $" $$ 0$' 0"# $-0 $ $ $/ *$ "),

)99)

CC

H C E T

N

3 0 , , H 0 ; 0 . 0 n Y . ,9 < E , ; < R , (6 . A . 0 ,

8 # n

LE O ÉC

Y L O P

$

4,'> &5 +4.*()9)*1 &) /.:)

T Y POL

N H EC

E U IQ

, # . G c˜kN .A 8 8 , * 0A

.

LE O ÉC *'1'.-':.1',*

' . 0 0 , K ;+6 , 0 xB = B −1 b ≥ 0 , . < P ? 0 * A 0 ;< m . 0 ,A − Idm 0 L 0M , ;< b ≤ 0A . (x0 , λ0 ) = (0, −b) 0 0 ;<

N H EC

E U IQ

1 A . . 0 y ∈ Rm A . . G k > 0A .

T Y POL inf

E L O

x≥0, y≥0 Ax+y=b

;-<

c·x+k·y

R 0 −1 , ? . b b ≥ 0 . (x0 , y 0 ) = (0, b) 0 0 0 , 0 ;< ;< , −∞A ;-< . ) y = 0 9 , ? ;-
ÉC

*2)0:',* &) -. /.:)

H C E T

N

E U IQ

5 ,. , , . ? 0 B k A H G 0 ;. 0 , B k 0 < @ 8 B k+1 ( B k 9 (A , j 6

LE O ÉC

Y L O P

% A

O P LE O ÉC

T Y L

B k+1 = B k E k

E k 8 ? .

(E k )−1

N H EC ⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ k . E = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎛

1

0

⎞

1

1

1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎠

0

−lj−1 1 −lj+1

T Y POL 0

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0

−l1

1

⎜ ⎜ ⎜ ⎜ 1 ⎜ = ⎜ lj ⎜ ⎜ ⎜ ⎜ ⎝

E U IQ

l1 1 lj ln

$

N H EC 1

−ln

E U IQ

1

@ 8 A 8 8 A

ÉC

>)0+'+)

E L O

(B k )−1 = (E k−1 )−1 (E k−2 )−1 · · · (E 0 )−1 (B 0 )−1 .

CC C $ * % * min

x1 ≥0, x2 ≥0, x3 ≥0, x4 ≥0, x5 ≥0

>)0+'+)

x1 + 2x2

⎧ ⎨ −3x1 + 2x2 + x3 = 2 −x1 + 2x2 + x4 = 4 ⎩ x1 + x2 + x5 = 5

CC C $ * % * min

x1 ≥0, x2 ≥0

2x1 − x2

H C E T

N

E U IQ

x1 + x2 ≤ 1 x2 − x1 ≤ 1/2 $ $ $

>)0+'+)

Y L O P

CC C $ * % *

LE O ÉC

min

x1 ≥0, x2 ≥0, x3 ≥0, x4 ≥0

3x3 − x4

$

N H EC

x1 − 3x3 + 3x4 = 6 x2 − 8x3 + 4x4 = 4

T Y POL

E U IQ

.

LE O ÉC

1 . B B% 0 -A . , A A , . , ; A A 0 # Xad < . ? , Xad I 0 ,? . / , 6 .-(,0'149) &) 10.<)+1,'0) +)*10.-) # . J A ? , 6 8 L0 M Y A /C , ? . 1 . inf

x∈Rn

" Ax=b, x≥0

c · x.

C E T LY

@ ) x > 0

ÉC

E L O

PO

π(x) = −

n

HN

E U IQ

;<

;<

log xi .

i=1

* µ > 0A 0 . min µπ(x) + c · x. ;< x∈Rn

" Ax=b, x>0

+ , x > 0 , , I . J ;< L,M 0 x > 0 8 LM π(x) . +∞ , I ;< /C . µ 9 (A µ . FA 0 ;< . ;<

E U IQ

>)0+'+) CC " ( Xad ( xµ 8 $ ( / x0 ( xµ * x0 µ #

*

Y L O P

H C E T

N

; I? . 6 < ' ? .

LE O ÉC

%

N H C " E T LY

; < inf

x∈Rn

O P LE O ÉC

Ax=b, x≥0

$

E U IQ

; <

c · x,

R A m × nA b ∈ Rm A c ∈ Rn * p ∈ Rm A ; < ;<

L(x, p) = c · x + p · (b − Ax),

R , L M , @ 8 G(p) = min L(x, p), x≥0

A A .

G(p) =

p·b −∞

A∗ p − c ≤ 0

0 ; < sup

N H EC

" A p−c≤0 ∗

T Y POL p∈Rm

;<

p · b.

E U IQ

;"<

, 0 0 ;"< Pad = {p ∈ Rm A∗ p − c ≤ 0} .

E L O

+ , 0 ; <

ÉC

Xad = {x ∈ Rn Ax = b, x ≥ 0} .

; < ;"< &5.-'1% , H . . 5

CC , ,= $ - )" *$' &" "$ )" , " p ∈ Pad "$ )" ,=

4%,0?9) x ∈ Xad

-*$"

c·x = c·x = p·b =

E U 8 ) ' " $" "$ )" , " ,= " $" IQ N -*$" $ "$ )"" Q #$ " A J H C E T $/ "$ Y L , ,= $ - )" $*$' / " )/+ " P - ,O LE O ÉC min

x∈Rn

" Ax=b, x≥0 x

p∈Rm

max

" A p−c≤0 ∗

p·b

;$<

p

Ax = b, x ≥ 0, A∗ p − c ≤ 0, x · (c − A∗ p) = 0.

;&<

$

N H EC

E U IQ

)9.0;5) CC 5 A x ∈ Xad p ∈ Pad 0 ; < ;"
T Y POL

1 HA x ∈ Xad p ∈ Pad . )

LE O ÉC

c·x= b·p

x ; < p ;"< ' . 8 0 . ; < ;"
%9,*:10.1',*C Xad Pad . x ∈ Xad p ∈ Pad ' x ≥ 0 A∗ p ≤ cA c · x ≥ A∗ p · x = p · Ax = p · b,

E U IQ

Ax = b 9 A . 0A A ) A , . , ;$< , ;&< 5

, 0 . ) * ) A , , 0 ; # 8 0 < 3A : , p ;"< Xad , . A . ) 7 Xad , . 2% & * p ∈ Rm A . Rm+1 b p ˜b = p˜ = . −b · p 1

ÉC

E L O

T Y POL

N H EC

@ . ) ˜b · p˜ = b · p − b · p ≤ 0A p ∈ Pad 1, A p ∈ Pad ) * A m+1 ∗ ˜ ˜ p˜ ∈ C = p˜ ∈ R p˜m+1 = 1, A p˜ ≤ 0 . A = . −c∗

E U IQ

' ˜b · p˜ ≤ 0 p˜ ∈ C A 2% & , ˜xA ,6?6 x˜ ∈ Xad , . x ˜ ∈ Rn x ˜ ≥ 0 ˜b = A˜ 2 A . 0 ; < −∞ Pad , . A x ∈ Xad p ∈ Pad A c · x ≥ b · p 9

Xad 0 b · p = −∞A 0 1 Pad . A . ; < ) A Xad .

LE O ÉC

Y L O P

H C E T

N

%

N H EC

$"

E U IQ

, H ; < 1, A , A H 8 0 ;"< ; m . 0 n , < 0 ; < ; n . 0A m , n , < 1, A : ; < 8

LE O ÉC >)0+'+)

T Y POL

CC H @+ A W

*

min

x1 ≥0, x2 ≥0, x3 ≥0, x4 ≥0

>)0+'+)

8x1 + 9x2 + 4x3 + 6x4

4x1 + x2 + x3 + 2x4 ≥ 1 x1 + 3x2 + 2x3 + x4 ≥ 1

E U IQ

CC < 2 $

N H EC

x ≥ 0( $ + $ * *

T Y POL

L(x, p, q) = c · x + p · (b − Ax) − q · x

q ∈ Rn q ≥ 0 0 ! q ' $ $: $ B + @ A x ≥ 0

ÉC

>)0+'+)

E L O

CC ! + 2

>)0+'+)

CC ) v ∈ Rn ( c ∈ Rn ( A m × n b ∈ Rm *

;-<

inf c · v .

v≥0 Av≤b

" ( q ∈ Rm ;<

sup b · q . q≥0 A∗ q≤c

E U IQ

) v q . /( " v q ( (

H C E T

(c − A∗ q) · v = 0 (b − Ac) · q .

Y L O P

N

; <

* 2; +,*&'1',*: &): %+.01: +,93-%9)*1.'0): ( ? 4 *

LE O ÉC

$$

43

N H EC

E U IQ

/ . I , 0 ,,31'9':.1',* +,*1'*5) J 8 ? ( 0A ,0 0 ) , , 0 ) A H ( 0 ,,31'9':.1',* +,9/'*.1,'0)A A 0 ,6 0 0 &':+0)1 3 A 0 ,(6 8 ,0I ,9 A ,0 0 ,0 n : 0 0 , A , ,0 0A ; n! 0 ,(
LE O ÉC

T Y POL

sup c · x ,

x∈P ∩Zn

N H EC

E U IQ

; <

R P # Rn A ,6?6 , 0 ) 6 A c ∈ Rn 8 ; < 0 ( 6 0 0 0 J 0 , ;
ÉC

E L O

T Y POL x∈P

0 A 8 ) 0

0)-R+4% ; < ;1 A 0 QA A 0 < 0 Q ; < :6 J : ; < . # P ; 3,'*1 )*1')0A ? < / R 0 ; < . ? Q ; < ' A . 0 A 8 , A 6 0 0 0 J A ( A 0 D ? G

! )) $ )

H C E T

N

E U IQ

. 0 0 0 3,'*1 )>10%9.-A I? 1) J , . K x x = (y+z)/2 y, z ∈ K P y = z = x @ K ,0 K + X 60 Rn A )*2)-,33) +,*2)>) X A ,

LE O ÉC

Y L O P

*

N H EC

E U IQ

$&

X A . X A . ) , ? ,0 0# , 0 ) , X ,)*2)-,33) +,*2)>) =)09%) X A X A . 8 X ? 8 X . 8

LE O ÉC 4%,0?9)

T Y P O L :$ )" $-&

CC $'*D,I:D'6

$/ )$" &" &,

Rn

" $-)) $-&

' : K = K K . Rn 3 8 A K = K K 8 . 7 %C% ,# , A , = 0 J # : H = {y ∈ Rn | c · y = α}A . c ∈ Rn A c = 0A α ∈ R 4B3)03-.* &!.335' , . K A x ∈ K A α = c · x ≤ c · y A y ∈ K / ,0. .

H " $ #!))$ )) $ $-& )$" &" H ∩ K " )$" &" K ,

)99)

CC

%9,*:10.1',*C

K ⊂ Rn '

N H EC

" "

E U IQ

H = {y ∈ Rn | c · y = α} . c ∈ Rn A c = 0A α ∈ RA # , K x = (y + z)/2 . y, z ∈ K A x ∈ K ∩ H A . α = c · x = (c · y + c · z)/2A α ≤ c · y α ≤ c · z A α = c · y = c · z A y, z ∈ K ∩ H , x K ∩ H A . x = y = z A x K

E L O

T Y POL

%9,*:10.1',* &5 14%,0?9) &) '*D,I:D' CC C @ . K = ∅ ; A . < + , . 6. ) , : , @ . 6 K ? Rn 6 :A K n n = 0A K ? A . ) . n − 1A x K 0# , 0 ) K x 8 K A ' 8 # , H K x ' K ∩H

. n − 1A # A x 0# , 0 ) K ∩ H A K , * x K A D : x ,0 D ∩ K 8 [y, z]A R y, z 8 K 1, A y z 0# , 0 ) K ' x H 0# y z A

ÉC

E U IQ

N H C ## ( ?3 V3 T E * ( W? " " Y L % % % * ' O "

" ( B(@( P LE O ÉC

$-

E U @(! $ " I Q % % * % % " %N ( H C E T Y OL P LE O ÉC •

/ 7 %C% 0 , 6 0 ; < 1 A 8 G J(x) = c · x A 9.>'9':.1',* 86 .A ( 9'*'9':.1',* 8 . ' / 0 X 6 0 A P ∩ Zn

0,3,:'1',*

CC $.>'9':.1',* &) =,*+1',* +,*2)>):6

$-& J : Rn → R' " ) " " $/ X ⊂ Rn '

" " $"$

sup J(x) = sup J(x) = sup J(x) , x∈ X

x∈X

x∈ X

" X " /$' sup J(x) = x∈X

sup

x∈ X

N H EC J(x) .

; <

E U IQ

; <

y ∈ X A y = i xi A . xi ∈ X A 1≤i≤k α αi ≥ 0A 1≤j≤k αj = 1 * J .A J(y) ≤ 1≤j≤k αj J(xj ) ≤ max1≤j≤k J(xj ) ≤ supx∈X J(x)A . y ∈ X A supx∈ X J(x) ≤ supx∈X J(x) * A z ∈ X A z = limk→∞ yk A . yk ∈ X ' 8 . Rn → R ;8 9 &
%9,*:10.1',*C

ÉC

E L O

T Y POL

*

,. . ,0 0 X = P ∩ Zn 0 ; <

$ )) )*2)-,33) )*1'?0) $ )!+ P ⊂ Rn ' $- )) $-& $/ )$" $" P ' $ $" Pe = (P ∩ Zn ), %@*'1',*

CC

E U IQ

L. M J , A

. 0I A , Pe ⊂ P

,0,--.'0)

CC

J : Rn → R

N H C " $-&' " T E" $ )!+ ' Y L O P P ⊂ Rn

sup J(x) = sup J(x) .

LE O ÉC

x∈P ∩Zn

x∈Pe

; "<

*

N H EC

$

E U IQ

3 A I 0 ; < 0 ,0 0 . J A 0 ? ; "< J : A , A # Pe # R . 8

LE O ÉC %@*'1',*

T Y P$O "L $ )!+

CC

P

" $ 3,-B?&0) )*1')0 P = Pe ,

/ : ; < , #

5

# , P = {x ∈ Rn | Ax ≤ b}

; $<

E U IQ

. A ∈ Rm×n b ∈ Rm . , # # Xad 0 ;8 1) < J (A Xad ) X 8 Rn A . I? Xad , 6. A I , # ;8 + &< Xad ; "< , .

T Y POL

N H EC

E :$ )$" &" )!+ *$ ) ,% " $ L $" "$O $ !"+ ' . " $ " $-/ 0$ 0$ )$ $" , É C ' " " -"

)99)

CC

n

A

b

A x = b

A

P

b

%9,*:10.1',*C x P A I(x) = {1 ≤ i ≤ m | Ai ·x =

bi } ;,0 . x
E U IQ

- , # , , 0 )

>)0+'+)

H C E T

N

CC " x P A x = b (

Y L O P

A b .( x

-

, R , #

LE O ÉC

&

E U

$ " $-/, "$ -$" " IQ N $" -$" O H C , G E , ) " " ' $ Y T , OL P LE O ÉC 0,3,:'1',*

CC

A∈Z

n×n

detA = ±1

A−1 b ∈ Zn

b ∈ Zn

%9,*:10.1',*C , ⇒ X 8 ' + 6 A A . ) , 7 ,0 A−1 ? : 9 b i6 . 0 Rn A . i6 A−1 A [ . A−1 bA ? : ' . 1 ≤ i ≤ nA A−1 ∈ Zn×n 1 detA−1 ∈ ZA 1 = detA detA−1 detA . 1A ,6?6 detA = ±1

$ " $ " A ∈ Zn×n " 5*'9,&5-.'0) $ detA = ±1' " $ " B ∈ Zm×n " 1,1.-)9)*1 5*'9,&5-.'0) $ " " " &"" B " "$$" ±1 0, %@*'1',*

CC

E U IQ

9 6 1 × 1A . : , . ±1 0 , . .

N H E C $"

T " ""$" $ ' ' Y ' " ,O 'L )$" &" & )!+ P E L$", $ )" ' " /$' $ O $" $$" C " "É $"$ $-& $ ' $

,0,--.'0) {+∞})n

CC

D ∈ Zm×n $ f ∈ (Z ∪ {−∞})m ' g ∈ (Z ∪

$31'9.-'1% &): :,-51',*: )*1'?0):6 f ∈ (Z ∪ {+∞})

m

g ∈ (Z ∪ {−∞})n

Q = {x ∈ Rn | f ≤ Dx ≤ f,

g ≤ x ≤ g}

Q = Qe '

Q

J

Rn

R

sup J(x) = x∈Q

; &<

" ) ; -<

sup J(x) . x∈Q∩Zn

%9,*:10.1',*C @ ; <

Q = {x ∈ Rn | Ax ≤ b} ,

R b . ) A 8 ±Di A . Di DA 8 ±ej A R ej j 6 . 0 Rn A 1 ≤ j ≤ n @ ,0 A M 6 k × k A 7 k detM ∈ {±1, 0} k = 1A X D . 6 A k − 1A M M ? . ±ej A . detM ? A A . M A detM = 0A

LE O ÉC

Y L O P

H C E T

N

E U IQ

*

N H EC

E U IQ

&

. A M [ A A . 6 DA D A detM ∈ {±1, 0}A . . A ' Q ;
LE O ÉC

T Y POL

>)0+'+) CC L $ * $ 0 0R : Xad = {x ∈ Rn | Ax = b, x ≥ 0}

* ( A ∈ Zm×n * m " I b ∈ Zm ( Xad X 2 m × m A ±1 0 ) D ∈ Zm×n ( Q = {x ∈ Rn | Dx ≤ b, x ≥ 0} 6 $ $ % OE N= I b ∈ Zm ( Q X D )9.0;5)

ÉC

E L O

T Y POL

N H EC

E U IQ

CC @ '

J A . 9 A J(x) = c·x A , . • G c

)9.0;5)

CC ' 1, A J A H ? A 0# • , )9.0;5) CC Q 0 , :6 Q = Qe ' J # 0A : 0 6 • 0 A ) , > T&U

H C E T

N

E U IQ

0 / 0 ? :

Y L O P

0,3,:'1',* CC $,'*+.0%6 A " $ " 6$" ±1 0' - ) $ 6$" 1 ) $$' " ) $ 6$" −1 ) $$' A " ""$" $ ,

LE O ÉC

&

N H EC

E U IQ

%9,*:10.1',*C ' . ) A 6 A :

. A A detA ∈ {±1, 0} A A detA = 0 A . : 6 A . ? A , detA ∈ {±1, 0} , R A : 1 : −1 J A A detA = 0

LE O ÉC

T Y POL

/ * " 0 D 6 . 1 , R , ? ?

>)0+'+)

CC $0,/-?9) &) +,52)0150)6 H $ % % * I ct B + $ t ∈ {1, . . . , T } H B : $ : k % ( $% i [αi , βi ]( 1 ≤ αi ≤ βi ≤ T ( + Si + + $ αi + $ βi uit = 1 αi ≤ t ≤ βi ( uit = 0 Y

T Y L

inf

O P E L ÉCO 1≤i≤k

N H EC

x ∈ Nk 1≤i≤k xi uit ≥ ct , ∀1 ≤ t ≤ T

E U IQ

;<

xi Si .

" $ $ $ $ : 2&( 4 D $ ( $ + + M 0, 1 1 " $ $ 0

! & C 3) ) 0 DA G = (N , A) J N ,0 *S5&:A A ⊂ N × N ,0 .0+: a i a j (i, j) @ (i, j) ∈ A , +.3.+'1% uij ∈ R+ ∪ {+∞} , +,M1 cij ∈ R ; L uM uij L 0 M< @ a D bi ∈ R ; bi < 0A , , D A 0 < @ T,1 8 x ∈ RA A (i, j) → xij A . ) -,' &): *S5&: &) L'0+4,A bi + xji = xij , ∀i ∈ N , ;< j∈N ,(j,i)∈A

.

OLE

Y L O P

0 ≤ xij ,

ÉC

j∈N ,(i,j)∈A

H C E T

∀(i, j) ∈ A .

N

E U IQ ; <

*

N H EC

E U IQ

&

9 a ;
LE O ÉC

T Y POL

i∈N

/ I ;< . ) D .&

9'::'/-) , 8

%@*'1',*

xij ≤ uij ,

CC

$

min

;<

∀(i, j) ∈ A .

$ )) 30,/-?9) &) T,1 E +,M1 9'*'959 )0

cij xij (11.31), (11.32), (11.34).

(i,j)∈A

;"<

E U IQ

0 D ? G 60 A 0 ,9 A 0 ,( ,9

T Y POL

N H EC

>)0+'+) CC ' Z + , $' / * %

E L O

8 0 D ? G 0 D A 30,/-?9) &) T,1 A ; G< G a 6

A s pA . :,50+) 35'1:A s , ;{i ∈ N | (i, s) ∈ A} = ∅
ÉC

bi =

v −v 0

i = sA i = pA

CC 30,/-?9) &5 T,1 9.>'9.- $" " - $ R" / s p - &, %@*'1',* >)0+'+)

N

E U IQ

CC " Z E

H C E T

Z + , L I 5 * % Z

Y L O P

9 A . , 0 D J A 0 ,9 A ? . . H A . 6 .

LE O ÉC

&

N H EC

E U IQ

0 D ? G 3) , ' A a B ( ;< , Ax = bA R A ∈ RN ×A A 9.10'+) &!'*+'&)*+) *S5&:.0+: G A )

T Y POL

LE O ÉC

−1 i = k A 1 i = j A 0

Ai,(j,k) =

A 0 ) A 8 R 0 A ,6 ?6 (j, k) j = k D 0 0 ) a B ( ;
0,3,:'1',*

$ ,

CC

E U IQ

;$<

" $ $ $S $ 0)# " ""$"

N H EC

T Y R" $"$" &0+$ L $" $"' " )" $" $"+ $*$' ' )$" & O " & $/ E ,3%P "$ / $ )/+ R" 4" L $ )" ' $/ R" / " /$ " $ $" O $", $$- 'C &" $ R" $" )", É

%9,*:10.1',*C ', * " ,0,--.'0)

CC

$31'9.-'1% &): T,1: )*1')0:6

bi

uij

%9,*:10.1',*C ', ' * -

>)0+'+) CC " ' Z + , ( $' >)0+'+) CC C $ E ( $' /2 5 n * R n ( ( aij % (i, j)( %% σ ∈ Sn ( max

aiσ(i) .

C E T Y L PO

σ∈Sn

1≤i≤n

HN

E U IQ ;&<

" max

LE O ÉC

x∈Bn ∩Zn×n

1≤i,j≤n

aij xij ,

;-<

*

N H EC

E U IQ

&"

4 Bn * $ % ( $ + $ x = (xij ) n × n

T Y L

∀i ∈ {1, ..., n},

O P LE O ÉC

∀j ∈ {1, ..., n},

1

=

xik ,

1≤k≤n

1

=

xkj ,

1≤k≤n

∀i, j ∈ {1, ..., n},

0

≤ xij

2 " . Z + , * % ' : Bn J + M $ E K 6 % : $ % , + V=%E 6 + ( : n * R n ( % * R

: + r ( % : + r * R( r ≥ 1( n $ B

5+ $ + $ % , + N1*

>)0+'+)

T Y POL

N H EC

E U IQ

CC 0 * % % Z (

, + 7 7= Z $ s + p * % G = (N , A) 6 & - (

$ uij +∞ - I, J ⊂ N ( x = (xij ) ∈ RA ( x(I, J) = i∈I, j∈J, (i,j)∈A xij $ N S S¯ s p s ∈ S p ∈ S¯( $ ¯ u(S, S) ¯ s p( Z x " (S, S) s + p v ( ¯ − x(S, ¯ S) = v . x(S, S)

ÉC

E L O

' Z s + p 5 s p 2 8 Z x s + p( Gr (x) = (N , Ar (x)) 4 Ar (x) * $ (i, j) (i, j) ∈ A xij < uij ( (j, i) ∈ A xji > 0 " $ % γ s + p * % Gr (x)( Z x s + p + x( xij (i, j) (j, i) % γ x ( % x $ $: % s + p Gr (x) ) S $ [ s Gr (x)( S¯ S N " ¯ s p * + Z x (S, S) 0 Z s + p * + $ s p 0$ % @Z \ A(

LE O ÉC

Y L O P

H C E T

N

E U IQ

&$

N H EC

E U IQ

@ ZQ\ A( 7 7= ! ' * % ( ( Z s + p O(v ∗ |A|)( 4 v ∗ $ Z s + p( |A| $

T Y POL

)9.0;5) CC , P LD c 6 M 2 2 % ; ,9 -< 6 , , ? 0 0 D . A . • 2 2 %

LE O ÉC

# ( J J 3 0 C) " ( L , % ( ( L ( ( 0 W - & . O(nm2 ) : n = |N | ' Y m = |A| ' 5 & & (" ( J J 3 ( s & p & ( s & p " ' 6 ( L + ( 8L V (9 O ' D; I)N! " O(n2 m) D ' L & -

> FH •

E L O

T Y POL

N H EC

E U IQ

4 3 +

ÉC

# A . + E "A

, 0 ; A 8 . 0 8 < , 0 , ) 0 ,%1.1 3 . A , # . , ? ? ? , . ; # ? , . 7%. ? 00 <

! # D

E U IQ

30,(0.99.1',* &B*.9';5) , EA , 0 J , . A ? . B . C A 6 A ? C A ? C 9 dXY X ? Y A 0

OLE

Y L O P

H C E T

dAB = min(dAC + dCB ) ,

ÉC

C

N

;<

% *

N H EC

E U IQ

&&

R min ,0 . C A ? B , ;< , # A %;5.1',* &) )--9.*A . . d dXY X Y . . , P J 0 dAB A A B . )A . . , . 0 dCB ; A dAC < . C . 0 G H ) 8 . J , 8 6 . / ,0 . 0A R P A 6 . 0 A 8 A 6

LE O ÉC

T Y POL

! & E (

E U IQ

' 0 A 6 0 X F ) G = (N , A) A , c : A → RA +,M1 + , +4)9'* a . A , a [ +'0+5'1 G , ) G 2 T ;, F
ÉC

E L O viT =

T Y POL min

(0 , . . . , T ) 0 = i

N H EC

;<

c0 ,1 + · · · + cT −1 ,T .

8 i → viT A , . v T ∈ (R ∪ {+∞})N A =,*+1',* 2.-)50 ; F T < / . . ;< min ∅ = +∞A . ? viT = +∞ ,# T i T = 0 J ;< @ . A # A G F 9 A vi0 = 0 . [. ;< ? 6

T A 0 P . T #6 . 8 A ,( #6 ; # ) + $$< , viT 8 . , F , A t vk 0 ≤ t ≤ T k ∈ N 9 (A i ∈ N A t ≥ 1A vit

OLE vi0

ÉC

=

Y L O P min

k∈N , (i,k)∈A

= 0 .

H C E T

(ci,k + vkt−1 ) ,

N

E U IQ ;< ; <

&-

N H EC

E U IQ

, ;< , E J G 6 iA t A 0 . i → k A 6 ? G .A ,6?6 ci,k G k F t− 1 ; < . J , ? I A # @ . , , 6

# A %;5.1',* &) )--9.*A ? 0 J . v 0 , v 1 , . . . ∈ (R ∪ {+∞})N 8 0 ) ; < ' 6 . 8 .

LE O ÉC

T Y POL

viT =

min

(0 , . . . , T ) 0 = i

;<

c0 ,1 + · · · + cT −1 ,T + φT ,

R φ ∈ (R ∪ {+∞})N . ? , ) φ = 0A . ;< φ 8 , j ∈ N A

k = j 0 φk = ;< +∞ A

N H EC

E U IQ

;< 8 ? , j A viT G T i ? j 8 . ;< . ) I ;
ÉC

E L O

T Y POL

;"<

v0 = φ .

', T . , E ;
LE O ÉC

Y L O P

H C E T

N

E U IQ

% * >)93-)

N H EC

E U IQ

&

CC * A , ( . 2 . 8 FA =A A 3 A E 0 ; ,R ( . ? . < ( 8 T A =A A cA , A φA 8 , ) @ A −∞ B ) . ) I B / 8 , , 0 A , # , . A −∞ +∞ J

LE O ÉC

T Y POL t vH

t−1 t−1 = max(3 + vH , 10 + vA )

t vA t vB

t−1 t−1 = max(12 + vH , 7 + vB ) t−1 t−1 = max(−5 + vH , −3 + vA )

0 vH

0 vA

= 0,

N H EC

0 vB

= 2,

= −∞ .

E U IQ

T Y POL G

' F

ÉC

E L O

B

C

C

H

@

/

W (

: ,. 1 vH 1 vA

= max(3 + 0, 10 + 2) = 12 = max(12 + 0, 7 + −∞) = 12

1 vB 2 vH

= max(−5 + 0, −3 + 2) = −1 = max(3 + 12, 10 + 12) = 22 ,

H C E T

N

;$< ;&< ;-< ;<

E U IQ

R , @ A A 0 I ,0 ;< ; . ,0 = ? 3
LE O ÉC

Y L O P

-

E U , " I Q ( ( % Y - & N H 8(9!

% 89! % K % C T 5E % &

" % & " Y "

% & L (

$ % - O * " ( ' & P E #! L O ÉC (N , A) A

1

G=

n = |N | m = |A| h ∈ {1, . . . , n}m c ∈ Rm m (i, j) ∈ A ck = ci,j

c t ∈ {1, . . . , n}m k

O(n + m)

hk = i tk = j O(n + m) vt v t−1 •

! & F G #H =

' 0 +4)9'* &) +,M1 9'*'959 A i j A ? . 0 i ? j G , , 0 I? . 6 J A G 8 9 A 8 vi = inf viT = T ∈N

inf

N H EC

;"<

c0 ,1 + · · · + cT −1 ,T + φT ,

T Y POL

(0 , . . . , T ) T ∈ N, 0 = i

E U IQ

R . φ ) j ;< @ φ ∈ (R ∪ {+∞})N A @ vi ? . R = R ∪ {±∞} ; ? ;"< . . +∞A A 0 0 8 A , 6

0 < 8 . v . ) . E vi = min φi , min (ci,k + vk ) , ∀i ∈ N . ;"<

ÉC

E L O

k∈N , (i,k)∈A

* . ;'9.-) , I

H C E T

N

E U IQ

$"$ - v *$ ) ,= " "$ & "$ K$ ,=,

4%,0?9)

CC

Y L O P

%9,*:10.1',*C @ I? v . ) ;"< v ∈ RN

LE O ÉC

;"< 7 v ≤ v (0 , . . . , T )

% *

N H EC v r

E U IQ

-

≤ cr ,r+1 + v r+1 A i ' v . ) ;"
T Y POL

vi ≤ c0 ,1 + · · · + cT −1 ,T + φT

, ) (0 , . . . , T ) iA vi ≤ vi 3 v ≤ v A v ;"<

LE O ÉC

>)0+'+) CC " $ V ! L I * % >)0+'+)

CC > * % φ [ i * %( % i + [ j φj = +∞ * % $ , * φ( $ V ! ( * + v - ( v ∈ RN $ V ! $

C = {i ∈ N | φi ≥

min

$ % % i ∈ C [ π(i)

O P E L ÉCO

HN

(ci,k +

C E T LY k∈N , (i,k)∈A

vk )}

,

E U IQ

vi = ci,π(i ) + vπ(i) .

π : C → N " i ∈ C ( k k πk (i)( $ + C 0 v ≥ v @ ;"< ) v = f (v)A . 6 ) . f : RN → RN 3 5

66 v ? , , ) * ? . )A . ? A ,# . 8 X ? , @ ( RN , A ) A A , 6 f : x → f (x) * f (φ) ≤ φA f r+1 (φ) ≤ f r (φ)A r ≥ 0 {f r (φ)}r≥0 P . . . v ∈ RN ; , . −∞ : v A . < @ f (v ) = v f 1, A v ) f A . 6 v ≤ φA f A v = f r (v ) ≤ f r (φ)A r ≥ 0A , ) r ≥ 0A v ≤ v A v ) f * , 5 A 8 . 6) f A . .

H C E T

Y " L O $"$ - *$ ) ,=, P E L O ÉC 4%,0?9)

CC $1%0.1',* :50 -): 2.-)50:6 v

N

{f r (φ)}r≥0

E U IQ

T" -

-

E U

I Q " % ( & & N

H " " " " C ! & " " % E T ! Y * '

L < (" ( 4V "

P O " % . @0AD10W 1 . E 5 > 6 66 ( 1 @ ? II/! %

L $ % . O É C % % f

f r (φ)

x = f (x)

φ f (φ) ≤ φ

φ

φ

/0 # (5* + , 2 G !!

!0 ! " f |N |−1 (φ) = v 1 ! G !! !0 ! " G !%!! φ & / !! 44$* ! f |N | (φ) < f |N |−1 (φ)

E U IQ

0 #C

( % $ x0 = φ xr = f (xr−1 ) r ≥ 0 % & (" xr = xr−1 " xr = v " % % ' " |N | − 1! % & r = |N | K " - * 0 "

" ( % < % % O 1 ( J @ " & ; < % 6

% "

T Y POL

N H EC

E L O G

7+ ( ) 1 , D 9 É C

G = (N , A) c : A → R F 9 r i, k ∈ N : 9 r ← 0 b ← % H i ∈ N vi ← φi I r < |N | b 9 b ← * H i ∈ N (i, k) ∈ A ci,k + vk < vi vi ← ci,k + vk b ← % " r < |N | v r = |N | 0 !

v ∈ (R ∪ {+∞})N b

% ' ( + J & % ( 0 ' H 5

* H & A % " 22! ( + & * ' vH = +∞ ' < 11 (" f & /! %

ÉC

Y L O P

H C E T

[f (x)]H

=

max(0, −8 + xH , −1 + xA )

[f (x)]A

=

max(2, 1 + xH , −4 + xB )

[f (x)]B

=

max(−16 + xH , −14 + xA ) .

OLE

N

E U IQ

% *

N H EC

% & x x

0

0

E U IQ

= φ x = f (x ) 1

0

=

φ = (0, 2, +∞)

x1H

=

max(0, −8 + 0, −1 + 2) = 1

x1A x1B

=

max(2, 1 + 0, −4 + −∞) = 2

T Y POL

-

E L O % " " C " " * É x

B

2

=

max(−16 + 0, −14 + 2) = −12

=

x1 = v,

H

,

A

A

/0 + % ! " <%=

! !! !0 " 8

6 ' % ( J @ " * * " Y i (i, k) ' ( ## K % " 8' ( 9 ( * ( J @ ( -

( " - * ! - % * . * ' " J @ " ( G + + & "

≤ " ( i & j i ≤ j 1 % Y ;& ! . % % FH $ " ( ' * ( " % A " NN ! , * Y v & +∞ + * ' vi ← min(φi , mink∈N , (i,k)∈A ci,k + vk ) (" i i ' * % " , " ' : - * K ( * 1 /! ( ? .;3 % FH • /0 > / 4 ' & ! !%'% >

! ( ! n 5 ! p1 , . . . , pn ∈ N !& u1 , . . . , un ∈ R P ∈ N

' ( xi = 1 i% 5 !%'% xi = 0 ( & ! ' ! max xi ui . /! x ∈ {0, 1}

ÉC

E L O

T Y POL

N H EC

n

1≤i≤n

xi pi ≤ P

1≤i≤n

H C E T

! ! 1 ≤ t ≤ n 0 ≤ Q ≤ P

Y L O P

max

x ∈ {0, 1}

OLE

1≤i≤t

ÉC

t

xi pi ≤ Q

1≤i≤t

E U IQ

xi ui ,

N

E U IQ /C!

-

N H EC

E U IQ

& ( v = 4 / v ! vt−1 ' " * / " 44* ? ! " 8 @

" ' & A max 8x1 + 2x2 + 9x3 . /#! t vQ t

t

T Y POL

t (vQ )0≤Q≤P

LE O É C " & x ∈ {0, 1}3 2x1 + 3x2 + 5x3 ≤ 6

/0 (5 8 * , 1 0

1

m−1 0

m

1

c ∈ RA

G =

!0 τ ∈ NA . ( . B ! s T ∈ N

!! ' & ! (0 , . . . , m ) " s 0 = s " c , + · · · + c , + φ ! ! T ! τ , + · · · + τ , ≤ T (

!! ! < " " % ! @

! A & " ! !0 B 1 B 6 10 " 44 3""

(N , A) φ ∈ (R ∪ {+∞})N

m

m−1

m

E U Q /0 + C D

I 5 N & ! & E @&" 3 & E ! H

! ! + ! !! &

C C " 5 4,,%4,T (E !F !

C ' / L

Y

" E &

& ! E! ' ! ! O

!

' 1 & 5 P ! E ' E

L

E

! ! 2 ! ' O ! ' !0 " !! É C ! & E " " 0

5

/0 " ( " 2

' ! &! (a, b) → a ⊕ b = min(a, b) ! (a, b) → a ⊗ b = a + b ! ' - 2 &

29G 4 3 ! " ! (R ∪ {+∞}, ⊕, ⊗)

/

&. H H

" &. a ⊕ a = a =

! I . 44$4 ! % viT =

/

Mi,k ⊗ vkT −1 ,

k∈N

N

E U IQ

&! Mi,k = ci,k (i, k) ∈ A Mi,k = +∞ 3 H !

% % !& v t = M vt−1 , v T = M T φ . ? ' =

444 !& v = Mv ⊕ φ . //!

LE O ÉC

Y L O P

H C E T

" % %

-"

E U Q ' 2 !0 ! "

I 2 44 " ' N : G & ) 44$* 2 H !! !0 " C ! !7! " ' G & E T ! / !! 44$* Y L O P /0 ! @

5 2J& . 5 ' 5 K !L KL E L ! ' M 5 " . ' ! ! O !! + " . A 5 ! C É "" ! ""

% t i v = M ∗φ G

··· limT →∞ M T = +∞

k (M t )i,k M∗ = M0 ⊕ M ⊕ M2 ⊕ +∞

!! 5 8 (

5 ! " & ! 9! A ! & ' min max & & A % ! & 5 "" N %

& 5 "" N % ! & 5 !

E U

" 5 " %

" IQ '' * "

( ?3 % N H

" * " 8 9 K C ' % E T ; (! Y L O P E L O É C-

% & "

•

#

@ , (-,51,* ;L #MA 6

0 . ? A , . 0 ;, . 0 L% M
. A ? O J , ? # , , H ,H . ? 8 . #6 A : ? . , ? , 0 8 ;. ? I + "" 6 < / 8 , 8

LE O ÉC

Y L O P

H C E T

N

E U IQ

-$

E U . A & I Q #& = N $ F H C E T Y OL P OLE

@ , ? 6 / V ,0 A E ,0 HA 60 ,0 ? V @ {i, j} ∈ E , # H i j / 0 6 A ,HA . a , ; L V M L E M . L. M L M< ;6 < +,**)>) . H ;6 < H +,93,:.*1): +,**)>):A ) , . R iRj , # i j @ =,0U1 ;6 < .0/0) 8H @ , 6 G . G G , H G ;6 < G = (V, E)A H , 8 G E → RA {i, j} → cij A 0 ,.0/0) +,520.*1 &) +,M1 9'*'959 ? . 0 . T G cij ,

ÉC

T Y POL {i,j}∈T

N H EC

E U IQ

' 0 , 0 Q0 A , . 0 ) , B % 8H @ 8H 6 H 3 A I ? 8HA H , I A G , 8H 0 .

ÉC

E L O

CC G = (V, E) " $ 0)# $$$" $$&' $ $ $"$ 4" /" c : E → R' 0"# Q J $" $ / -$" 4" $ ,

4%,0?9)

3) 5 "A O 0 .A . ) ,

E U IQ

)99) CC $ )99) &!%+4.*()6 T " $ / -$" $ 0)# G ' " i " j $" & " G ' &" $ $ #$ T $" i j , $ "' {i, j} " $ E" G $))"$" ) T ' $ /"$" $ $ / -$" $ ) $ E" $ #$ T $" i j ) {i, j},

Y L O P

H C E T

N

/ . ,

" G " c $ A#+ ,=,' " " -$" G , "$ -$" $" -$",

)99)

CC

LE O ÉC

T

$ /

" % %

N H EC

E U IQ

-&

, T " 4" $ G , ) # E" {i, j} $" ) $ T ' $ #$ T $" i " j " E" $" # $ " 4" $ 0 cij G 3, ) # E" {r, s} T ' ) " " )$" $$& C 0)# /"$ $ "$" {r, s} T ' " ) " " E" {i, j} $" $ &"" "

$ " $ C ' crs ≤ cij ,

T Y POL

LE O ÉC

%9,*:10.1',*C , ; <⇒; < , " J

{i, j} H , T A {r, s} H , T i j A cij < crs A 0 {i, j} ? {r, s} T . 0 . G 8 ? T / ; <⇒; < + ,0

0 {r, s} T A C C = V \ C , H {i, j} . i ∈ C j ∈ C A cij < crs , i ? j T {r, s}A ; < , 8 / ) ;<⇒;< T 0 . ) ;
ÉC

E L O

T Y POL

N H EC

E U IQ

%9,*:10.1',* &5 4%,0?9)

CC C , B 6 %A ? A . 0 . ' 0 . ) , "

6 ' ( W 3

E U Q ( ; I* ;& ( ' N K % ' & *'" 'C H %

' T & E & * % & (" * ' ( " Y L & ; * % " " " ' % ;& * PO

( " * LE O ÉC O(|E|| log E |)

: |E| ' ( % FH

E

E

E

•

--

E U & '! % ( I Q '

( N % H

/ " " C ( T ( T E T %

FH Y L O P LE O ÉC •

.

: + +(

1 , 0 +@A Q 6 P : ; A DA

A < , 8 A A H . 0 6 A # , K @ 0 G ? A ? 6 0 A . ? I + $& 1 , A X A ? 0 , 0A . , .A A , ? ,

E U IQ

N H C & 7 $ 6& E T Y L O P E L O ÉC ' 0 0 min J(x) ,

x∈X

;"$<

. X ) ;
min

s

s

J(s ) .

;"&<

; L M A ,0 . A 8 0< G s . 0 ;"$< ; A s A ;"&< [ . ;"$<
Y L O P

H C E T

N

E U IQ ;"-<

, . ) a ,0 / . , A ; < /,0*) '*=%0')50) /,0*) b(s)

LE O ÉC

'

N H EC

E U IQ

-

*,*+% &) -!.-(,0'149) &) :%3.0.1',* )1 %2.-5.1',*

, . ? a ,0A 9 , A mA G I? .A @ , . m = +∞ . 8 a s A . 0 b(s) b(s) ≥ mA ? , 0 ) a sA G 8 ,# . , G A , a s ) . , ,0 ; . , +,53) 0 ,0 a s< b(s) < mA 0 0 s m J , ,0A a ) s . ? 8 ,0A 0 x ∈ X ;"$
LE O ÉC

T Y POL

T Y POL

N H EC

E U IQ

--5:10.1',* 7 )>)93-) &5 2,B.()50 &) +,99)0+)

E L O

3) A 0 .# A I? . ,9 / . 6

.# A 6 +,93-)1 G = (V, E) ; ) E 8 V < @ V = {1, . . . , n}A . n = |V| / ? {i, j} , tij ; 0. t{i,j} A tij = tji 6 < 0 . A ,6?6 1 , . . . , n G 8 A t 1 2 + t 2 3 + · · · + t n 1 ;"<

ÉC

9 , . ;"< A I , 1A 1 = 1 ) 2 , . . . , n−1 :%3.0.1',* 0 . ? 6 , * A 2 ∈ V \ {1 } A 3 ∈ V \ {1 , 2 } A I ,? n−1 ,0 ? 6 (1 = 1 , 2 , . . . , k )A . k ≤ n − 2A L M , 0 8 G ;"&
LE O ÉC

Y L O P

H C E T

N

E U IQ

N H EC

E U IQ

1 , . . . , k @ 0 [.

T Y POL

b1 (1 , . . . , k )

LE O ÉC

= t1 2 + · · · + tk−1 k + min

m∈V\{2 ,...,k }

min

j∈V\{1 ,...,k−1 }

tm1 + (n − k − 2)

t k j +

min

j, m ∈ V \ {1 , . . . , k } j = m

tjm . ;$<

9 (A ;"< (1 , 2 , . . . , k )A t1 2 + · · · + tk−1 k A ,H {k , k+1 }A , ;$

ÉC

E L O

T Y POL 1

N H EC

E U IQ

2

3

4 5

W .# 7 ,0 2 @

a (1)A ? . 1 . , ? . . , . . A H 2A 3A 4A 5 ' 2A a (1, 2) . ? ,0A . ' ,. 0 , A m = +∞A b1 (1, 2) < m . )A b1 (1, 2) * . 8 ,0 J . a (1, 2, 3) ; . ? .
LE O ÉC

Y L O P

H C E T

N

E U IQ

'

N H EC

E U IQ

A ? 8 .A (1, 2, 3, 5, 4)A 18 m . a (1, 2, 4)A , b1 (1, 2, 4) = 13 < m J ) (1, 2, 4)A 8 x = (1, 2, 4, 5, 3)A 14A m J m = 14 , ) (1, 2, 4) 8 14A (1, 2, 4, 3, 5) . a . . (1, 2, 5)A . b1 (1, 2, 5) = 17 ≥ m J 60 (1, 2, 5)A . a . (1, 3) / ) A x = (1, 2, 4, 5, 3) 14

LE O ÉC

T Y POL (1)

(1, 3), b1 = 12

(1, 2)

E L O

T Y POL

(1, 2, 4), b1 = 13

(1, 2, 3)

ÉC

E U IQ

(1, 4)

N H EC

(1, 5)

(1, 2, 5), b1 = 17

16 18 14 14 x = (1, 2, 3, 4, 5), m = 16 x = (1, 2, 4, 5, 3), m = 14

W ,0 . A 0 .# 2 )9.0;5)

H C E T

N

E U IQ

CC 0 .# 7 .# A tij i ? j A A · 1

.# O A , , . ; . # , A 0 .# A

LE O ÉC

Y L O P

N H EC

E U IQ

· ∞ A < 0 .# ; <

0 /*6 : ;. + $$<

T Y POL

) -!'93,01.*+) &) -. ;5.-'1% &) -. /,0*)

0 b1 0A .# ,

0 / . , A .#6

7A 8 . 0 n > I # A 0 a ,0 . A 8 n J

LE O ÉC n

n = 15 I? *' / . , ,)>3-,:',* +,9/'*.1,'0) + .A 0 b1 P J n − k A . (1 , . . . , k ) (n − k − 2) 8 H A . ? , n @ . ,. A 0 ,. 0 : L# M 0

E L O

T Y POL

N H EC

E U IQ

,0*) &5 1.0/0) 3,50 -) 2,B.()50 &) +,99)0+)

ÉC

1 , 0 .# 6 (1 , 2 , . . . , k ) G A . k ≤ n−2A

G 0 V = V \ {1 , . . . , k }A ,6?6

,0 V ,0 ,H E = {{i, j} ∈ E | i, j ∈ V } / (G ) G , 0 . G ;+ ,0 . ) "A R . . , 0 . G O(|E| log |E|) , B %< @ 0 8 . b2 (1 , . . . , k ) = t1 2 + · · · + tk−1 k +

min

j∈V\{1 ,...,k−1 }

t k j +

min

m∈V\{2 ,...,k }

E U IQ

tm1 + (G ) ;. $<

H C E T

N

1 R A k = 1A : 0 b2 A H {1 , 2 } {k , 1 }

. H ;?
LE O ÉC b2 (1 )

=

Y L O P

min

j,m∈V\{1 }, j=m

(t1 j + tm1 ) + (G ) .

;$ <

'

N H EC

E U IQ

' 0A A /,0*) &5 1.0/0) ; 160 , 6 8 , , 0 . L 1MA , H 1 , < * A 2 A ,6

B % 8 160 G 2 " G 160 b2 (1) = 13 9 . ? ,0 2 A . b1 b2 . a (1, 3)A b2 (1, 3) = 14 ; 8 A < ? . m = 14 . . (1, 3)

LE O ÉC

T Y POL 1

2 3 4

5

N H EC

E U IQ

" W E 160 b2 (1)A 0 .# 2 ,0 . G A H 0 1

T Y POL

5 +4,'> &) -!.0/0) )1 &) -!,0&0) &!)>3-,0.1',* &): /0.*+4):

E L O

# ,0 6 0 X A .

. ? 80 0 3 A . 6 $ , .#6

. 0 A {i, j} ∈ E . 0 xij . 1 ,H {i, j} 0 3. A 0 6 ,. 0 A ? ) . , . 0 xij ? 0 ? 1 , . J I , X 0 ; A ,0< , , D G A 0 0 0 ,0 1 H A m . . , 0 A +∞A 8 , ? X 0

ÉC

H C E T

" ) & &

N

E U IQ

# ,0 0 8 . G , 0 0 ? 0)-R+4)0 ; < 0A ,6?6 ? ,0 0 ? 0 0 8 A 8 0 8 0 ,0

LE O ÉC

Y L O P

N H EC

E U IQ

0 A Q , 0 / . I? . . 0 160 .# J 160 . /

LE O ÉC

T Y POL

)-.>.1',*: +,*1'*5):

,:

. 0 0 0 J 0 0 8 Q , * .# 6 1F A 2 %A N "A . ? , 0 ? 49 . 6 I , ? . ; , A ? L@ 8 . 0MA 1 3 A + E 0#A > '.dA \ '%A 8 $" B"#,A 9 . '7 -A 6 $"6$"$A ? < / .# 6 J G = (V, E) ; E H V
ÉC

E L O

T Y POL

N H EC

E U IQ

k∈V,{k,j}∈E

;$< : ? A ,0 H {i, j} xij = 1 H J 8 A . x ∈ {0, 1}E ;$< I 3) , A I .A :,5:1,50A xkm ≤ |S| − 1 . ;$< S ⊂ V, S = ∅ S = V, k,m∈S,{k,m}∈E

H C E T

N

E U IQ

@ . 8 x ∈ {0, 1}E . ) ;$< ;$< A A J (A . ? . ) A A x ∈ {0, 1}E . ) ;$< I A : S ,0 , 6 0 k,m∈S,{k,m}∈E xkm = |S|A x . )

LE O ÉC

Y L O P

'

N H EC

"

E U IQ

6 ;$
LE O ÉC

T Y POL {i,j}∈E

0 X 0 8 . 0 (V C) : min tij xij 0 ≤ xij ≤ 1, (11.63), (11.64). {i,j}∈E

7 0 ;$
(V C) A ;$< . x1 . ? :%3.0)0 x1 P A ,6?6 ? . 6 H 1 x1 ∈ H 1 A P ⊂ H 1 A , . . ? t . # P 2 = P 1 ∩ H 1 A 0 L M P 1 H , 80 #6 . , LM P . R t 1 . H 8 0 (V C)A , 0 . , A 0 A , : 0 X Pe P

ÉC

E L O

T Y POL

N H EC

E U IQ

>)0+'+)

E U IQ

CC " (V C) (

H C E T

S ⊂ V, S = ∅ S = V,

N

xkm ≥ 2 .

;$"<

k∈V, m∈S\V,{k,m}∈E

Y L O P

" ( % 7 7= J $' . $ x ∈ [0, 1]E ! + $ $ * % Z

LE O ÉC

$

)-.>.1',*: .(0.*(')**):

N H EC

' 0

LE O ÉC

T Y POL

E U IQ

J(x) J x ∈ X, Fi (x) ≤ 0, i = 1, . . . , m, Fi (x) = 0, i = m + 1, . . . , m + q,

;$$<

R J, F1 , . . . , Fm+q 8 Rn RA X 60 6 . Rn ' L : X × Λ → RA . Λ = (R+ )m × Rq L(x, λ) = J(x) + λ1 F1 (x) + · · · + λm+q Fm+q (x) .

J ∗ . ;$$< / . I? + , I , 8 0 J ∗ = inf sup L(x, λ) ≥ sup inf L(x, λ) = sup D(λ) , x∈X λ∈Λ

R

λ∈Λ x∈X

λ∈Λ

HN

D : Λ → R ∪ {−∞}; D(λ) = inf L(x, λ)

C E T LY x∈X

;$&<

E U IQ

;$-<

8 0)-.>.1',* .(0.*(')**) ? 6 # 0 ;$&< 0 8 . J ∗

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0 D(λ) ≤ J ∗ A 0 0A . ? D @ DA ) 8 :A . A , 0 0 A X ) A D ) ) 8 :A 8 *,*&'A%0)*1'./-) ; 8 0 G < 7 D . ,,31'9':.1',* *,* &'A%0)*1'./-)A 8 . 6 ( 0 ; # A 8 . 6 ( 0< / 0 . A 9%14,&) &) :,5: (0.&')*1A 6 " 8 ,0 ) :,5:(0.&')*1 ; 8 .<

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8 . Rn R ∪ {−∞}A ) J = {x ∈ Rn | J(x) > −∞} , 8 .A

. # 8 . ? # ? 8 ? 0

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Y L O P

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CC J " $ $"$ $-& R $ )) :,5:&'A%0)*1')- J $ x $/

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C E T Y $" )) , $"$ 5$" $" L O " 0 $"P $ $"$ $- $ $" *$ ! " $" E $ $-$" $0" $ ,%, L O ÉC ∂J(x) = {p ∈ Rn | J(y) − J(x) ≥ p · (y − x),

:,5:(0.&')*1:

∂J(x)

Rn

∀y ∈ Rn } .

;$<

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X 1) ;$< 6 ( ∂J(x) . 8 Rn 1, A 0 ∈ ∂J(x)A x . J

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T Y POL

N H EC

E U IQ

J(y) ≥ J(x) + p · (y − x) + o(y − x), y → x .

J .A . ? p ∈ ∂J(x)

E L O

•

3 ? 8 D D ? Rm+q D(λ) = −∞A λ ∈ ΛA ) 0 8 . Rm+q R ∪ {−∞} X ) A D = Λ 6 8 D A ? , ,0. .

ÉC

CC ))$ X *$' " D $"$ *$ ) ,%' " ) " " λ ∈ Λ' )$ Γ(λ) = x∈X L(x, λ) = {x ∈ X | L(x, λ) = D(x)}, ' ) " " x ∈ Γ(λ)' F (x) = (Fi (x))1≤i≤m+q " $ 0 $" D )$" λ,

0,3,:'1',*

%9,*:10.1',*C * x ∈ Γ(λ)A µ ∈ ΛA D(µ) − D(λ) ≥

L(x, µ) − L(x, λ) = F (x) · (µ − λ)A F (x) 6 D λ

)9.0;5)

E U IQ

CC * $ . 8 0 .

H C E T

N

6 ( 8 . J Rn R ∪ {+∞} 8 J(x) = supi∈I Ji (x)A R I 0 ) A x → Ji (x) .A i ∈ I A A x ∈ J A + ∂J(x) = ∂Ji (x) . ;&<

LE O ÉC

Y L O P i∈I Ji (x) = J(x)

-

HN

E U IQ

. 8 ,0I ,9 $"

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•

PΛ I Rm+q ΛA λ → ,.-(,0'149) &) :50(0.&')*1 ;I< 8 . D ? ρk pk , λk+1 = PΛ λk + ;&< pk

O P LE O ÉC

+ (λ+ 1 , . . . , λm , λm+1 , . . . , λm+q )

R λ0 ∈ Λ 0 A R pk 6 D

λk A R ρk 8 ρk → 0, ρi = +∞ . ;& < i

. A . λk+1 , 0 ) pk = 0 pk = 0A , 6 ,H J λk J ; ) H 6 , 6 8 < , G . A I? ,9 $ # /

G = (N , A)A , (i, j) G cij τij * ) A G 10A a s = 1 a p = 6A

ÉC

T Y POL

E L O

2

1, 8 1 8, 1

1, 4

4

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N H EC 2, 2 6

3, 6 3

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2, 3

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@ . c τ A A A , (1, 2) G c12 = 1 τ12 = 8 8 ,0 8 0 8 ;$$< * A . E x ∈ RA xij = 1 (i, j) A xij = 0 0 , s ? p T , (i,j)∈A cij xij J x ∈ {0, 1}A , ;&< x s ? p, (i,j)∈A τij xij ≤ T .

Y L O P

H C E T

N

E U IQ

# 0 G , ;$$< 3 A , .6 . L x s ? pM

LE O ÉC

'

N H EC

E U IQ

8 / . a B ( ;< 7 a . J ? ) L M 0 ? 0 A 0 8 0 A , , 8 QA , 0 A 0 6 0 G 9 A 8 D : R+ → R ∪ {−∞} , ( D(λ) = inf cij xij + λ τij xij − T , A

LE O ÉC

T Y POL

x ∈ {0, 1} x s p

= −λT +

(i,j)∈A

inf

x ∈ {0, 1}A x s p

(i,j)∈A

(cij + λτij )xij .

(i,j)∈A

E U IQ

3 λ )A D(λ) . ? 0 G :.*: +,*10.'*1): &) 1)93: G cij + λτij A 8 # A H , A , , ;&< 3 , 6 ;&< * λ1 = 0A 8 . G 1 → 6 G cij +0τij

ÉC

E L O

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2

1

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4

2

1

2 6

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2

5

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3. ρ1 = 1A ;&< λ2 = 1A . G cij + 1τij 5

2 9

4

3

1

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3

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9

9

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7

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ρ2 = 1A λ3 = 2A

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1

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G . ? ; 6 < 0 @ ,4)50':1';5): .(0.*(')**): * A . 6 0 .# 6 G = (V, E)A , 8 t : E → R+ , {i, j} → tij @ 0 .# 8 0 A . ? (V C)A min {i,j}∈E tij xij E x ∈ {0, 1} , k∈V,{k,j}∈E xkj = 2, j ∈ V A x ? |V| H .

ÉC

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N H EC

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1

1 ∈ V A j ∈ V \ {1}A Q k∈V,{k,j}∈E xkj = 2A 0 8 D : RV\{1} → R D(λ) = min {i,j}∈E tij xij + j∈V\{1} λj ( k∈V, {k,j}∈E xkj − 2) x ∈ {0, 1}E k∈V,{k,1}∈E xk1 = 2 x ? |V| H . 3 − j∈V\{1} 2λj A P D(λ) G , 160 D = (V, E)A H {i, j} G tij +λi +λj A i, j ∈ V \{1}A H {i, 1}A i ∈ V \{1}A G t1i +λi ' ,. I? 6 $A , B % " I 160 G O(|E| log |E|)A a . : ,

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6 ;&< 0 . J supλ∈RV\{1} D(λ) ' 0 0 0 .# A ? = B 0 160 . 6 $ [ . . D(0) 0 8 # k∈V,{k,j}∈E xkj = 2A j ∈ V \ {1}

T Y POL

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>)0+'+) CC 0 D(λ) &(

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>)0+'+) CC @ A : * + * % G = (N , A)( $ t : A → R+ ( + %% % [ * *( % λ * *( D(λ) + $ E ' O N ( : * >)0+'+)

T Y POL

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CC > &;( 4 Ji

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Rn R $ +∞ $ $ ( $ 0% ;( + > C &; " C ⊂ ∂J(x) 2 0 ∈ ∂J(x) ' *

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t

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N

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( & A$ ( " ' " ; ' ' " % " ( % & * + K ( (" " % FCH! ' % F/H , ( % ( " " & % ' 1 " " ' ( " " % %

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/ 0 . = 0 ; A .# T$U< * ) 6 A , = 0 R

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5 ) PK A 6 I ,0 . K A PK x = xK @ . ) PK 8 0 A ,6?6

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PK x − PK y ≤ x − y ∀ x, y ∈ V .

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N H EC

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" " yn 5 ( 5

V B' yn % % xK

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C É 0 % %

x − xK 2 ≤ x − (xK + θ(y − xK ))2 .

x − xK 2 ≤ x − xK 2 + θ2 y − xK 2 − 2θ x − xK , y − xK ,

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0 ≥ −2 x − xK , y − xK + θz2 .

0 * θ % ' ! A " xK " % y ∈ K x − y2 = x − xK 2 + xK − y2 + 2 x − xK , xK − y ≥ x − xK 2 ,

" % " xK ' ; ( x K

C E T Y L PO

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x, en en .

n≥1

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E L O

n )n≥1

T Y POL

N H EC

E U IQ

Sp z ≤ z∀z ∈ V.

5

Sp z (en )1≤n≤p " (z − Sp z) ( & ( (en )1≤n≤p % * "

ÉC

Sp z =

p

z, en en .

n=1

p

Sp y = y y ' (en )n≥1 " Sp x − x ≤ Sp (x − y) + y − x ≤ 2x − y ≤ 2.

$ % Sp x % x . % " #! 2 2 lim Sp x = x ,

p→+∞

" " *

C! %

H C E T

N

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, , 0 0 00 , = 0 . :6 , , 0 0 00

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V

vn

[v1 , · · · , vn ] = [e1 , · · · , en ]

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(vn )n≥1

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V

W

$ A V $ W " " $"$ &" $ $"$" C " AxW ≤ CxV

∀x ∈ V.

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N H EC

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$ ) @/" , $ V " $/ V ' " $/ ))"$ $ $ R, *$"$' $ $ $" L ∈ V " LV =

|L(x)| . x∈V,x=0 x sup

1 = 0 Q

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N H C

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x∈V,x =0

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x = z0

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1

H C E T

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% 5 K Ln ' % ( L *' " L = 0 5

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@ # 0 ? . x ∈ Rn ; < , 0 . Ax = b,

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;<

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xi =

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Y L O P

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sup x∈Cn ,x=0

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x∈Cn ,x=1

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x∈Cn ,x≤1

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H C E T

N

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@ Ap 0 ? . Cn n 1/p ) p ≥ 1 xp = ( i=1 |xi |p ) A p = +∞ x∞ = max1≤i≤n |xi | @ 0 ;1 A∗ I A<

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>)0+'+) C C "

Y L O P

*

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A2 = A 2 = * A( 2 A1 = max1≤j≤n ( ni=1 |aij |) ( ∗

n

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A∞ = max1≤i≤n

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Mn (R)A Mn (C) R ⊂ C · C . Cn A ) · R ? Rn . Rn * A ∈ Mn (R)A ) 0 AC AR AC =

sup x∈Cn ,x=0

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3 ) . H 4Q 8 ,9 A , [ xC x1 A x2 A x∞ ' A , . . AC > AR * A . * & 0 ) C 0 H ', C 1) 0 •

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# ρ(A) , Mn (C) 9 (A . ρ(A) = 0 . A = 0 ; A A . F < ' A 6 , ,0

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U " $ " $" U ∗ = U −1' $ U A2 = AU 2 = A2 , $ $"' A " $ " $ A∗ A = AA∗ ' A2 = ρ(A),

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%9,*:10.1',*C ' U ∗ U = IdA U A22 =

U Ax22 U ∗ U Ax, Ax = A22 . = sup 2 x, x x∈Cn ,x=0 x2 x∈Cn ,x=0 sup

E U IQ

1, A . 0 y = U x . ) x2 = y2 A

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AU x22 Ay22 Ay22 2 AU 22 = sup = sup = sup 2 2 = A2 . −1 y2 x∈Cn ,x=0 x2 y∈Cn ,y=0 U y∈Cn ,y=0 y2 2

Y L O P

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A A 0 0 . A2 = diag(λi )2 = ρ(A)

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·

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ρ(A) ≤ A.

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A ≤ ρ(A) + .

%9,*:10.1',*C λ ∈ C . A ρ(A) = |λ|A x0 = 0

. ;Ax0 = λx0 < @

λx0 = ρ(A)x0 = Ax0 ≤ Ax0 ,

,R , ρ(A) ≤ A ' . x0 H A , . Cn H ;8 + < + A U . 0 T = U −1 AU * δ > 0 ) Dδ = diag(1, δ, δ 2 , · · · , δ n−1 ) Tδ )

N H EC

Tδ = (U Dδ )−1 A(U Dδ ) = Dδ−1 T Dδ

. )

⎛

⎜ ⎜ Tδ = ⎜ ⎜ ⎝

t11

ÉC

δt12

··· δ

E L O 0 0

···

T Y POL

0

n−1

t1n

δtn−1n tnn

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

⎛

⎜ ⎜ . T = ⎜ ⎜ ⎝

t11 0 0

t12 ···

E U IQ

···

0

t1n tnn

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

9 > 0A δ : 6 Tδ A A 1 ≤ i ≤ n−1A n

δ j−i |tij | ≤ .

j=i+1

3 , B → (U Dδ )−1 B(U Dδ )∞ 0 ; 6 A < . ) 0 ; <

)99) C C

E U IQ

" A $ " Mn (C), " $ "$ -$" $"

N

-$" , limi→+∞ Ai = 0' , limi→+∞ Ai x = 0 ) " " -" x ∈ Cn ' 3, ρ(A) < 1' 9, &" $ $ $ " / $$ " A < 1,

LE O ÉC

Y L O P

H C E T

*

N H EC

E U IQ

%9,*:10.1',*C 7 ,0 ;< ; < , Ai x ≤ Ai x

T Y POL

limi→+∞ Ai x = 0 9 A ; < ;< A ρ(A) ≥ 1A λ x = 0 Ax = λx |λ| = ρ(A)A A A Ai x = λi x . . ' L;< ;<M * &A ,? ;< ;< * A 0 A < 1A

LE O ÉC

Ai ≤ Ai → 0 i → +∞,

Ai .

'

&

E U IQ

3. # A 8 . 0 0 , 9 (A ,# A ? 0 0 0 J ,0

$ 0 ; 8 ? - $ ( ) 8< 8 8 . 0 , ? , # , ) 0 9 A ? 8 )J+.+): )1 :1./-): ' ) * ) A ,

ÉC

E L O

T Y POL

N H EC

%@*'1',* C C " $ $ " / $$ $ $" A - 8*$"$ 3,,, $ )) $ "$$$" $ " A ∈ Mn (C)' " "" $' - *$ ) cond(A) = A.A−1 .

' . , ) 6 ; 0 <

E U IQ

" A $ " $-/, " b = 0 $ -" $$ $ , , " x " x + δx "$ )"- !"+ Ax = b ' " A(x + δx) = b + δb. $

0,3,:'1',* C C

LE O ÉC

Y L O P

H C E T

δb δx ≤ cond(A) . x b

N

;<

*

E U "$ )"- !"+I Q N ' " C H E T Y POL

, " x " x + δx

Ax = b

$

LE O ÉC

(A + δA)(x + δx) = b.

δA δx ≤ cond(A) . x + δx A

;<

8 ) ' $0" $" )",

)9.0;5) C C @ , 0 6 ; . < , 3 * A 6 8 8 , # • %9,*:10.1',*C * A Aδx = δbA

E U IQ

δx ≤ A−1 .δb @A b ≤ AxA ;< ' . J AA δb x ; A< ;< 8 9 (A , 0 ;. < x b = Ax δb δx = A−1 δb * 0 ;< Aδx + δA(x + δx) = 0A δx ≤ A−1 δAx + δxA ;< * , A . A 0 δA 0 b # 4Q y = 0 A−1 y = A−1 y @ δA = Id b = (A + δA)y @ . ) y = y + δx δx = −A−1 y A δA = || 0 , ;<

ÉC

E L O

T Y POL

N H EC

condp (A) = Ap A−1 p p = 1, 2, +∞,

R Ap ) @ . ) 8 0

>)0+'+) C C ) A ∈ Mn (C) cond(A) = cond(A−1 ) ≥ 1( cond(αA) = cond(A) ∀α = 0(

H C E T

N

E U IQ

(A) ( 4 µ1 (A), µn (A) 2 ( cond2 (A) = µµn1 (A) * * A(

Y L O P

n (A)| ( cond2 (A) = |λ |λ1 (A)| ( 4 |λ1 (A)|, |λn (A)| * A( U ( cond2 (U ) = 1(

LE O ÉC

*

N H EC

E U IQ

"

! U ( cond2 (AU ) = cond2 (U A) = cond2 (A)

>)0+'+) C C " * Kh (

T Y POL

2 % P1 (

LE O ÉC

cond2 (Kh ) ≈

4 . π 2 h2

Kh λk = 4h−2 sin2

kπ 2(n + 1)

;"<

1 ≤ k ≤ n,

uk

ukj

= sin

jkπ n+1

1 ≤ j, k ≤ n.

E U IQ

)9.0;5) C C , ;"< 6

N H EC

Kh 0 A . 9 (A ) . h = 1/(n + 1) . F 3 A . H 0 7 A . 0 , 0 ,H ) uh ( 2 A , 0 bh # Kh Uh = bh , * , , A ,O , 0 . b . Kh ? . λn 1, ,96 A . 8 ; , ? , < 0 bh L M ;,6?6 , 0 K . uk Kh
ÉC

E L O

T Y POL

δb δx ≤ C(K) , x b

R C(K) n ', . A I ) , ) , , •

5

Y L O P

%14,&) &!%-'9'*.1',* &) #.5::

H C E T

N

E U IQ

, ? , # 9 (A , # A

LE O ÉC

*

$

N H EC

E U IQ

T x = bA R T . 0A 8 0 . 9 (A #

T Y POL

⎧ t1,1 x1 + t1,2 x2 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ t2,2 x2 + ⎨

ÉC

OLE ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

···

···

t1,n xn = b1

··· tn−1,n−1 xn−1 +

t2,n xn = b2

.

tn−1,n xn = bn−1 tn,n xn = bn

,0 xn = bn /tn,n A xn−1 A I ,? x1 @ 0)9,*1%) ; , 8 A x1 ? xn &):+)*1)< + , # T x = b . T 1 H A , 4 . # Ax = b , . A , 4 J ; < J , M . 0 M A = T 6

A ; < ? I 0 J M bA ; < 0 J # T x = M b 6 , , M . . . , , , 6 4

ÉC

E L O

T Y POL

N H EC

E U IQ

" A $ " $-/ $$, &" $ $ " $-/ M " " T = M A " "$0 ) ,

0,3,:'1',* C C

Ak 1 ≤ k ≤ n (k − 1) E % A1 = A & An = T " $ akij 1≤i,j≤n Ak % Ak akkk

Ak & Ak+1 ' " % akkk 1 k % % & k k * ' % k + % $ Ak P k ' A˜k = P k Ak " % a˜kkk 1 akkk = 0 P k = Id 1 akik = 0 % i ≥ k + 1 k % i P k = (e1 , ..., ek−1 , ei , ek+1 , ..., ei−1 , ek , ei+1 , ..., en ) k akik % i ≥ k

LE O ÉC

Y L O P

H C E T

N

E U IQ

* & * S! ⎛

ÉC

OLE

˜k A

E U I Q N H EC T E

&

k

⎞

1

Y L PO ⎜ ⎜ 0 ⎜ ⎜ ⎜ ⎜ k E =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

1 −

a ˜k k+1,k a ˜k

1

k,k

a ˜k

− a˜n,k k

0

CN!

1

k,k

" k $ ⎛

a ˜111

···

⎜ ⎜ 0 ⎜ ⎜ ⎜ k+1 k ˜k A =E A =⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0

···

···

···

a ˜11n

a ˜kk,k

a ˜kk,k+1

···

a ˜kk,n

0

ak+1 k+1,k+1

···

ak+1 k+1,n

0

ak+1 n,k+1

···

ak+1 n,n

C E T LY

O P E L ÉCO

...

H N

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

E U IQ

a ˜ % ak+1 = a ˜kij − a˜ a ˜kk,j k + 1 ≤ i, j ≤ n Ak+1 ' * ij % k " E (n − 1) An % An = M A % M = E n−1 P n−1 ...E 1 P 1 M %' detP i = ±1 detE i = 1 $ & ; ' & M b! * & " P k E k $ ' (bk )1≤k≤n b1 = b, bk+1 = E k P k bk 1 ≤ k ≤ n − 1, ' bn = M b Ax = b " & An x = M b : An = T k i,k k k,k

# 6 " " " (

O $ ; M S $ ' E k P k M b An 1 A %' % % An = T % % T x = M b ; ' C k " k + 1 & n k + 1 & n # 5

O

A 0 + detA = ± detT ' +

LE O ÉC

Y L O P

H C E T

N

E U IQ

*

-

E U / ' '

" I Q

( % % N * ( % 'C H % ' ? " % & % E $ " * Y % T % ' L

* ! $P O " * % % ' * LE ! O ÉC a ˜kkk

akkk

a ˜kkk

k

(n − k) × (n − k)

•

%14,&) &) -. =.+1,0':.1',* ? 8 A A = LU A R L 8 ;L LCM < U ;U L M < , 8 H , 4 R ,* *) 3'2,1) <.9.': 8 0 8 AA # Ax = b . ? # Ly = b U x = y

T Y POL

N H EC

E U IQ

" $ " A = (aij )1≤i,j≤n n " " " " 0$ k' *$ )

0,3,:'1',* C C

ÉC

E L O

⎛

⎜ ∆k = ⎝

a11

···

a1k

ak1

···

akk

,, ,

,,,

,, ,

⎞ ⎟ ⎠,

$" $-/, &" $ $ ) " (L, U )' - U "$0 ) ' " L "$0 $ !$" $ 0$ ' " A = LU.

)9.0;5) C C ,# * " , 0 9 (A . A A A ) . 9 (A ∆k , . 0A . xk ∈ Ker∆k F . x = (xk , 0) . ) Ax · x = 0A ) 8 A •

H C E T

N

E U IQ

%9,*:10.1',*C , , 4 ,# 06 8 .A ,6?6 . 6 akkk 3A . * An = E n−1 ...E 1 A . E k ) ;$< @ U = An L = (E 1 )−1 ...(E n−1 )−1

LE O ÉC

Y L O P

*

N H EC

E U IQ

3 A = LU ? . ) L 0 86 8 (E k )−1 ,0 8 ? E k 6

A ,6?6 A lik = aki,k /akk,k A k + 1 ≤ i ≤ nA ⎞ ⎛ ⎛ 1 1 ⎟ ⎜ ⎜ ⎟ ⎜ 0 ⎜ 0 ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ 1 1 ⎟ , (E k )−1 = ⎜ Ek = ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ −lk+1,k +lk+1,k ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎠ ⎝ ⎝

ÉC

OLE

T Y POL −ln,k

0

1

0

+ln,k

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎠ 1

L 8 k 6 H (E k )−1 ⎞ ⎛ 1 0 ... 0 ⎜ ⎟ ⎜ l2,1 ⎟ ⎟. ⎜ L=⎜ ⎟ ⎝ 0 ⎠ ln,1 . . . ln,n−1 1

T Y POL

N H EC

E U IQ

8 . ) . , ,# 8 ∆k @ . ) . a11 ? ∆1 . 0 @ . I ,? , k − 1 7 . . akkk ' k − 1 . A 0 Ak @ , (E 1 )−1 ...(E k−1 )−1 Ak = A 8 , 0 k k k L11 0 ∆ A12 U11 Ak12 = , Lk21 Id Ak21 Ak22 A21 A22

ÉC

E L O

k . U11 A Lk11 A ∆k 0 k A Ak22 A IdA A22 0 n − k 9 0A 0 k Lk11 U11 = ∆k ,

E U IQ

k R U11 A Lk11 86 k . @ U11 = (Lk11 )−1 ∆k . 0 . 0 @ k # k detU11 = akii = 0,

LE O ÉC

.

akkk

Y L O P

? , k

i=1

H C E T

N

*

N H EC

E U IQ

,? . ) , 6 −1 −1 A = L1 U1 = L2 U2 @ L−1 2 L1 = U2 U1 A R L2 L1 −1 8 U2 U1 . & 9 A L−1 2 L1 −1 A L−1 2 L1 = U2 U1 = Id

LE O ÉC

T Y POL

" T $ " "$0 $ , $ $- &" " $ " "$0 $ " $" 0$ & $" $ - $" 0$ & T , " T $ " " "$0 $ , ) " T T " "$0 $ ' " $" 0$ & $" ) " $" 0$ & T " T ,

)99) C C

/ &

E U IQ

.-+5- 30.1';5) &) -. =.+1,0':.1',* C @ 8 LU

; < , A ) A LU 9 A = (aij )1≤i,j≤n A ⎛

1

⎜ ⎜ l2,1 L=⎜ ⎜ ⎝ ln,1

ÉC

0

...

PO

. . . ln,n−1

E L O

N H EC

⎛ ⎞ u1,1 0 ⎜ ⎟ ⎜ ⎟ ⎟, U = ⎜ 0 ⎜ ⎟ ⎝ 0 ⎠ 1 0

T Y L

...

...

u2,2 ...

0

u1,n un,n

⎞

⎟ ⎟ ⎟, ⎟ ⎠

L 8 U A 1 ≤ i, j ≤ n . min(i,j) n ai,j = li,k uk,j = li,k uk,j . k=1

k=1

9 ) A L U 3 A . (j − 1) L U 8 (j − 1) AA j 6 A ai,j = ai,j =

i k=1 j

li,k uk,j li,k uk,j

⇒ ui,j = ai,j − ⇒ li,j =

k=1

ai,j −

i−1

li,k uk,j

k=1 j−1 k=1 li,k uk,j

ujj

Y L O P

1 ≤ i ≤ j,

H C E T

N

E U IQ

j + 1 ≤ i ≤ n.

@ j j 6 U n − j j 6 L 8 (j − 1) @ . . ujj H V

LE O ÉC

*

N H EC

E U IQ

-(,0'149) *59%0';5)C @ 6 8 ,

? @ . . , A J ? , k k 6 LA 8 P F k 6 ( 0 k 6 . k + 1 ? n ' ? , k k k − 1 )A % H 0 A AA Ak Lk = (E 1 )−1 ...(E k−1 )−1 ; % . Lk ? F Ak k − 1 < 3 ) 0 L U ;L 86 ; < U <

LE O ÉC

T Y POL

←− ) k ( k = 1, n − 1 ( i = k + 1, n ←− i ik aik = aakk ←− * ( j = k + 1, n aij = aij − aik akj ←− i k + j + i + k

T Y POL

N H EC

E U IQ

,931) &!,3%0.1',*:C * ,: , 6

0 , ? ; ? , < @ 0 , A .6 # n 1 A ) . ; 0 H < W 9 J 0 , Nop

ÉC

E L O

Nop =

n−1

n

n

(1 +

j=1 i=j+1

1),

k=j+1

A A Nop ≈ n3 /3 W 0 ; 6 # < J 0 , Nop 8 Nop = 2

n

j,

j=1

H C E T

N

E U IQ

A A Nop ≈ n 3 , # Ax = b 8 Nop ≈ n3 /3 n2 0 . n3 n

Y L O P 2

LE O ÉC

*

N H EC

E U IQ

)9.0;5) C C @ 8 , . , * 0 A−1 A A 8 n # . 0 . 0 (ei )1≤i≤n ; 0 A . 0 ei 0 A , . L< 0 , A−1

ÉC

OLE

T Y POL

n2 n3 j 2 + + n( ) ≈ n3 . 3 2 2 j=1 n

Nop ≈

* AA A 8 U ; L .
%14,&) &) 4,-):DB

E U IQ

', , , # A ) . 9 ? 8 A 8 A = BB ∗ R B

8 ; B ∗ I <

T Y POL

N H EC

0,3,:'1',* C C " A $ " !" ' *$ )"-, &"

$ $ " B "$0 $ ' " " $" 0 $ & $" )"' " -*

ÉC

E L O

A = BB ∗ .

%9,*:10.1',*C * * "A

(L, U ) A = LU . U L 8 # @ D ) D = √ diag( uii ) 0 " U 0 ki=1 uii = det∆k > 0A R ∆k 6 , k AA uii 8 @ B = LD C = D−1 U . ) A = BC ' A = A∗ A C(B ∗ )−1 = B −1 (C ∗ ) 9 . & C(B ∗ )−1 A B −1 C ∗ 8 9 1 A B C HA B −1 C ∗ , A ,6?6 C(B ∗ )−1 = B −1 C ∗ = IdA C = B ∗ * , '%#A , 8 A = B1 B1∗ = B2 B2∗ A ,R B2−1 B1 = B2∗ (B1∗ )−1 1 &A B2−1 B1 = D = diag(d1 , ..., dn )A A = B2 B2∗ = B2 (DD∗ )B2∗ ' B2 . 0A . D2 = Id di = ±1 @ : , '%# 8 # 1 di = 1A B1 = B2

LE O ÉC

Y L O P

H C E T

N

E U IQ

*

N H EC

E U IQ

.-+5- 30.1';5) &) -. =.+1,0':.1',* &) 4,-):DBC 9 A 86

'%# B ) , A = BB ∗ A = (aij )1≤i,j≤n A B = (bij )1≤i,j≤n . bij = 0 i < j * 1 ≤ i, j ≤ nA .

LE O ÉC

T Y POL aij =

n

min(i,j)

bik bjk =

k=1

bik bjk .

k=1

9 ) A ; A .

H A # < B 3 A . (j − 1) B 8 (j − 1) AA j 6 A 1 2 j j−1 2 2 ajj = (bjk ) ⇒ bjj = 3ajj − (bjk )2 ai,j =

k=1 j

k=1

bjk bi,k

⇒ bi,j =

ai,j −

j−1

k=1

k=1 bjk bi,k

bjj

E U IQ

j + 1 ≤ i ≤ n.

N H EC

@ j 6 B 8 (j − 1) 3 A G A A # ) .A 8 3 A A , 2 ) .A . ajj − j−1 (b ) ≤ 0 j A k=1 jk H ,

E L O

T Y POL

-(,0'149) *59%0';5)C , '%# , 8O 6

ÉC

0 A ? 8 B + , : % 8 A A #

( j = 1, n ( k = 1, j − 1

ajj = ajj − (bjk )2 + k √ ajj = ajj ( i = j + 1, n ( k = 1, j − 1 aij = aij − bjk bik

+ k aij =

aij ajj

+ i + j

Y L O P

H C E T

N

E U IQ

,931) &!,3%0.1',*:C * ,: '%#

LE O ÉC

0 , ; < ? 6

*

N H EC

E U IQ

0 n 0 , W 2 '%# J 0 , Nop

ÉC

OLE

T Y POL Nop =

n

⎛

⎝(j − 1) +

j=1

n

⎞

j⎠ ,

i=j+1

A A Nop ≈ n3 /6

0 J 8 ( # ? B B ∗ 0 , Nop ≈ n2 '%# . &)5> =,': 3-5: 0.3'&) 4 # ) . W

.10'+): /.*&): )1 9.10'+): +0)5:):

N H EC

E U IQ

, 0 : A , +0)5:) ? A /.*&) * # ; )
# '

T Y POL

E L :$ " " " " /$ ' 0 O /$ # 0$ $" -*$" ) , C É /$ " , 0 %@*'1',* C C

A ∈ Mn (R) p∈N 2p + 1

ai,j = 0

|i − j| > p

, H 0 . .

>)0+'+) C C " H 0%=: )9.0;5) C C 8 '%# . 0 A , H 9 A A ;H ? , , 0
H C E T

N

E U IQ

, . ) , # ? 0

Y L O P

>)0+'+) C C " ( $ n * p( $ H O(np2 /3)

0%=: O(np2 /6)

LE O ÉC

*

N H EC

E U IQ

"

* 9 ,0 % ' % A 0 0 / 6 % A :1,+D.() ,0:)A A , , ⎛ ⎞ 9 0 −3 0 ⎜ 7 −1 0 4 ⎟ A=⎝ ⎠. 0 5 2 0 1 0 −1 2

LE O ÉC

T Y POL

A %A A 0 ? @ ) 0 0 A J ; ;i<< i @ 0 , 0

% J ai,j % (k)A

(k) = j 0 A ? .

;

. < 0 A ? . 1 A

N H EC

ÉC

E L O

T Y POL 9 −3 7 −1 4 5 2 1 −1 2

1 3 1 2 4 2 3 1 3 4

E U IQ

1 3 6 8

9 A 8 '%# , 8 L M . , ; 6 L M : 0 < 9 A . 6 . . 6 6 9 (A 6. 8 ? . . # %

E U IQ

N H C $ " " ( T E " Y % % A " C)! T

OL6

% " P " % E L O ÉC ;5'2.-)*+) &!,3%0.1',*: )1 .-(,0'149) &) 10.::)* n3

n

*

$

E U " ' I Q" % ' & % ' " ! N H C # ! E ! ! !6! !1 T ! Y

! ! O L ! P !

E L 1 ' ( O " % ? " ( É C ' " O(nα )

α≥2

O(nα )

A−1

I(n) C α≥2 AB

I(n) ≤ Cn

α

C

P (n)

" P (n) ≤ C nα % α $ " "

Id 0 0

A Id 0

0 B Id

−1

=

Id 0 0

−A Id 0

AB −B Id

.

E U IQ

" AB ' % C * .

N H EC

P (n) ≤ I(3n) ≤ C3α nα .

1 P (n) ' AB ( " C α % P (n) ≤ Cnα ? " ( A−1 ' I(n) " C " I(n) ≤ C nα % α $ " "

E L O

ÉC

A C

B D

−1

T Y POL

=

A−1 + A−1 B∆−1 CA−1 −∆−1 CA−1

−A−1 B∆−1 ∆−1

.

% ∆ = D − CA−1 B * 1( ! $ I(2n) ≤ 2I(n) + 6P (n),

0 * n = 2k ' I(2k ) ≤ 2k I(1) + 6

k−1

2k−i−1 P (2i ) ≤ C

2k +

i=0

5

α ≥ 2

k−1

2k−i−1+αi

.

i=0

I(2k ) ≤ C 2αk .

E U IQ

1 n = 2k k k " 2k < n < 2k+1 $ A

2k+1

:

Id

A 0

0 Id

Y L O P

2k+1 − n $ '

LE O ÉC

I(n) ≤ C (2

H C E T

) ≤ C 2α nα ,

k+1 α

N

*

&

E U IQ

N % ! H $ "

C ? % 1 % * E T INI " Y % 0 + 1 ( L

" ' 6 O ' ( P % CG! LE O & * K % ( É C5 " ' % "

n

n3

n

Nop (n) = O(nlog2 7 )

log 2 7 ∼ 2, 81.

n

; % ( ' & "

α " Nop (n) = O(nα )! 0%

( ' ( 1 +% ! ?( ( % ' '

" ! " " ( 1

% ' ' S

E U Q # # (+ - , ! # I N C ! < < ! H ; C E , Y " T OL P E L % ÉCO

a c

b d

α γ

β δ

=

m1 m2 m3 m4

m1 + m2 − m4 + m6 m6 + m7

= (b − d)(γ + δ) = (a + d)(α + δ) = (a − c)(α + β) = (a + b)δ

m4 + m5 m2 − m3 + m5 − m7

,

m5 = a(β − δ) m6 = d(γ − α) m7 = (c + d)α

$ ' G )

CC " 1 %' '

% 0 % ' 1 n = 2k $ # ' k−1 2 " 1 1

' Nop (n) +

% k k−1 k−1 2 Nop (2 ) = 7Nop (2

) + 18(2

) ,

H C E T

N

E U IQ

n " n2 ,

Y L O $ * " 'P E L O ÉC k−1

Nop (2k ) = 7k Nop (1) + 18

7i 4k−1−i ≤ 7k (Nop (1) + 6) .

i=0

Nop (n)

% ' CG!

*

-

5 $

N H EC

E U IQ

. 9 (A . . G % 0 8 ; 8 '%# , n3 < 'O .

LE O ÉC

T Y POL

" A $ " $-/, $ $" " $ )"$ 0 + A $ $0 1)""$02' " $ ) " (M, N ) - M $-/ " $- $ )" " A = M − N , "# ""- / )""$0 (M, N ) " *$ )

%@*'1',* C C

x0 $$ $ Rn , M xk+1 = N xk + b ∀k ≥ 1.

;-<

E U IQ

xk . . x k . , ) A A ? ;-
N H EC

(M − N )x = Ax = b.

T Y POL

* A . A 8 # 1, . A 8 . H A ,6 ?6 ? xk : x ' P xA ,H x − xk ≤ R * P Ax ; . b
ÉC

E L O

x − xk ≤ A−1 b − Axk ≤ A−1

H

%@*'1',* C C $ " $ "# ""- " $-0$" ' ;5)- ;5) :,'1 -) +4,'> &5 2)+1)50 '*'1'.- x0 ∈ Rn ' " "$ ))# xk

$-0 - "$ &" x,

H C E T

N

E U IQ

@ : . , . ? , # , ;. 1) " # <

Y L O P

"# ""- *$ ) 3, $-0 " $" !$ )" " ""$ M −1 N -* ρ(M −1 N ) < 1,

)99) C C

LE O ÉC

*

N H EC

%9,*:10.1',*C @ ) , ek = xk − x @

E U IQ

ek = (M −1 N xk−1 + M −1 b) − (M −1 N x + M −1 b) = M −1 N ek−1 = (M −1 N )k e0 .

T Y POL

* -A ek . A e0 A ρ(M −1 N ) < 1

LE O ÉC

. " & * % ! 5 " %

" # " A ! ! & " ! ; A & A = M − N M 2 ! (M ∗ + N ) ! ' (M ∗ + N ) & ρ(M −1 N ) < 1.

D ' " M ∗

∗

∗

∗

+N

∗

' ( K

E U IQ

(M + N ) = M + N = (A + N ) + N = A + N ∗ + N = M ∗ + N.

∗

$ % |x|A = Ax, x Rn " ' A %! $ . ' & |.|A $ % " M −1 N < 1 " " P & CG $

N H EC

T Y $

C O L P E L 5

' O C É

M −1 N 2 = max |M −1 N v|2A . |v|A =1

v

|M

−1

N v|2A

N = M −A |M

−1

N v|2A

= M

−1

w=M

= = = =

= = =

−1

M −1 N

" |v|A = 1

2

N .

−1

Av

−1

AM N v, M N v

AM −1 (M − A)v, M −1 (M − A)v

(Av − AM −1 Av), (I − M −1 A)v

Av, v − AM −1 Av, v + AM −1 Av, M −1 Av − Av, M −1 Av 1 − M −1 Av, M M −1 Av + AM −1 Av, M −1 Av − M M −1 Av, M −1 Av 1 − w, M w + Aw, w − M w, w 1 − (M ∗ + N )w, w.

$ (M ∗ + N )w, w > 0 (M ∗ + N ) % w = 0 A M %' . −1 2 ∗ M

N = 1 − (M + N )w, w < 1,

N

E U IQ

" " ( % &

( % " % ( * % % * J (

%

LE O ÉC

Y L O P

H C E T

*

E U # " " ! ; & I Q " !! N H %

! 7 C ! $< T ! E E Y L O ! P L E

O ÉC A = M−N Ax = b k ∈ Rn

A x ∈ Rn

b ∈ Rn

k

M

xk+1

xk+1 = M −1 N xk + M −1 b + k .

ρ(M −1 N ) < 1 k≥0

k ≤ .

2 K M −1 N lim sup xk − x ≤ K. k→+∞

$

M −1 N ek + k

$

ek = M −1 N

k

e0 +

ek = x k − x

k−1

M −1 N

i=0

i

k−i−1 .

N H EC

ek+1 =

E CI! U IQ

CG ' · s " M −1 N s < 1 " ρ(M −1 N ) < 1 $ *2 % $ % Rn " % K C ≥ 1 " " M −1 N "

ÉC

E L O

T Y POL

C −1 y ≤ ys ≤ Cy

∀y ∈ Rn .

0 ; CI! %

ek s ≤ M −1 N ks e0 s +

k−1

M −1 N is C ≤ M −1 N ks e0 s +

i=0

C 1 − M −1 N s

: ' % K = C 2 /(1 − M −1 N s )

. 0 diag(aii ) 0$

)"$

E U IQ

" A = (aij )1≤i,j≤n , $ $" D = A, $ )) "# / "# ""-

%@*'1',* C C $9%14,&) &) V.+,/'6 M = D,

H C E T

N = D − A.

N

, . * , 0 ) A 8 0 G D . 0 * &A R A # A N0 . A 2D − A ) .

LE O ÉC

Y L O P

*

E U , $ "I Q ) . H N " 0$' " )" "$0 $ ""$"' " " )" "$0 ) C E"# ? "# ""- ""$" , $ )) T Y )"$L O P LE O ÉC %@*'1',* C C $9%14,&) &) #.5:: )'&)-6

A = (aij )1≤i,j≤n D = diag(aii ) −E −F

A = D−E −F

A

A

M = D − E,

N = F.

* 4 6 0 ) A 8 D − E . 0A ,6?6 D . 0 ; (D−E) 8 ? . < * &A A # ) .A 4 6 .

" ω ∈ R+ , $ )) "# &"$ C $ $0 ) 1 - - C&"$2' ) )+" ω' "# ""- )"$

%@*'1',* C C $9%14,&) &) 0)-.>.1',* $ 66

D − E, M= ω

1−ω N= D+F ω

T Y POL

N H EC

E U IQ

* 0 ) A 8 D . 0 * ω = 1A . 4 6 ω < 1A , , , 6 A ω > 1 ?

6 9 A ωopt # , M −1 N A . .

ÉC

E L O

>)0+'+) C C ) A % " ω ∈ ]0, 2[( % *

>)0+'+) C C " ( % ( 5 ρ(M −1 N ) ≥ |1 − ω| , ∀ω = 0,

$ * 0 < ω < 2

E U IQ

" $ )+" α = 0, $ )) "# 0 $" "# ""- )"$

%@*'1',* C C $9%14,&) &5 (0.&')*16 M=

1 Id α

"

N=

1 Id − A . α

Y L O P

H C E T

N

0 . 6 A 8 f (x) = 12 Ax · x − b · x 0

LE O ÉC

*

)99) C C ... ≤ λn ,

E U )) " $ " 0$/ - I$Q$-0 N ' "# H 0 $" ) $ ' C "# 0 $" $-0 " E )"' $ ' " )+" ' " T Y P O L "

LE O ÉC

λ1 ≤ λ2 ≤

A

λ1 ≤ 0 ≤ λn - α, 0 < λ1 ≤ $" 0 < α < 2/λn αopt =

... ≤ λn

ρ(M −1 N )

α

2 λ1 + λn

min ρ(M −1 N ) = α

λn − λ1 . λn + λ1

)9.0;5) C C λ1 ≤ ... ≤ λn < 0A #

) 8 α −α * αopt A # , 8 λn /λ1 A I A , cond2 (A) A * A A 0 A . •

E U IQ

%9,*:10.1',*C 1, $A . ρ(M −1 N ) < 1 @ M −1 N = ( Id − αA)A

N H EC

ρ(M −1 N ) < 1 ⇔ |1 − αλi | < 1 ⇔ −1 < 1 − αλi < 1 , ∀i.

T Y POL

' αλi > 0 1 ≤ i ≤ n * . A . H H α . λ1 ≤ 0 ≤ λn A α A A 0 < λ1 ≤ ... ≤ λn A , 8 0 < α < 2/λn * αopt A 8 λ → |1 − αλ| ] − ∞, 1/α] [1/α, +∞[A

ÉC

E L O

ρ(M −1 N ) = max{|1 − αλ1 |, |1 − αλn |}.

* A 8 α → ρ(M −1 N ) , 6 2 αopt = λ1 +λ n

5 I I

. # # ) . , , ; 0 . A . 1) < * I

A , B#.

Y L O P

H C E T

N

E U IQ

" r0 $ -" $ Rn , $ )) ) Q!- -" r0 ' " $ $" Kk ' ) -" Rn $0$ ) k + 1 -" {r0 , Ar0 , ..., Ak r0 },

%@*'1',* C C

LE O ÉC

*

N H EC

E U IQ

B#. (Kk )k≥0 8 6 .6 Kk ⊂ Kk+1 ∀k ≥ 0 ' Kk ⊂ Rn A . 8 6 ? , * A . ) , k0 A . 0 ≤ k0 ≤ n − 1A

LE O ÉC

T Y POL

dimKk = k + 1 0 ≤ k ≤ k0 , dimKk = k0 + 1 k0 ≤ k.

9 r0 = b−Ax0 A 8 . A A , xk ? , : [x0 + Kk−1 ] ; ) ,0 . x (x − x0 ) ∈ Kk−1
A ? , . x0 ∈ Rn r0 = b − Ax0 A ? . xk ∈ [x0 + Kk−1 ] rk = b − Axk 6 Kk−1 A k ≥ 1

E U Q I " $ " !" *$ N )"- , " ' ' " " ) H Q!- , "# 0 $" $> 0 " *$' ) E C ' ) T "Y L O P $ $ -" $$ ) 3,, 8 ) ' "" " " ' &" /$ E "# $-0L - "$ !"+ $ $' ) ' ""$, O ÉC

)99) C C

A x0 ∈ Rn r0 = b − Ax0

(Kk )k≥0

xk ∈ [x0 + Kk−1 ]

n r0

k≥1

;<

rk = b − Axk ⊥Kk−1 .

k≥1

xk

Ax = b

n

)9.0;5) C C $ , 6 I

, O . 8 , . 0 ) , ; k0 + 1 R k0 B#.< ' A , . . L M 0 k0 + 1 6 . 8 . I

9 (A rk ? 6 Kk • %9,*:10.1',*C 7 ,0 , 0 xk 6 8 # ' A ) .A ) x, yA = Ax · y Rn @ xk 8 xk = x0 + yk . yk ∈ Kk−1 A , rk . A−1 r0 − yk , yA = 0

H C E T

∀ y ∈ Kk−1 ,

Y L O P

N

E U IQ

, , yk I A−1 r0 6 Kk−1 ; , A < ' . , , xk

LE O ÉC

*

N H EC

E U IQ

k0 B#.A ,6?6 A k ≥ k0 A dimKk = k0 + 1 9 A AKk0 ⊂ Kk0 +1 = Kk0 A rk0 +1 = b − Axk0 +1 = r0 − Ayk0 +1 ? Kk0 * A rk0 +1 = 0 xk0 +1 #

LE O ÉC

T Y POL

)9.0;5) C C I

, ? , B#. / . A .A , 0 * A xk ∈ [x0 + Kk−1 ] [x0 + Kk−1 ] f (x) =

1 Ax · x − b · x, 2

0 rk = b − Axk Kk 8 g(r) =

1 −1 A r·r 2

. r = b − Ax

N H EC

E U IQ

•

>)0+'+) C C ) A : ) (xk )0≤k≤n

T Y POL

% % * 5* rk = b − Axk dk = xk+1 − xk " $ N: Kk * +

ÉC

E L O

Kk = [r0 , ..., rk ] = [d0 , ..., dk ],

(rk )0≤k≤n−1 %* rk · rl = 0 0 ≤ l < k ≤ n − 1,

(dk )0≤k≤n−1 5* + A Adk · dl = 0 0 ≤ l < k ≤ n − 1.

) , . I

9 (A , , rk ? Kk−1 A xk . 8 .

HN

E U IQ

" A $ " !" *$ )"-' " x0 " (xk , rk , pk ) " " *$ ) "$ $

0,3,:'1',* C C

C E T Y L " ) O P E OL

p0 = r0 = b − Ax0 ,

ÉC

⎧ ⎨ xk+1 = xk + αk pk 0≤k rk+1 = rk − αk Apk ⎩ pk+1 = rk+1 + βk pk

∈ Rn ,

;<

*

- αk =

N H EC

rk 2 Apk · pk

T Y POL

" βk = rrk+12

E U IQ

"

2

.

k

' (xk )0≤k≤k +1 " " "$ ))# "# 0 $" $> 0 *$ ) 3,, 0

LE O ÉC

6 * " r0 = b − Ax0

rk+1 = rk − αk Apk xk+1 = xk + αk pk

" " rk ' & % rk = b−Axk , * " r0 = p 0

rk = rk−1 − αk−1 Apk−1 pk = rk + βk−1 pk−1

E U IQ

" " pk rk & W % Kk k ≥ 0 $ xk+1 = xk + αk pk " xk+1 ' & [x0 + Kk ] * " rk+1 ( & Kk D ' " rk+1 ( & rj 0 ≤ j ≤ k " pk+1 ; & pj 0 ≤ j ≤ k & " Apk+1 · pj = 0 2

T Y POL

N H EC

r1 · r0 = r0 − α0 Ap0 · r0 = 0

p0 = r0

E L $ " ; " & É C O

Ap1 · p0 = (r1 + β0 p0 ) · Ap0 = α−1 0 (r1 + β0 r0 ) · (r0 − r1 ) = 0. k

Apk · pj = 0 0 ≤ j ≤ k − 1. ? " % & k + 1 * " xk+1 rk · rj = 0

0 ≤ j ≤ k − 1,

rk+1 · rj = rk · rj − αk Apk · rj ,

& rj = pj − βj−1 pj−1 '

rk+1 · rj = rk · rj − αk Apk · pj + αk βj−1 Apk · pj−1 .

( ( * " rk+1 · rj = 0 j ≤ k − 1 " * αk " " rk+1 · rk = 0 . * " pk+1 & Apk+1 · pj = pk+1 · Apj = rk+1 · Apj + βk pk · Apj ,

Apj = (rj − rj+1 )/αj

Y L O P

H C E T

N

E U IQ

Apk+1 · pj = α−1 j rk+1 · (rj − rj+1 ) + βk pk · Apj .

j ≤ k −1 ( ( ( rk+1 " % '! % " Apk+1 · pj = 0 j = k ' Apk+1 · pk = 0 P *

LE O ÉC

*

$

HN

E U I Q (

αk βk 5 5

* (rk )0≤k≤k ' " rk = 0 $ rk ∈ Kk " = Kk = [r0 , ..., rk ] " rk+1 ' ( & Kk xk ' ;

O P LE O ÉC

C E T LY

0

# $ $ " (p ) 6+ & A k

( x, yA = Ax · y 5 " & (

* C! 8%9 0 * " & CCI 0 C)

" dk = xk+1 −xk ; & A " (d0 , ..., dk ) T % Kk $ " dk K " ( O 1( & (r0 , ..., Ak r0 ) x, yA % pk " & dk % " dk = αk pk ! OP & A * O 1( " pk ' ' " " • ! * C!

N H EC

E U IQ

-(,0'149) *59%0';5) 1 A , . , I

A 8 ;< * .

x0 r0 = p0 = b − Ax0

ÉC

E L O

T Y POL

) k ≥ 1

⎧ rk−1 2 ⎪ αk−1 = Apk−1 ⎪ ·pk−1 ⎪ ⎪ ⎪ ⎨ xk = xk−1 + αk−1 pk−1 rk = rk−1 − αk−1 Apk−1 ⎪ ⎪ rk 2 ⎪ ⎪ ⎪ βk−1 = rk−1 2 ⎩ pk = rk + βk−1 pk−1

1 rk = 0A , . A ,6?6 xk # Ax = b @ . k0 +1 A R k0 ≤ n−1 B#. ; , P < ' A I I ? , A . rk0 +1 = 0 ', A e e ;# 10−4 10−8
H C E T

N

E U IQ

* A # ; n k0 e eA , 104 ? 106
.A ,6?6 , . A 6 A 0 , 0 8 ? k0 + 1 ;8 * 6 <

LE O ÉC

Y L O P

* )9.0;5) C C

N H EC

E U IQ

&

9 A , , A , I

x0 = 0 0 ( A x0

T Y POL

3 , 0 8 , 6. A ? . Apk A rk 8 rk = b − Axk

LE O ÉC

* . I

A , % A 0 . Ay . y

I

: 9 0 6 . A # ) .

E U IQ

•

>)0+'+) C C ) % * 5* %

N H EC

( ( k0 = n − 1( $ : Nop = n3 (1 + o(1))

T Y L "P O

$ " !" *$ )"-, " "$ &"L E !"+ , " " "$ ))# 0 $" $> 0 , ÉCO / . . ;. TU<

0,3,:'1',* C C

A

x

Ax = b

xk − x2 ≤ 2

xk

cond2 (A)

cond2 (A) − 1

cond2 (A) + 1

k

x0 − x2 .

@ , # ) .A 8 cond2 (A) = λn /λ1 A R λ1 , λn 6 . . A * . I

. 0 . @ 5 ,0 A I

8 0 . 9 (A H 8 n . A , x xk , , 1, A . . AA 6 H I

. 0 . ; .6

,H < 9)A . , cond2 (A) A ,6?6 A 0

LE O ÉC

Y L O P

H C E T

N

E U IQ

*

-

0%+,*&'1',**)9)*1

N H EC

E U IQ

' . . I

AA , # Ax = b C −1 (C −1 A) A 9 C LM A 8 ? .

LE O É C "$$$"

T Y POL

" !"+ $ Ax = b, $ )) )$ A' $ " C $- " cond2 (C −1 A) " ) )"" cond2 (A), $ )) !"+ )$ "$$ !"+ -$" C −1 Ax = C −1 b, %@*'1',* C C

9 A C −1 A , # A 0 I

', . L# M . ; )< C # ) . '%# C = BB ∗ @ # Ax = b # . ˜x = ˜b, R A˜ = B −1 AB −∗ , ˜b = B −1 b x˜ = B ∗ x. A˜ ; <

T Y POL

N H EC

E U IQ

A˜ # ) .A , 6 I

# / A P I 8 '%# C ; 0 . 8 , < # 8 , I

# ; < R C P ; 8 B <

ÉC

E L O

>)0+'+) C C ˜· + $ * % rk = * 5* : 2 ) xk = B −∗ x˜k ( rk = B˜ b − Axk ( pk = B −∗ p˜k " $ * % * 5* 2 $

⎧ ⎨ x0 r0 = b − Ax0 ⎩ p0 = z0 = C −1 r0

) k ≥ 1

⎧ z ·rk−1 αk−1 = Apk−1 ⎪ k−1 ·pk−1 ⎪ ⎪ ⎪ xk = xk−1 + αk−1 pk−1 ⎪ ⎪ ⎨ rk = rk−1 − αk−1 Apk−1 zk = C −1 rk ⎪ ⎪ ⎪ zk ·rk ⎪ βk−1 = zk−1 ⎪ ·rk−1 ⎪ ⎩ pk = zk + βk−1 pk−1

Y L O P

H C E T

N

E U IQ

: . / 0 C

LE O ÉC

*

N H EC

E U IQ

L 6

M J ? C = diag(A) :A 8 . L @+M ; # @+< 9 D = diag(A) , # A −E 8 A = D − E − E ∗ A ω ∈]0, 2[A D D ω −1 ∗ −E D −E . Cω = 2−ω ω ω

LE O ÉC

T Y POL

@ . ) A A ) .A C , # Cz = r 8 ? C I? 8 8 . 8 , . C . ? ( . . ; @+
>)0+'+) C C

) A $ n N = 1 $ h = 1/(n + 1) ⎛

2

⎜ ⎜ −1 ⎜ −1 ⎜ A=h ⎜ 0 ⎜ ⎜ ⎝ 0

E L O

−1

0

2

−1

···

···

⎟ ⎟ ⎟

E U IQ

⎟ . 0 ⎟ ⎟ ⎟ −1 2 −1 ⎠ 0 −1 2

T Y POL

⎞

N H EC

0

"

ÉC

ωopt =

2 π ) π 2(1 − 1 + 2 sin 2n n

Cω−1 A 5 cond2 (Cω−1 A) ≤

1 1 + π , 2 2 sin 2n

( n * ( * * n * L '%# M C 8 BB ∗ R B 8 L M 8 '%# A ;. * < ' 8 B 0 , 8 '%# ? A 8O , bij = 0 aij = 0 ' ) , A , 8 B AA , 8 0 ; < 8 A ; ) < '%# . :

LE O ÉC

Y L O P

H C E T

N

E U IQ

*

N H EC

E U IQ

% %

1 . . , # # , ) , . 1 A . . # 6I ;. ;& << * . , A #X det(A − λ Id) * . A [. , L :M 8 #X , J 4 30 , ; A A < , #X ? " * , . A , #X nA P (λ) = (−1)n λn + a1 λn−1 + a2 λn−2 + · · · + an−1 λ + an ,

LE O ÉC

T Y POL

E U IQ

#X ; . ? < ⎞ ⎛ −a1 −a2 · · · · · · −an ⎜ 1 0 ··· ··· 0 ⎟ ⎟ ⎜ ⎜ ⎟ ⎟. 0 A=⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ 0 ··· 0 1 0

ÉC

E L O

T Y POL

N H EC

* A ;,6?6 0 ) , < . V , . . ; . < . . . , Q 0 : , # 2 A # ; , : < 0 6 I / # ; # , A . : < 0 4 .6=

; < . . ? . 2 A FA L0M

I

A ? 0 0 . :

Y L O P

5

H C E T

N

E U IQ

. ; < , ; . <

LE O ÉC

$ $

N H EC

E U IQ

. H , H ; ? A ,6?6 6 < A # , nA . (λ1 , · · · , λn ) . λn > |λi | 1 ≤ i ≤ n − 1 . λn ) , 6

LE O ÉC

T Y POL

# , x0 ∈ Rn x0 = 1

,) k ≥ 1 # yk = Axk−1 xk = yk /yk

xk − xk−1 ≤ ε- .

1 . ε 0 A # ? 10−6 δk = xk − xk−1 A xk . A . yk Axk − yk xk = Aδk

N H EC

E U IQ

$ )) " A " !" ' - )) (λ1 , · · · , λn )' $ / "#$ -" )) (e1 , · · · , en )' " - )) ) 0$ λn " ) " )"-' " |λ1 |, · · · , |λn−1 | < λn , $ )) -" $" x0 $" ) "#0$ en, "# ) $ $-0' " 0,3,:'1',* CC

ÉC

E L O

T Y POL

lim yk = λn ,

k→+∞

lim xk = x∞

k→+∞

- x∞ = ±en.

-" $-0$ " ))"$$ ))" |λn−1 |/|λn | λn−1 2k , |yk − λn | ≤ C λn

λn−1 k . xk − x∞ ≤ C λn

)9.0;5) CC . . yk

. xk ; < 8 #6 A . yk • n

%9,*:10.1',*C x0 = ? Ak x0 =

i=1 βi ei . A n k i=1 βi λi ei A ,R .

H C E T k

n−1 βn en + i=1 βi λλni ei xk = . n−1 2 λi 2k 1/2 2 βn + i=1 βi λn

LE O ÉC

Y L O P

E U IQ

. βn = 0 . xk

N

*

N H EC

E U IQ

' |λi | < λn xk . . sign(βn )en 1 HA n−1 2 λi 2(k+1) 1/2 2 βn + i=1 βi λn yk+1 = λn , n−1 2 λi 2k 1/2 2 βn + i=1 βi λn

LE O ÉC

T Y POL

. . λn

9 ; . , 0
# A . . 0 < λ1 < |λi | 2 ≤ i ≤ n

# , x0 ∈ Rn x0 = 1

,) k ≥ 1 # ) Ayk = xk−1 xk = yk /yk

xk − xk−1 ≤ ε- .

δk = xk − xk−1 A xk−1 . . 1/yk Axk−1 − xyk−1 = −Aδk k

E L O

T Y POL

N H EC

E U IQ

$ )) " A " !" ' - )) (λ1 , · · · , λn )' $ / "#$ -" )) (e1 , · · · , en )' " - )) ) )"" λ1 " ) " ""$" )"-' " 0 < λ1 < |λ2 |, · · · , |λn |, $ )) -" $ " x0 $" ) "#0$ e1 , "# ) $ $- $-0' " 0,3,:'1',* CC

ÉC

1 = |λ1 |, k→+∞ yk lim

lim xk = x∞

k→+∞

- x∞ = ±e1.

-" $-0$ " ))"$$ ))" λ1 /|λ2 | 2k yk −1 − λ1 ≤ C λ1 , λ2

k λ1 xk − x∞ ≤ C . λ2

E U IQ

? * ,

H C E T

N

)9.0;5) CC + , ? . * . A I ? A , A − σ Id . σ λ1 •

LE O ÉC

Y L O P

$ $

E U 5 -$ =8 I Q N H C E T Y L O P LE " $ " !" O C &" " "#0$ " É

4 .6= . J ,0 , = # A ; ,( 0 ) ,
)99) CC $&) ,5:)4,-&)06

A

(n − 2)

Hk

n,

T = (H1 H2 · · · Hn−2 )∗ A(H1 H2 · · · Hn−2 )

" " 0$, K$ 4' A " T $" E - )),

A (Ak )1≤k≤n−1 A1 = A Ak+1 = Hk∗ Ak Hk % Hk ( ( Ak ' %

Ak =

Tk Ek

Ek∗ Mk

N H EC

" "

E U IQ

: Tk k Mk n − k Ek & (n − k) k ak ∈ Rn−k

T Y O L P E OL ⎛

×

⎜ ⎜ × ⎝

⎞

×

Tk = ⎜

C 6 É "

× ×

×

⎟ ⎟ ⎟, ⎠

⎛ ⎜ ⎜ ⎝

Ek = ⎜

0

0

···

···

0

ak,1

ak,2

0

⎞

⎟ ⎟ ⎟. ⎠

ak,n−k

An−1 * $ " " A ' * k = 1 1 Hk

Hk =

Idk 0

0 H˜k

,

% Idk k H˜k B ( n − k vk (vk )∗ , % vk = ak + ak e1 , CC! H˜k = Idn−k − 2 vk 2 : e1 % ' " Rn−k A " " H˜k ak = −ak e1 " Hk ( " 4 " H˜k ' " vk = 0

2 k Ak ;& Hk = Idn , "

N H C T E ⎛

Ak+1 = Hk∗ Ak Hk =

ÉC

Tk H˜k Ek

Y L O P

(H˜k Ek )∗ H˜k Mk H˜k

OLE

%

⎜

⎜ H˜k Ek = ⎜ ⎝

0

0

···

···

E U IQ

0

−ak

0

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⎟ ⎟ ⎟, ⎠

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9 , # ⎛ b1 ⎜ ⎜ c1 A=⎜ ⎜ ⎝ 0

O P LE O ÉC

HN

E U IQ

0 4 . 6

c1

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0

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cn−1 bn

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R A A ci = 0 1 ≤ i ≤ n − 1 9 (A , i ci = 0A . 8 det(A − λ Id) = det(Ai − λ Id) det(An−i − λ Id)

R Ai An−i H # A , i n − i . 1 ,0

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E U "I Q ) N H

1 ≤ i ≤ n' $ *$" $ " Ai

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b1 ⎜ ⎜ c1 Ai = ⎜ ⎜ ⎝ 0

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⎟ ⎟ ⎟. ⎟ ⎠

E L O $ )!$H "" , " ci−1

ci−1 bi

" pi (λ) = det(Ai − λI) $

i

pi

-*

pi (λ) = (bi − λ)pi−1 (λ) − c2i−1 pi−2 (λ) ∀i ≥ 2,

- p1 (λ) = b1 − λ " p0 (λ) = 1, 8 ) ' ) " " i ≥ 1' )!$H pi )+ ))" -$" lim pi (λ) = +∞' , λ→−∞ , pi(λ0 ) = 0' pi−1 (λ0 )pi+1 (λ0 ) < 0' 3, pi " i $ "$" )$" ""$" (i + 1) $ pi+1 ,

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0 % det(Ai −λ Id) & ' * % < " % " * " pi (λ0 ) = 0 2

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H C E T

pi+1 (λ0 ) = −ci pi−1 (λ0 ).

N

5

ci = 0 pi−1 (λ0 )pi+1(λ0 ) ≤ 0 5 * pi−1 (λ0 ) = pi+1 (λ0 ) = 0 " " pk (λ0 ) = 0 0 ≤ k ≤ i + 1 " ' " p0 (λ0 ) = 1

LE O ÉC

$ $

"

E U

I " Q " " " ? N H D ' " C T E % Y L O P " $ L " E ? " O C É $ < pi (λ)

λi1 ≤ ... ≤ λii

i

i

Ai

pi

pi−1

i=2

p2 (λ) = (b2 − λ)(b1 − λ) − c21

pi+1

qi

(λ21 , λ22 ) pi (λ) i+1 2i

λ11 = b1

p1 (λ)

i

λ21 < λ11 < λ22 pi−1

pi

qi (λ) = pi−1 (λ)pi+1 (λ).

$ = ;& i − 1 qi pi−1 ! i % λ pi ! " qi (λ) < 0 & " qi (λi−1 k ) = 0 1 ≤ k ≤ i − 1,

%

qi (λik ) < 0 1 ≤ k ≤ i,

i λi1 < λi−1 < λi2 < ... < λi−1 1 i−1 < λi .

E U IQ

0 λik λik+1 qi γk = λi−1 % k qi pi+1 qi " λi−1 k

i−1

' % qi λi−1 k $ λk i−1 pi−1 λk pi+1 ? & % " λi−1 < pj % 0 ≤ j ≤ i + 1 " k ' p0 = 1 " % " (" λik , λik+1 γk = λi−1 < qi pi+1 k % % (i − 1) pi+1 " pi qi (λi1 ) < 0 qi (λii ) < 0 "

ÉC

E L O

T Y POL

N H EC

lim qi (λ) = +∞.

λ→±∞

$ qi pi+1 " pi

" " µ ∈ R' $ *$"

0$ pi(µ) pi (µ) = 0, sgnpi (µ) = 0$ pi−1 (µ) pi (µ) = 0. " N (i, µ) $/ #$0$" 0$ $" $" $ " $$ E(i, µ) = {+1, sgnp1 (µ), sgnp2 (µ), ..., sgnpi (µ)}, ' N (i, µ) " $ / $ pi $" ""$" $ µ,

)99) CC

E U IQ

N H C &

CN $ E % T Y L O P E L O ÉC $ " ' " sgnp (µ) ' T " i

pi (µ) = 0 i

pi−1 (µ) = 0 i=1

µ ≤ b1 ⇒ E(1, µ) = {+1, +1} ⇒ N (1, µ) = 0, µ > b1 ⇒ E(1, µ) = {+1, −1} ⇒ N (1, µ) = 1.

*

$

$ % ; " & i 1 pi+1 $

E U I Q N H

(λik )1≤k≤i

C E T LY

pi

(λi+1 k )1≤k≤i+1

λi1 < ... < λiN(i,µ) < µ ≤ λiN(i,µ)+1 < ... < λii ,

O P E C

CN 6 ' % L O É C 11

i λiN(i,µ) < λi+1 N(i,µ)+1 < λN(i,µ)+1 ,

λiN(i,µ) < µ ≤ λi+1 sgnpi+1 (µ) = sgnpi (µ) N (i + 1, µ) = N (i, µ) N(i,µ)+1 i λi+1 < µ < λ sgnp (µ) = − sgnp (µ) N (i+1, µ) = N (i, µ)+1 i+1 i N(i,µ)+1 N(i,µ)+1 1 µ = λiN(i,µ)+1 sgnpi (µ) = sgnpi−1 (µ) = − sgnpi+1 (µ) N (i + 1, µ) = N (i, µ) + 1 &

CN . N (i + 1, µ) ' ' pi+1 * & µ

#

-(,0'149) 30.1';5) &) #'2)*:C @ λ1 ≤ · · · ≤ λn . A

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* i6 . λi A . [a0 , b0 ] G λi . ; 0 −a0 = b0 = A2 < @ 0 N (n, a0 +b 2 ) ) & a0 +b0 ; . pj ( 2 )A 1 ≤ j ≤ nA 8 0 $< , . N (n, a0 +b 2 ) ≥ iA a0 +b0 0 λi ? , . [a0 , 2 [ . N (n, a0 +b 2 ) < iA a0 +b0 λi ? , . [ 2 , b0 ] 1 . , . λi * A ,6?6 . , . λi A . λi .

ÉC

E L O

T Y POL

N H EC

5 E F . , #6 , B#.A I? ? , I

' ; 0 6 < : , 1 A A # , nA r0 = 0 ∈ Rn . A Kk , B#. A . {r0 , Ar0 , · · · , Ak r0 } + , k0 ≤ n − 1A B#.A A k ≤ k0 A 8 (r0 , Ar0 , · · · , Ak r0 ) 0 dimKk = k + 1A k > k0 Kk = Kk0 , F ? . (vj )1≤j≤k0 +1 8 .A 2 ≤ j ≤ k0 + 1A

LE O ÉC

Y L O P

H C E T

vˆj = Avj−1 − (Avj−1 · vj−1 )vj−1 − ˆ vj−1 vj−2 ,

vj =

vˆj , ˆ vj

N

E U IQ ;<

$ $

N H EC

&

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. v0 = 0 v1 = r0 /r0 * k ≤ k0 + 1A ) Vk n × k . (v1 , · · · , vk )A , # 10'&'.(,*.-) Tk k × k

T Y POL

(Tk )i,i = Avi · vi , (Tk )i,i+1 = (Tk )i+1,i = ˆ vi+1 , (Tk )i,j = 0 |i − j| ≥ 2.

E L O É C" "

3. A F . ) 0

" (vj )1≤j≤k0 +1 " /$ *$ ) 3,9 ˆvj = 0 ) 1 ≤ j ≤ k0 +1' "$ vˆk0 +2 = 0, 1 ≤ k ≤ k0 +1' (v1 , · · · , vk+1 ) D$ - / "#$ Kk $" " ) ) ?# " )) (r0 , Ar0 , · · · , Ak r0 ), 8 ) ' ) 1 ≤ k ≤ k0 + 1' $

)99) CC

AVk = Vk Tk + vˆk+1 e∗k ,

;"<

. ek " k+ -" / $$ Rk ' Vk∗ AVk = Tk " Vk∗ Vk = Idk , . Idk " " $"" " k × k,

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;$<

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$ ' C#! (vj )1≤j≤k +1 2 % " " % & C#!!

T Y POL

vˆj = Avj−1 −

E L O

j−1

(Avj−1 · vi )vi ,

vj =

i=1

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0

CG!

j ≥ 2,

% v1 = r0 /r0 @ - CG! " ˆvj = 0 1 ˆvj = 0 " ( & j vj ( & vi 1 ≤ i ≤ j − 1 % * " vj ∈ Kj−1 5

W % Kj j ≤ k0 +1 " " ( % (v1 , · · · , vj ) * ' ( Kj−1 " vj ( & (v1 , · · · , vj−1 ) ( & Kj−2 0 * (v1 , · · · , vj ) CG! T % ' ( Kj−1 O 1( " & * (r0 , Ar0 , · · · , Aj−1 r0 ) 5 % " ( & " W % k0 & " ˆvj = 0 " j ≤ k0 + 1 vˆk +2 = 0 ? " C#! CG! (vj ) " 5

A "

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0

Avj−1 · vi = vj−1 · Avi = vj−1 · vˆi+1 +

i k=1

(Avi · vk )(vj−1 · vk ).

H C E T

N

E U IQ

OP ( (vk ) " Avj−1 · vi = 0 1 ≤ i ≤ " Avj−1 · vj−2 = ˆvj−1 . C#! CG! T J C/! " C#! vˆj Vk∗ Vk = Idk ( * (v1 , · · · , vk ) " Vk∗ AVk = Tk ' C/! & ( Vk∗ Vk∗ vˆk+1 = 0

j − 3

LE O ÉC

Y L O P

*

-

N H EC

E U IQ

)9.0;5) CC , F P 6 8 , = . / A , F , A n A , , . vj •

LE O ÉC

T Y POL

/ . . A Tk0 +1 ; H < @ λ1 < λ2 < · · · < λm . A ;. 1 ≤ m ≤ n
m

λi Pi , Id =

i=1

m

Pi , Pi Pj = 0 i = j.

;-<

i=1

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)99) CC

0

)9.0;5) CC

N H EC

E U IQ 0

1 R Pi r0 = 0 iA A Tk0 +1 H . A . . ( . Tk0 +1 @ . 6 , . A Tk0 +1 r0 0 9 (A r0 . AA k0 = 0 Tk0 +1 . ? r0 •

ÉC

E L O

T Y POL

1 λ y

∈ Rk0 +1 = 0

% % "

Tk0 +1 y = λy 5

vˆk0 +2 C/! % k = k0 + 1 AVk0 +1 = Vk0 +1 Tk0 +1 % y ' A (Vk0 +1 y) = λ (Vk0 +1 y) % Vk0 +1 y y = 0 Vk0 +1 ' " Vk0 +1 y ' % A & % λ " * & λi A " % Em Rn % (P1 r0 , · · · , Pm r0 ) 1 Pi r0 = 0 1 ≤ i ≤ m % ' ; Pi & ( " Em m 4 " m = k0 + 1 C)! Ak r0 = mi=1 λki Pi r0 & " Ak r0 ∈ Em W % % Kk ⊂ Em k ≥ 0 0 " " dimKk0 = k0 + 1 ≤ m ' (P1 r0 , · · · , Pm r0 ) Em % Ak r0 (λk1 , · · · , λkm ) * (r0 , Ar0 , . . . , Am−1 r0 ) Em ' (P1 r0 , · · · , Pm r0 )

M

⎛ ⎜

M =⎝

1

LE O ÉC 1

Y L O P

H C E T

λ1

λ21

···

λm−1 1

λm

λ2m

···

λm−1 m

⎞

⎟ ⎠.

N

E U IQ

$ $

N H EC

E U I" Q %'

M > . ? m λi . * (r0 , Ar0 , . . . , Am−1 r0 ) ' " (P1 r0 , · · · , Pm r0 ) 5 " " dimKm−1 = m m − 1 ≤ k0 . : " m = k0 + 1 Em = Kk OP & * C)! A(Pir0 ) = λi (Pi r0 ) 5

Pi r0 ' % A & % λi 5

Em = Kk " Vk +1 * ' Kk " % yi ∈ Rm " Pi r0 = Vk +1 yi $ CN! yi '

T Y POL 0

LE O ÉC 0

0

0

0

Tk0 +1 yi = Vk∗0 +1 AVk0 +1 yi = Vk∗0 +1 APi r0 = λi Vk∗0 +1 Pi r0 = λi Vk∗0 +1 Vk0 +1 yi = λi yi ,

yi % Tk +1 % λi 5 " 0

, 8 6 F I ,? , k0 + 1A . Tk0 +1 ) , . . A ' 6 F 0 ? 4 .6= ; k0 , nA , < 1 A F 0 ? , . vj . 0 , ;. + < 2 A . , , 8 0 , F 0 . Tk 0 A ;. k 0 k0 n<

ÉC

E L O

T Y POL

N H EC

E U IQ

" $ $" 1 ≤ k ≤ k0 + 1, " λ $ - )) Tk " y ∈ Rk $ -" )) $$ $ , &" $ - )) λi A " 0,3,:'1',* CC

|λ − λi | ≤ ˆ vk+1

|ek · y| ≤ ˆ vk+1 , y

. ek " k+ -" / $$ Rk , )9.0;5) CC * A

E U IQ

ˆ vk+1 A . Tk 0 . A J , . Tk A . 0 , . A •

Y L O P

1 % y C/! y '

LE O ÉC

H C E T

∈ Rk

AVk y = Vk Tk y + (ek · y)ˆ vk+1 ,

" Tk y

N

= λy

?

*

"

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CI! N H $ ' % C E T Y L O P E CI! % & C)! $ L O ÉC C ! :

A(Vk y) − λ(Vk y) = (ek · y)ˆ vk+1 .

Vk y

A

Vk y =

m

Pi (Vk y).

i=1

Vk y

m

(λi − λ) |Pi (Vk y)|2 = (ek · y) (ˆ vk+1 · Vk y).

i=1

0 ; ( 5 ( 1(VE % 2 min |λi − λ| Vk y ≤ y ˆ vk+1 Vk y.

1≤i≤m

E U IQ

5

Vk ( Vk y = y '

N H EC

min |λi − λ| ≤ ˆ vk+1 .

1≤i≤m

5 " 5 (1(VE < ek , y > C ! . %

E L O

T Y POL

min |λi − λ| ≤ ˆ vk+1

1≤i≤m

|ek · y| , y

"

ÉC

LE O ÉC

Y L O P

H C E T

N

E U IQ

T Y POL

É

N H EC

E U IQ

$%&' LE O C

TU 3=N3 +BA 734/3/5 5A @+/ NEA "NJ + .A /C N# ;<

RNA * =A 6

0+/ $ $ , ; " &'

T U 33+9 4A B3E9+ 7A 9 A * ; < TU E9/37 7A 9 B3+@ /A A * ; "<

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$ "' , *#6

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TU E9+/3+1 'A 7313Z ZA +3*955 2A 8""$ -"$$ )/+ & " )" ' 7 3 "A A * ; <

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T"U E@//3/ NA 4E9+5 N6'A 973+9'=3 'A 343 5b3E3 'A ) ""$ $ ' 7 3 &A A * ;&<

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T$U E@/Z N67A ; $!, A# "/ "$ , *# A * ; <

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" $! < '

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T Y POL

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T "U 4@>9+ 2A 34/3 7A A/ #' B CA E ;&< T $U 4'=3+195 3A $"0"$, $! #/"$$' *# A 96 A 7% A * ;-< T &U 319/E3'= 2A ; 5$" " $"0' , *#6 A * ; < T -U 9735+9 NA '=3E@'=9 N6A B$ " & ' 1 A * ;-"< T U @/ N6A V "# "$ )/+ & " $$ $' 1 A * ;$< TU @/ N6A 7349/9 9A /+ & " $$ #0+$ " )) "$' 1 A * ;&<

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TU '/ EA *+@//93 @A $" "$ $"* ' 7A * ;$< T U 7@=37731 EA 3h3' N6=A " "$ $ ' 1 A * ; < TU *31E9+4 7A $ )"7"$ $ &"$$A A E ;<

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TU *+@//93 @A B"# $" *$ ) R ' 7A * ;--< T"U +3>3+5 *63A 5=@73 N67A $" "$ $! $ "$ & - )"' 7A * ;-<

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N H A#! $E $ C $"0 )0$0 T Y L -"J P O "$, A#! $ $ $!'

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37

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I ( & 2 E -" ? ) ? I

A I B % -$ "& 6 ,FC 0 -$ & H -$ 0 6 I

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H C E T

N

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0 " 8 "& 0 . A $& , 8 . $ "A $A $

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$" ( $$ , &A $ A ) & = & & 4 $ A & . . $$ $ . E && ( A $ ! A & . "$ , $& " , -$A A $A "A "" E - = 0 0. -$ , $A A A &$

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I &

8 '%# D & 8 G 0I 8 - 4 $ & . && 8H -$ 8 $ 8 8 0 " 8 . & 8 4 $-A

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""

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"A "$A &"A " - 8 "$ 0 " "A "A "-A , 8 $ 4 6 4 .6= N0 F $ /C & I

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& 0 $ , , " . $

0 ,A " "A $ $ $$ # &A $ # $ # & # ; < " "A $ A A -A A $A "A $$ 6 . . &A 0 &A $& 0 & ' # &A $& "A $ "A $ $$ " # &$ I ? . ) A A $A $&

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& . A 0 " " 0 " 0 "" 8 $$ 8 0 " - " . " "$A -$ 6 ( & 6 & " 0 A &A & % A & 0 $-A - - # #

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B 5 % 67 & 7 %C% $& + & -A & &&A " &"

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A∗ I X . . X $$ C k (Ω), C k (Ω) 8 k 8 G .0 $$ Cc∞ (Ω) 8 ) .0 ? - A Cc∞ (Ω) 8 ) .0 ? 0 - D(Ω) 8 ) .0 ? D (Ω) 0 δij #0 B% "dx . ; 0 < $ds 8 ; 0 < $∂ α . , α ∂Ω 0 , . Ω $Epi(J) , 8 J K 0 , . K $$ H 1 (Ω) 0. -$ H01 (Ω) 0. ? H m (Ω) 0. , m H 1/2 (∂Ω) H(div) . J ( , 8 J L2 (Ω) 0 L∞ (Ω) 0 - n $|N | , 0 ) N &Pe . P $Pk #X k && Qk #X . 0 k " RN + 6 V , V $

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Analyse numerique

Analyse numerique

Acceleration de la Convergence en Analyse Numerique

Acceleration de la Convergence en Analyse Numerique

Analyse numerique pour ingenieurs

Analyse numerique pour ingenieurs

Analyse numerique matricielle

Analyse numerique matricielle

Analyse convexe et optimisation

Analyse convexe et optimisation

Variation et optimisation de formes: Une Analyse Geometrique

Variation et optimisation de formes: Une Analyse Geometrique

INTRODUCTION a la PEINTURE de la CHINE et du JAPON

INTRODUCTION a la PEINTURE de la CHINE et du JAPON

Analyse mathematique: Integration et theorie spectrale, analyse harmonique

Analyse mathematique: Integration et theorie spectrale, analyse harmonique

Optimisation et contrôle stochastique appliqués à la finance (Mathématiques et Applications) (French Edition)

Optimisation et contrôle stochastique appliqués à la finance (Mathématiques et Applications) (French Edition)

A la une, a la deux, a la mort

A la une, a la deux, a la mort

La Faim et la Santé

La Faim et la Santé

La pensée et la langue

La pensée et la langue

La raison et la religion

La raison et la religion

Nagios 3 pour la supervision et la metrologie : Deploiement, configuration et optimisation

Nagios 3 pour la supervision et la metrologie : Deploiement, configuration et optimisation

Principaux indicateurs economiques : Analyse methodologique comparative : Indices des prix a la consommation et a la production

Principaux indicateurs economiques : Analyse methodologique comparative : Indices des prix a la consommation et a la production

Grammaire et analyse : Analyse grammaticale et analyse logique

Grammaire et analyse : Analyse grammaticale et analyse logique

L'Ecriture et la différence

L'Ecriture et la différence

Nietzsche et la philosophie

Nietzsche et la philosophie

La rage et l'orgueil

La rage et l'orgueil

Guillaume et la meteorite

Guillaume et la meteorite

Pierre a griller (la) sauces et garnitures

Pierre a griller (la) sauces et garnitures

La transparence et l'énonciation

La transparence et l'énonciation

La névrose et vous

La névrose et vous

La nation et l'internationalisme

La nation et l'internationalisme

Charlie et la chocolaterie

Charlie et la chocolaterie

Charlie et la chocolaterie

Charlie et la chocolaterie

La Science et l'hypothèse

La Science et l'hypothèse

La science et l'hypothese

La science et l'hypothese

Charlie et la chocolaterie

Charlie et la chocolaterie

Wittgenstein : la rime et la raison. Science, éthique et esthétique

Wittgenstein : la rime et la raison. Science, éthique et esthétique

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