Korikache R., Methode d'elements finis mixtes.pdf

8/11/2019 Korikache R., Methode d'elements finis mixtes.pdf

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Universite de Valenciennes et du Hainaut Cambresis N dordre: 07/30LAMAV

Methode delements nis mixtes :application aux equations de la chaleur

et de Stokes instationnaires

TH ESE

presentee et soutenue publiquement le 15 Novembre 2007

pour lobtention du

Doctorat de lUniversite de Valencienneset du Hainaut-Cambresis

(specialite mathematiques appliquees)

par

Reda KORIKACHE

Composition du jury

Rapporteurs : Christine Bernardi Universite Pierre-et-Marie-CurieJean-Claude Nedelec Ecole Polytechnique, Palaiseau

Examinateurs : Van Casteren Universite de Antwerp BelgiqueEmmanuel Creuse Universite de ValenciennesSerge Nicaise Universite de Valenciennes

Directeur de These : Luc Paquet Universite de Valenciennes

Laboratoire de Mathematiques et leurs Applications de Valenciennes EA 4015


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P 1 P 2


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p (t) := u (t) t (t) := u (t) t

R 2 u (t)

t

u (t) t t [0, T ]

u(t) K T h


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0

u (t) (T h )h> 0

h h

u (t) p (t) (t) := u (t)

t (t)

0 K

p (t)

u (t)

h

1

D

K(x, ) x D t K u (t)


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( ph (t) , u h (t))

( ph (t) , u h (t)) ph (t)

uh (t) K

( ph (t ) , u h (t))

K N

D

1


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R 2

u

u 0

H 2 u t (s) s

[0, t ] H 3 u(t)


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H 2,

u p

[ ]


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R2. T > 0 Q := ]0, T [ := ]0, T [ : f L2 (0, T ; L2()) , g H 1(), u H 1 (0, T ; L2()) L2(0, T ; H 1())

u t(x, t ) u(x, t ) = f (x, t ) Q

u(x, t ) = 0

u(x, 0) = g(x) x .

u H 1 (0, T, L 2()) H 1 (0, T, L 2()) C ([0, T ];L2()) u(., 0) = g(.) H 1()

p = u

:

div p(x, t ) = u (x, t )

t f (x, t )

p L2(0, T ; H (div , )) u H 1 (0, T ; L2())

H (div , ) := q L2()2; divq L2()

u H 1 0, T, L 2( ) L2(0, T, H 1())

u H 1 (0, T, L 2())

A H = L2() D (A) = {v H 1(); v L2()}et Av = v v D (A).


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A ( m )m0

W m D (A) m

AW m = m W m .

W m W m 0, = 1 0, L2()

A

t u(t)

f L2(0, T ; L

2())

g H 1()

u(t ) =m =+

m =1{e m t (g, W m ) + t0 e(ts) m (f (s), W m )ds}W m .

u t(t) =m =+

m =1

e m t ( m )(g, W m )W m +m =+

m =1{(f (t), W m ) t0 e(ts) m m (f (s), W m )ds}W m .

m =+

m =1

e m t ( m )(g, W m )W m2

0,

=m =+

m =1

e2 m t 2m |(g, W m )|2

m =+

m =1

e m t ( m )(g, W m )W m2

L 2 (0 ,T ;L 2 ())

= T

0

m =+

m =1

e2 m t 2m |(g, W m )|2 dt

+ 0 m =+ m =1 e2 m t 2m |(g, W m )|2 dt=

12

m =+

m =1

m |(g, W m )|2 g 2H 1 ()


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0 T :

+

m =1

t

0e(ts) m m (f (s), W m )ds W m

2

L 2 (0 ,T,L 2 ())

T

0

+

m =1

m t

0e(ts) m |(f (s), W m )|2 dsdt

=+

m =1

m T 0 t0 e(ts) m |(f (s), W m )|2 dsdt

+

m =1

m T 0 |(f (s), W m )|2 + s e(ts) m dtds

s= T

s=0

m =+

m =1|(f (s), W m )|2 ds

= s= T s=0 f (s) 2L2 () ds = f 2L 2 (0 ,T ;L 2 ()) .

u t (.) 2L2 (0 ,T ;L2 ()) c g 2H 1 () + 2 f 2L 2 (0 ,T ;L 2 ())

u t (.) L 2 (0 ,T ;L 2 ()) g H 1 () + f L 2 (0 ,T ;L 2 ())

X := H (div , ); M := L2()

I [0, T ]

p = u, p = ux 1 , ux 1 ,


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u t (x, t )

div p(x, t ) = f (x, t ) Q ,

u(x, t ) = 0

p(x, t ) = u(x, t )

u(x, 0) = g(x) x .

q X

p(t). q dx + u(t) divq dx = ( u(t). q + u(t) div q )dx.= u(t) q.n ds, t I,

u L2(0, T ; H 1()) , u(t )| = 0 t I

p(t). q dx + u(t) div q dx = 0 q X, t I. u t (t) div p(t) = f (t),

v (u t (t) div p(t))dx = f (t) v dx, v M, t I

v div p(t)dx = (f (t) u t (t)) v dx, v M, t I 1.9 (1.8)

u H 1 (0, T ; L2())L2(0, T ; H ()) (1.1) ( p := u, u )L2(0, T ; H (div , )) H 1(0, T ; L2())


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| p(t )|

2 dx + 12

ddt

u(t)2dx = 0 t I.

ddt u(t)2dx = 2 | p(t)|2 dx 0 , t I

t u(t)2dx

u(0)

2dx = 0

u(t)2dx = 0 t I

(1.11)

| p(t)|2 dx = 0 t I = p(t) = 0 t I. p = 0 L2(0, T ; H (div; ))

: t u(t)2dx

(t) = 2 u t (t) u(t) dx

T

0 | (t)|dt 2 T

0( |u t (t)| |u(t)|dx ) dt.

T 0 ( |u t (t)|2 + |u(t)|2 dx ) dt= T 0 ( |u t (t)|2 dx ) dt + T 0 ( |u(t)|2 dx ) dt= T 0 u t (t) 20, dt + T 0 u(t) 20, dt = u 2H 1 (0 ,T ;L 2 ()) < + .


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L1([0, T ]) L1([0, T ]) (0) = 0

(t) = t

0 (s) ds

0 0 (0) = 0 (t) = 0 t I i.e. u(t)2dx = 0 t I

g H 1() p, u )L2(0, T ; H (div , )) H 1(0, T ; L2())

p(t).q dx + u(t) divq dx = 0 , q X, t I . v div p(t)dx = (f (t) u t (t)) v dx, v M, t I,

u(0) = g H 1() .

R 2

R2 := N j =1 j ,

j j = 1 , 2,...,N

. H 2, ()

H 1 () r , r

R 2


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u (1.1) . >

1 u L2 (0 ,T ;H 2 , ()) c f L 2 (0 ,T ;L 2 ()) + u H 1 (0 ,T ;L 2 ()) .

A L2 () ,

D (A) := {v H 10 (); v L2()}, Av = v, v D (A). D (A) H 2, () > 1

v H 2 , () c v L 2 () .

1.1.1 [0, T ] f L 2(0, T ; L2()) g H 1() u H 1(0, T ; L2()) L2(0, T ; H 1())

u t (x, t ) u(x, t ) = f (x, t ), ]0, T [

u(x, t ) = 0 ,

u(x, 0) = g(x) x .

u(t) = f (t) + u t (t) , t [0, T ] ,

u(t) L2(), t ]0, T [ .

(1.12) t ]0, T [ : u(t) H 2, ()

u(t) H 2 , () c f (t) L 2 () + u t (t) L2 () .


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:= A(a1, a 2) B (b1 + a1, b2 + a2), C (c1 + a1, c2 + a2)

F K : K K

( x1, x2) a1

a2+

b1 c1

b2 c2 x1 x2.

K K K

K = x = ( x1, x2) R 2; 0 x1 1, 0 x2 1 x1 .

B K = b1 c1

b2 c2.

det B K > 0.

K

K Piola :

v K

K

v (x) = 1

det B K B K v(F 1K (x)) , x K .

v K

K

v(x) = det B K B 1K v (F K (x)) , x K .

(T h )h> 0


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(T h )h > 0 0

h > 0, K T h , K := h

K K 0 .

(1.9) : ( ph , u h ) L2(0, T ; X h )H 1(0, T ; M h )

ph (t).q h dx +

uh (t) div q h dx = 0 , q h X h , t I ,

vh div ph (t) dx = (f (t) uh,t (t)) vh dx , vh M h , t I, uh (0) = gh M h ,

X h := q h H (div , ); K T h : q h/K RT 0(K )

M h := vh L2(); vh/K P 0 , K T h

RT 0(K ) = P 0(K )2 P 0(K ) x1x2

0 K RT 0(K ) D1 (K ) P 0

K.

(1.15)

( ph , u h ) L2(0, T ; X h ) H 1(0, T ; M h ). ph H 1(0, T ; X h )


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gh M h L2() gh H 10 () q

(1)h , . . . ,q

(J )h X h , v

(1)h , . . . , v

(K )h

M h . ph (t) ( uh (t)) q ( j )h j =1 ,...,J

X h ( v(k)h k=1 ,...,K

M h ) :

ph (t) =J

j =1

j (t) q ( j )h , uh (t) =

K

k=1

k(t)v(k)h .

(1.15)

J j =1 j (t) q

( j )h .q

( j )h dx +

K k=1 k(t)v

(k)h divq

( j )h dx = 0 , j = 1 , 2,...,J

v(k )h ( J j =1 j (t)div q ( j )h ) dx = (f (t) K k=1 k(t)v(k)h ) v(k )h dx, k = 1 , 2,...,K

J j =1 (

q

( j )h . q

( j )h dx) j (t) +

K k=1 (

v

(k)h div q

( j )h dx) k(t) = 0 ,

j = 1 , 2, ...,J,

J j =1 ( v(k )h divq ( j )h dx) j (t) = f (t)v(k )h dx + K k=1 ( v(k)h v(k )h dx) k(t),

k = 1 , 2,...,K.

a kk = v(k)h v(k )h dx , b jj = q ( j )h q ( j )h dx , c j k = (divq ( j )h )v(k )h dx j, j = 1 , 2, ...,J, ; k, k = 1 , 2,...,K.


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J j =1 b j j j (t) +

K k=1 c j k k(t) = 0 , j = 1 , 2,..,J,

J j =1 (C )k j j (t) = f (t)v(k )h dx + K k=1 akk k(t)

k = 1 , 2,...,K.

A = ( akk )1k ,kK A RK K

B = ( b j j )1 j ,j J B

R J J ;

C = ( c j k)1 j J, 1kK C R J K

(t) =

1(t)

K (t)

RK , (t) =

1(t)

J (t)

RJ , F (t) =

(f (t)v(1)h dx

f (t)v

(K )h dx

RK .

A (t) = C (t) + F (t),

B (t) + C (t) = 0 .

(t) = B 1C (t). (1.17)

A (t ) = C B 1C (t) + F (t ),

(t) = B 1C (t).


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A (t) + C B 1C (t) = F (t) , F

L2(0, T ; R K )

(0) = 0 (i.c. )

0 R K R K K

k=1

( 0)kv(k)h = gh M h .

K K (t) = A1C B 1C (t) + A1F (t).

(t) = eA 1 C B 1 C t 0 + t0 eA 1 C B 1 C ( t )A1F ( ) d .

C ([0, T ];R K ) et L2(0, T ; R K ). (t) = B 1C (t) C ([0, T ];R J ) L2(0, T ; R J )

uh H 1(0, T ; M h ) ph H 1(0, T ; X h ).

( p , u)

(1.13) ( ph , uh ) (1.15)

( p (t ) , u (t))

t


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F (t) = (f (t) u t (t)) v

(1)h dx

(f (t) u t (t)) v(K )h dxR K .

F (t ) L2(0, T ; R K ).

(1.19)

(t) = B 1C (t), t I,

C (t) + F (t) = 0 . t I .

(C B 1C ) (t) = F (t).

C B 1C R K K

R K

\{0

}

( C B 1C , ) = ( B 1C, C ) 1

max (B )C 2 ,

max (B ) B C B 1C R K B 1C R J .

(C B 1C, ) 0 R K \{0} C B 1C

(C ) R J R K \{0}. C = 0

(divq ( j )h )( K k=1 v(k)h k) dx = 0 .


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q (1)h , q (2)h , . . . . . ,q

(J )h X h .

q h X h

(divq h )(K

k=1

v(k)h k) dx = 0 .

K k=1 kv

(k)h M h (1.2)

K

k=1

kv(k)h = 0

1 = 2 = . . . . . . = K = 0 = 0 .

C B 1C

(t) = ( C B 1C )1 F (t)

F L2(0, T ; R K ), L2(0, T ; R K ) L2(0, T ; R J ) (1.19).

uh L2(0, T ; M h ) et ph L2(0, T ; X h ).

(1.13)

f H 1(0, T ; L2()) g + f (0) H 10 ().

u t H 1(0, T ; L2()) L2(0, T ; H 2, ()) L2(0, T ; H 1()) ,

uh H 1(0, T ; M h ) ph H 1(0, T ; X h ).


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df

dt L2(0, T ; L2()) . v

H 1(0, T ; L2())

L2(0, T ; H 1())

ddt

(v(t)) = ( v(t)) + df dt

(t) Q

v(0) = g + f (0)

g + f (0) H 1() et df

dt L2(0, T ; L2()) ,

1.1.1, v

(1.14)

v L2(0, T ; H 2, ()) .

u(t) =

t

0v(s)ds + g.

u (1.1). dudt = v.

v (1.20).

(t) = ( C B 1C )1 F (t),

ddt (t) = ( C B 1C )1 ddt F (t)

= ( C B 1C )1 ( ddt f (t) ddt u t (t)) v(1)h dx

( ddt f (t) ddt u t (t)) v(K )h dx.


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ddt

(t) L2(0, T ; R K ).

ddt

(t) L2(0, T ; R J ),

(t) = B 1C (t)

uh H 1(0, T ; M h ) ph H 1(0, T ; X h ).

( ph (t), uh (t))

u(t ) = div p(t) = f (t) u t (t) L2(),

{T h}

1 w , 1

(i) hK h 11

K T h (ii ) hK (inf x K r (x)) h K T h

c > 0 h

p(t) ph (t) 0, c h |u(t)|H 2 , () ,

u(t) uh (t) 0, c h |u(t)|H 1 () + |u(t)|H 2 , () , 1 w , 1 {T h}

1.4.6

h t I :

uh (t) P h u(t) 0, 1 p(t) ph (t) ,


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P h M M h

(1.9), (1.15)

p(t). q dx + u(t) divq dx = 0 , q X, v div p(t)dx = (f (t) u t (t)) v dx, v M,

ph (t). q h dx +

uh (t) divq h dx = 0 , q h X h ,

vh div ph (t)dx = (f (t) uh,t (t)) vh dx, vh M h ,

q = q h (1.9)(i)

p(t).q h dx + u(t) divq h dx = 0 . div q h K T h , q h X h ,

u(t) divq h dx = K T h K u(t)divq h dx=

K T hdivq h |K K P h (u(t)) dx

= P h (u(t))div q h dx (1.25)

p(t). q h dx + P h (u(t)) div q h dx = 0 q h X h , t I, (1.26) (1.15)(i)

( p(t) ph (t)) . q h dx + (P h u(t) uh (t)) divq h dx = 0 , q h X h , t I .


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1.15 inf sup (1.27)

uh (t)

P h u(t)

0, 1

p(t)

ph (t)

t I,

inf sup (1.24)

{T h} , c > 0 h t I :

u(t) P h u(t) 0, c h |u(t)|H 1 () . {T h} ,

(i) (ii ) 1.4.6 1 w , 1 , c > 0 h

t I :

u(t) uh (t) 0, c h |u(t)|H 1 () + 1

ch|u(t)|H 2 , () + ph (t) ph (t) . (1.28) (1.24)

u(t) uh (t) 0, u(t) P h u(t) 0, + P h u(t) uh (t) 0, t I,

ch

|u(t)

|H 1 () +

1

p(t)

ph (t)

0,

ch|u(t)|H 1 () + 1

p(t) ph (t) 0, + ph (t) ph (t) 0, . (1.22) t I

u(t) uh (t) 0, c h|u(t)|H 1 () + 1

ch|u(t)|H 2 , () + ph (t) ph (t) 0, .


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h (t) := ph (t)

ph (t) h (t) := uh (t)

uh (t).

h (t). q h dx + h (t ) div q h dx = 0 , t I , q h X h . t I

h H 1(0, T ; X h ), h H 1(0, T ; M h ) [0, T ] X h M h

t

ddt h (t).q h dx + ddt h (t) divq h dx = 0 , t I , q h X h . q h = 2 h (t)

2 ddt h (t). h (t) dx + 2 ddt h (t) div h (t) dx = 0 t I ,

ddt |

h (t)|2

dx + 2 ddt h (t) div h (t) dx = 0 t I.

(1.15)(ii ) (1.18)(ii )

vh div h (t)dx = ddt (uh u)( t) vh dx, vh M h , t I (1.34) (1.33),

vh = 2dh (t)

dt

2dhdt (t) div( h (t))dx = 2 ddt (uh u)( t) dhdt (t) dx t I = 2 ddt (uh uh )( t) dhdt (t) dx + 2 ddt (uh u)( t) dhdt (t) dx= 2 dhdt (t) 2 dx + 2 ddt (uh u)( t) dhdt (t)dx.


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(1.33) (1.35)

ddt |

h (t)|2

dx + 2 dhdt (t)

2

dx = 2 ddt (uh u)( t)

dhdt (t) dx t I

2 ddt (uh u)( t) 21/ 2

dhdt (t) 2 dx1/ 2

ddt (uh u)( t) 2 + dhdt (t) 2 dx.

ddt | h (t)|2 dx ddt (uh u)( t) 2 dx t I.

0 t

| h (t)|2 dx | h (0)|2 dx + t0 ddt (uh u)( t) 2 dx dt. uh (0) = uh (0) h (0) = uh (0) uh (0) = 0 (1.32)

t = 0

h (0) .q h dx = 0 , q h X h . q h = h (0)

| h (0)|2 dx = 0 h (0) = 0 .

(1.36)

| h (t)|2 dx t0 ddt (u uh )( t) 2 dx dt t0 dudt (t) dudt (t) h 2 dx


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ddt (.)h

|h (t)|

2

dx t

0

dudt (t)

dudt (t) h

2

0, dt

dudt (t) dudt (t) h 0, .

f H 1(0, T ; L2()) g + f (0) H 1(), 1.4.5

u t H 1(0, T ; L2()) L2(0, T ; H 1()) L2(0, T ; H 2, ()) . u t (t)

u t (t) t c > 0 h

dudt

(t) dudt

(t)

h 0,c h

du (t)dt H 1 ()

+du (t)

dt H 2 , ().

h (t) 0, c hdudt L 2 (0 ,T ;H 2 , ())

, t [0, T ].

1.4.6

p(t) ph (t) 0, p(t) ph (t) + ph (t) ph (t)

ch |u(t)|H 2 , () +dudt L 2 (0 ,T ;H 2 , ())

.

(1.29) h (t)

u(t) uh (t) 0, c h |u(t)|H 1 () + |u(t)|H 2 , () +dudt L 2 (0 ,T ;H 2 , ())

.


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[0, T ] N [tn1, t n ] n

0 = t 0 tn < tN = T ,

t = tn tn1 unh u tn = n t M h ut

tn

,

u nh = (unh un1h )

t

un1h u tn1.

( p nh , unh )n N

p nh .q h dx + unh div q h dx = 0 , q h X h , n 0 vh div p nh dx = (f (t n ) u nh ) vh dx, vh M h , n 1

u0h (c.i. ),

u0h = uh (0) , (1.37)

( pnh , u nh )n N X h M h .

(1.37) ( p nh , unh )n N X h M h .


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F (vh ) := 1 t ( t f (t

n ) + un1h ) vh dx.

(1.37) ( p nh , unh )n N X h M h

p nh .q h dx + unh div q h dx = 0 , q h X h , n 0 vh div p nh dx 1 t unh vh dx = F (vh ) vh M h , n 1

u0h (

c.i.),

h X h M h ( ph , u h ) X h M h h ( ph , u h )

h ( ph , u h ) := ( q h ph .q h dx + uh divq h dx,vh vh div ph dx 1 t uh vh dx).

h

( ph , u h ) X h M h

ph . q h dx + uh div q h dx = 0 , q h X h , vh div ph dx 1 t uh vh dx = 0 , vh M h .

q h =

ph

(1.40)

, vh =

uh

(1.41)

,

| ph |2 dx + 1 t |uh |dx = 0 , ph = 0 uh = 0 .

h n =

1, 2, 3, ...


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n = 0 u0h p 0h

(1.39)(i)

p0h . q h dx + u

0h div q h dx = 0 , q h X h . X h ,

L2. X h

p 0h X h

p 0h . q h dx = u0h div q h dx, q h X h .

(1.37)

sup0mN

umh 0, 2exp(T ) u0h 0, + N n =1

t f (tn ) 20,

t 12 . vh = unh (1.37)(ii ) alors :

unh div p nh dx = (f (tn ) u nh ) unh dx,= u nh unh dx f (tn ) unh dx,=

1 t unh un1h unh dx f (tn ) unh dx,

u nh = unh u n

1h

t

(1.37)(i) q h = pn

h

unh div p nh dx = | p nh |2 dx.


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(1.43) (1.44) ,

1

t |

unh |

2 dx

1

t

un1h

unh dx +

|

p nh |

2 dx =

f (tn ) unh dx.

1 t |unh |2 dx + | p nh |2 dx = f (tn ) unh dx + 1 t un1h unh dx,

f (tn ) unh dx + 12 t |unh | 2 dx + 12 t un1h 2 dx.

12 t |unh |2 dx + | p nh |2 dx f (tn ) unh dx + 12 t un1h 2 dx.

12 t |unh |2 dx f (tn ) unh dx + 12 t un1h 2 dx,

unh20, un1h

20, 2 t f (tn ) 0, unh 0, .

n = 1

m,

m

n=1

unh20, un1h

20, 2 t

m

n =1

f (tn ) 0, unh 0, .

umh20, u0h

20, + 2 t

m

n =1

f (tn ) 0, unh 0, .

umh 20, u0h 20, + tm

n=1

f (tn ) 20, + tm

n =1

unh 20, .

umh20,

(1 t) umh 20, u0h20, + t

m

n =1

f (tn ) 20, + tm1

n =1

unh20,


45/195


46/195

q h = p nh (1.37)(i) vh = unh (1.37)(ii )

| pnh |2 dx + (u

nh f (tn )) unh dx = 0 . t,

t | p nh |2 dx + t (u nh f (tn )) unh dx = 0 . n = 1 , 2, 3,...,N,

N

n =1

t

|

p nh

|

2 dx +N

n =1

(unh

un1h )unh dx

N

n =1

t f (tn ) unh dx = 0 .

(unh un1h )unh dx = (unh ) 2 dx un1h unh dx unh 20,

12

un1h20,

12

unh20,

= 12 unh 20, 12 un1h 20, .

N

n =1 (unh un1h )unh dx N

n =1

12

unh20,

12

un1h20, =

12

uN h20,

12

u0h20, .

(1.51)

N

n =1

t | pn

h |2 dx + 12 uN h 20, 12 u0h 20, N

n =1

t f (tn ) 0, unh 0,

C N

n =1

t f (tn ) 20,

12

u0h 0,

+N

n =1

t f (tn ) 20, .


47/195

(1.52) (1.53) N

n =1

t f (tn )0,

unh 0,

=N

n =1

( t)12 f (tn )

0, ( t)

12 un

h 0,

N

n =1

t f (tn ) 20,12 N

n =1

t unh 20,12

N

n =1

t f (tn ) 20,12

Cste N

n =1

t u0h20,

+N

n=1

tN

n =1

t

f (t

n) 2

0,

12

(1.42)

Cste N

n =1

t f (tn ) 20,12

u0h 0, +N

n =1

t f (tn ) 20, .

(1.53)

(1.53)

ab a2 + b2 N

n =1

t f (tn ) 20,

12

u0h 0, u0h 20, +N

n =1

t f (tn ) 20, . (1.55) (1.53)

N

n =1

t p nh20, +

12

uN h20, Cste u0h

20, +

N

n =1

t f (tn ) 20, . (1.49)

(1.56) .

1.49 L2

p nh .

(1.37)

pN h 0, p 0h 0, + T 2 maxn =1 ,...,N f (tn ) 0, .


48/195

(1.37)(i)

pn

h . q h dx + unh div q h dx = 0 , q h X h , n 1.

q h = p nh (1.58) vh = unh (1.37)(ii )

p nh . p nh dx + u nh div p nh dx = 0 , u nh div p nh dx + f (tn ) unh dx u nh 2 dx = 0 .

(1.59)(ii ) (1.59)(i)

p nh . p nh dx + u nh 2 = f (tn ) unh dx.

p nh20, p n1h

20, + 2 t u

nh

2

2 t f (tn ) u nh dx, (1.60) n = 1 , 2, 3,...,N,

p N h20, p 0h

20, +

N

n =1

2 t u nh2

N

n=1

2 t f (tn ) u nh dx 2 t

N

n =1

f (tn ) 0, unh 0,

t

2

N

n =1

f (tn ) 20, + 2 tN

n =1

u nh20,

T

2 maxn =1 ,...,N

f (tn ) 20, + 2 tN

n =1

u nh20, .

p N h 0, p 0h 0, + T 2 maxn =1 ,...,N f (tn ) 0, .


49/195

f H 1(0, T ; L2()) g + f (0) H 1() u t H 1(0, T ; L2()) f ut [0, T ]

L2().

u(t),

unh , unh uh (tn ) 0, , ( ph (tn ), uh (tn )) X h M h tn

( ph (tn ), uh (tn )) X h M h

ph (tn ).q h dx + uh (tn ) divq h dx = 0 , q h X h , vh div ph (tn )dx = (f (tn ) u t (tn )) vh dx, vh M h .

(1.37) nh :=

unh uh (tn ) nh := p nh ph (tn ),

nh .q h dx +

nh divq h dx = 0 , q h X h ,

vh div nh dx + (u t (tn ) u nh ) vh dx = 0 , vh M h .

c > 0

unh uh (tn ) 0, c h tn0 u t (s) H 2 , () ds + t tn0 u tt (s) ds . q h nh (1.62)(i)

| nh |2 +

nh div n

h = 0 ,

vh nh (1.62)(ii )

nh div nh dx + (u t (tn ) u nh ) nh dx = 0 ,

| nh |2 + ( uh (tn ) u t (tn )) nh dx = nh nh dx.


50/195

(nh )2 dx

n1h nh dx =

t

|

nh

|2

t

nh ( uh (tn )

u t (tn )) dx

t nh ( uh (tn ) u t (tn )) dx t nh 0, uh (tn ) u t (tn ) 0, .

nh2

0,

n1h

nh dx + t nh

0, uh (tn )

u t (tn )

0,

n1h 0, nh 0, + t nh uh (tn ) u t (tn ) 0, . nh

nh 0, n1h 0, + t uh (tn ) u t (tn ) 0, .

n = u (tn ) h u t(tn ) u (tn ) h = uh (tn ) u (tn )

u (tn ) h = uh (tn )

(t j )

ph (t j ). q h dx + uh (t j ) div q h dx = 0 , q h X h ,

vh div ph (t j )dx =

(f (t j )

u t (t j )) vh dx,

vh

M h .

( ph (tn ), uh (tn )) .

ph (t j ), uh (t j )

ph (t j ). q h dx + uh (t j )div q h dx = 0 , q h X h , vh div ph (t j ) dx = f (t j ) u t (t j ) vh dx, vh M h .


51/195

f (t j )

u t (t j ) =

f (t j ) f (t j1) t

ut (t j ) u t (t j1) t

= ( u(t j )) ( u(t j1))

t= u(t j ) = u (t j ).

ph (t j ). q h dx + uh (t j )div q h dx = 0 q h X h ,

vh div

ph (t j ) dx =

u (t j ) vh dx vh M h .

u (t j ) h = uh (t j ).

(1.67)

n := n1 + n2

n1 = u (tn ) h u (tn ),

n2 = u (tn ) u t (tn ) .

n1

n1 0, = u (tn ) h u (tn ) 0,

= (R h I ) u (tn ) 0,

= (R h I ) u(tn ) u(tn1)

t 0,

= 1 t

(R h I ) tn

t n 1u t (s) ds

0,

1 t tntn 1 (R h I ) u t (s) 0, ds 1 t ch tntn 1 u t (s) H 2 , () ds ,

R h de

M h X h M h


52/195

t

n

j =1 j

1 0, chn

j =1 tj

t j 1 u t(s) H 2 , () ds

ch tnt0 =0 u t (s) H 2 , () ds . n2

n2 0, = u (tn ) u t (tn ) 0,

= 1 t u(tn ) u(tn1) t u t (tn ) 0,

= 1 t tntn 1 (tn1 s) u tt (s) ds 0,

tntn 1 u tt (s) 0, ds.

t

n

j =1

j2 0, t

tn

t0 =0 u tt (s) 0, ds .

(1.66)

nh 0, 0h 0, + ch tnt0 =0 u t (s) H 2 , () ds + t tnt0 =0 u tt (s) 0, ds.

0h = u0h

uh (0) = 0 ,

nh 0, c h tnt0 =0 u t (s) H 2 , () ds + t tnt0 =0 u tt (s) 0, ds .

u(tn ) unh 0, .


53/195

{T h} , (i) (ii ) 1.4.6 1 w , 1 ,

c > 0 h n 1 :u(tn ) unh 0, c h |u(tn )|H 1 () + |u(tn )|H 2 , () + tn0 u t (s) H 2 , () ds

+ t tn0 u tt (s) ds .

u(tn ) unh 0, u(t) uh (tn ) 0, + uh (tn ) unh 0, . (1.63) (1.23)

(1.73)

p nh .

ph (tn ) p nh0,

nh20, 2 nh , nh .

nh20, 2 nh , nh =

nh20, n1h 20,

t 2 n

h , nh n1h

t

= nh

20,

t n1h 20,

t + 2

nh ,n1h

t

= 1 t

2 nh ,n1h nh 20, n1h 20,

= 1 t

nh20, +

n1h 20, 2 nh , n1h .


54/195

2 nh ,n1h 2 nh 0, n1h 0, nh 20, + n1h 20,.

nh20, +

n1h 20, 2 nh , n1h 0 . (1.75) (1.76)

nh20, 2 nh , nh .

nh20, n 20,.

(1.62)

nh .q h dx + nh divq h dx = 0 , q h X h . q

h = n

h ,

nh . nh dx = (div nh ) nh dx= (n nh ) nh dx ( (1.62)(ii ) (1.67))= n nh dx nh 20, n 0, nh 0, nh 20,

1

2n 20, +

12

nh20, nh 20,

1

2n 20,

12

nh20,.


55/195

nh20, n 20, nh 20, .

nh20, n 20,.

nh2 n 2

1h

c > 0

1h 0, c h u t L 2 (0 , t ;H 2 , ()) + t u tt L2 (0 , t ;L 2 ()) .

u0h = uh (0) 0h = 0 u0h ,

1h = 1h 0h

t =

1h t

.

n = 1

1h .q h + 1h divq h = 0 , q h X h , vh div 1h (1 + 1h ) vh = 0 , vh M h .


56/195


57/195

u (t1)

h u (t1)

0,= (R h

I ) u (t1)

0,

= (R h I ) u(t1) u0

t 0,

= 1 t

(R h I ) t1 = t0 u t (s) ds 0,

1 t

t

0(R h I ) u t (s) 0, ds

1

t t0 ch u t (s) H 2 , () ds = ch t t0 u t (s) H 2 , () ds

ch t

t t0 u t (s) 2H 2 , () ds12

= ch t u t () L 2 (0 , t ;H 2 , ()) .

t u (t1) h

u (t1)0,

c h u t (

) L 2 (0 , t ;H 2 , ()) .

u (t1) u t (t1) 0, =u(t1) u(t0)

t u t (t1) 0,

= 1 t

u(t1) u(t0) t u t(t1) 0, .

u(t0) = u(t1) tu t (t1) + t0 =0

t 1 = tu tt (s)( t0 s)ds

= u(t1) tu t (t1) + t10 u tt (s) s dsu(t1) u(t0) tu t (t1) = t10 u tt (s) sds.


58/195

u (t1)

u t (t1)

0, 1

t t1

0u tt (s)

0, sds

1 t t10 u tt (s) 20, ds

12 t10 s2ds

12

1 t

t3

3

12

u tt () L2 (0 , t ;L 2 ())

t u tt (s) L 2 (0 , t ;L 2 ()) .

t u (t1) u t (t1) 0, t u tt (s) L 2 (0 , t ;L 2 ()) . (1.81) (1.82) (1.83)

1h 0, c h u t L 2 (0 , t ;H 2 , ()) + t u tt L 2 (0 , t ;L2 ()) .

nh 0,

c > 0

nh 0, c h u t L 2 (0 ,t n ;H 2 , ()) + t u tt L2 (0 ,t n ;L 2 ())

nh20, n 20,

2h2

1h2

t 22

3h2

2h2

t 32

nh2 n1h

2

t n 2


59/195

n

h

2

0, 1

h

2

0, t 2 2

0, + + n 2

0, .

nh20, 1h

20, + t

j = n

j =2

j 20, .

j = j1 + j2

j1et

j2 (1.70) ,

t j = n

j =2

j 20,

2 t

j = n

j =2

j120, + 2 t

j = n

j =2

j220, .

j1 = ( R h I ) u (t j ) = 1 t

(R h I ) tjt j 1 u t (s) ds=

1 t tjt j 1 (R h I ) u t (s) ds,

j1 0, 1 t tjt j 1 (R h I ) u t (s) 0, ds

h t tjt j 1 u t (s) H 2 , () ds.

t j = n

j =2

j120, t h

2

t2

j = n

j =2 tj

t j 1u t (s) H 2 , () ds

2

th 2

t

j = n

j =2 tjt j 1 u t (s) 2H 2 , () ds= h2 tnt1 u t (s) 2H 2 , () ds .


60/195

j2

t

j = n

j =2 j

2

2

0, = t

j = n

j =2

u (t j ) u t (t j )

2

0,

= 1 t

j = n

j =2

u(t j ) u(t j1) tu t (t j ) 20, .

u(t j1) = u(t j ) + u t (t j )( t j1 t j ) + tj 1t j u tt (s)( t j1 s) ds.

u(t j ) u(t j1) tu t (t j ) 20, = tj 1t j u tt (s)( t j1 s) ds2

0,

t jt j 1 u tt (s) 0, |(t j1 s) |ds2

t jt j 1 u tt (s) 20, ds t33 .

t j = n

j =2

j22

t2

3 tnt1 u tt (s) 20, ds.

t j = n

j =2

j 20, 2 t j = n

j =2

j120, + 2 t

j = n

j =2

j220,

2h2

tn

t 1u t (s) 2

H 2 ,

() ds + t2

tn

t1u tt (s) 2

0,ds .

(1.85)

nh20, 1h

20, + t

j = n

j =2

j 20,

ch2 tnt 0 =0 u t(s) 2H 2 , () ds + 2 t2 tnt0 =0 u tt (s) 20, ds,


61/195

nh 0, c h u t L2 (0 ,t n ;H 2 , ()) + t u tt L 2 (0 ,t n ;L2 ()) .

1.4.6

p(t)

ph (t)

0,

c h

|u(t)

|H 2 , () , t I.

1.84

p(tn ) p nh 0, n 1.

{T h} , (i) (ii ) (1.4.6) 1 w , 1 , c >

0 h t n 1 :

p(tn ) p nh 0, c h |u(tn )|H 2 , () + u t L 2 (0 ,t n ;H 2 , ()) + t u tt L2 (0 ,t n ;L 2 ()) .

tn12 = tn + tn1

2 , p n

12

h = p nh + p

n1h2

, un12

h = unh + u

n1h2

.


62/195


63/195

u nh un12h =

1

2t

unh

un1h

unh + u

n1h = 1

2t (unh )

2 ,

un1h = 0 . n = 1 (1.92) u0h = 0 u1h = 0 n = 2 , 3, .. .

(1.89) , u0h = 0 p 0h = 0 . p12

h = 0 , (1.91)

n = 1 , p 1h = 0 p 2h = 0 (1.91) n = 2

n 1.

p 0h

(1.5.2)) . (1.88) n = 1

p 1h u1h u

0h p

0h

p 1h .q h dx + u1h div q h dx = p 0h .q h dx + u0h div q h dx, q h X h , vh div p 1h dx 2 t u1h vh dx = vh div p 0h dx 2 t u0h vh dx 2 f t1/ 2 vh dx,vh M h .

h

X h M h

X h M h

p 1h , u1h q h p 1h .q h dx + u1h divq h dx,

vh vh div p 1h dx 2 t u1h vh dx . h

X h M h X h M h X h M h h

( p1

h , u1h )

p 1h .q h dx + u1h divq h dx = 0 , q h X h vh div p 1h dx 2 t u1h vh dx = 0 , vh M h .

(1.94) , q h = p 1h vh = u1h (1.95) ,

p 1h 2 dx + 2 t u1h 2 dx = 0 .


64/195

p 1h = 0 u1h = 0 . h .

1h (1.93) .

( p 2h , u 2h ) (1.88) ,

n = 2

t 12 , c > 0

uN h 0, u0h 0, +

t p 0h 0, + N n =1

t f (t n12 )2

0,

p

nh .q h dx +

u

nh divq h dx = 0 , q h X h , n 1

vh div p n 12

h dx = (f (tn12 ) u nh ) vh dx, vh M h , n 1

vh = unh

unh div p nh + p n1h2 dx = (f (tn12 ) u nh ) unh dx=

1 t

unh un1h unh dx

f (tn12 ) unh dx.

(1.97) ,

unh div p nh dx = | p nh |2 dx,

unh div p n1h dx = p n p n1h dx.


65/195

(1.98) (1.99) (1.100) ,

1

2 |

p nh

|2 dx +

1

2

p n

p n1h dx +

1

t |

unh

|2 dx

1

t

un1h unh dx =

f (tn

1

2)unh dx.

1 t |unh |2 dx + 12 | p nh |2 dx = 12 p nh p n1h dx + 1 t un1h unh dx + f (tn12 )unh dx,

12 t |unh |2 dx + 14 t | p nh |2 dx 14 p n1h 2 dx + 12 t un1h 2 dx

+12 f (t

n12 )2

dx + 12 |u

nh |2 dx.

unh20, +

t2 p

nh

20,

t2

p n1h20, + u

n1h20, + t u

nh

20, + t f (tn12 )

2

0,.

unh20, un1h

20, +

t2

p nh20, p n1h

20, t unh 20, + t f (tn12 )

2

0,.

N

n =1

unh20, un1h

20, +

t2

N

n =1

p nh20, p n1h

20, t

N

n =1

unh20,

+ tN

n=1

f (tn12 )2

0,.

uN h 20, + t2 p N h 20, u0h 20, + t2 p 0h 20, + t

N

n =1

unh 20, + tN

n =1

f (tn12 )20,

.

t uN h20,

(1 t) uN h20,+

t2

p N h20, u0h

20,+

t2

p 0h20,+ t

N

n =1

unh20,+ t

N

n =1

f (tn12 )2

0,.


66/195

t 12 ,

uN h

2

0, + t pN h

2

0, 2 u0h

2

0, + t p0h

2

0, + 2 t

N

n =1 unh

20, + 2 t

N

n =1 f (tn12 )2

0, .

n = unh20, + t p

nh

20,

m 0 = 2

m 1 = = m N 1 = 2 tC = 2 u0h

20, + t p

0h

20, + 2 t

N n =1 f (tn12 )

2

0,,

uN h20, + t p

N h

20,

2 u0h20, + t p

0h

20, + 2 t

N

n =1

f (tn12 )2

0,exp

N 1

t=0

m t

= 2 u0h20, + t p

0h

20, + 2 t

N

n =1

f (tn12 )2

0, exp (2 + 2( N 1) t) .

uN h20,+ t p

N h

20, exp(2) 2 u0h

20, + t p

0h

20, + 2 t

N

n =1

f (tn12 )2

0,(exp( T ))2 .

uN h20, + t p

N h

20, u

0h

20, + t p

0h

20, +

N

n =1

t f (tn12 )2

0,,

uN h 20, u0h 20, + t p 0h 20, +N

n =1 t f (tn12 )

2

0,.

(1.37)

pN h 0, p 0h 0, + T 2 maxn =1 ,...,N f (tn ) 0, .


67/195


68/195

(1.61)

n n 1

ph (tn )+ ph (tn 1 )2 . q h dx + u h (tn )+u h (t n 1 )2 divq h dx = 0 , q h X h , vh div ph (tn )+ ph (tn 1 )2 dx = ( f (tn )+ f (tn 1 )2 ut (tn )+ u t (tn 1 )2 ) vh dx, vh M h .

pn12h . q h dx + u

n12h divq h dx = 0 , q h X h , n 1

vh div p n12

h dx = (f (tn12 ) u nh ) vh dx, vh M h , n 1u0h (c.i. ),

nh := unh uh (tn ) nh := p nh ph (tn ).

nh +

n 1h

2 . q h dx + nh +

n 1h

2 divq h dx = 0 , q h X h ,

vh div nh + n 1

h2 dx = (f (tn12 ) f (tn )+ f (tn 1 )2 u nh ut (tn )+ u t (tn 1 )2 ) vh dx,vh M h .

f, df dt H 1(0, T ; L2()) , g + f (0) H 1() ( g + f (0)) +

df

dt (0) H

1

() .

u ttt L2(0, T ; L2()) w H 1(0, T ; L2()) L2(0, T ; L2())

dwdt (t) = w(t) +

d2 f dt 2 (t), t [0, T ]

w(0) = ( g + f (0)) + df dt (0) .


69/195

v(t) = t0 w(s) ds + g + f (0) , dvdt (t) = w(t) v(0) = g + f (0) H 1()

dwdt (s) = w(s) +

d2 f dt (s), s [0, T ] 0 t,

w(t) w(0) = ( v(t) g f (0)) + df dt

(t) df dt

(0)

dvdt

(t) ( g + f (0)) df dt

(0) = v(t) ( g + f (0)) + df dt

(t) df dt

(0) .

v

dvdt (t) = v(t) +

d2 f dt 2 (t)

v(0) = g + f (0) H 1().

1.4.5 v = dudt .

d2 udt 2 =

dvdt = w H 2(0, T ; L2()) .

u ttt L2(0, T ; L2()) .

unh uh (tn ) 0, .

c > 0

unh uh (tn ) 0, ch u0 H 2 , () + tn0 u t (s) H 2 , () ds +

2 t2 tn0 u ttt (s) 0, ds + tn0 f tt (s) 0, ds .

nh

n1h 0, n 0, . vh = nh + n1h (1.107) q h = nh +

n1h

nh + n1h 22 dx = f (tn12 ) f (tn ) + f (tn1)2 u nh ut (tn ) + u t (tn1)2 nh + n1h dx.


70/195

u nh ut (tn )+ u t (tn 1 )2 ;

u nh

ut (tn ) + u t (tn1)2

= nh + u(t

n)

u(t

n1)

ut (tn ) + u t (t n1)

2

= nh + ( R h I ) u(tn ) + u (tn ) ut(tn ) + u t (tn1)

2 .

u nh ut (tn ) + u t(tn1)

2 +

f (tn ) + f (tn1)2 f (tn12 )

= nh + ( R h I ) u (tn ) + u (tn ) u t (tn12 ) + u t (tn12 ) ut (tn ) + u t (tn1)

2

+ f (tn ) + f (tn1)2 f (tn12 )

= nh + ( R h I ) u (tn ) + u (tn ) u t (tn12 ) + u(tn12 ) 12

(u(tn ) + u(tn1)) .

u(t) + f (t) = u t (t) t > 0.

n := n1 + n2 + n3 ,

n1 : = (R h I ) u (tn ),

n2 : = u (tn ) u t (tn12 ) ,

n3 : = u(tn12 ) 12

(u(tn ) + u(tn1)) .

(1.110) (1.109)

nh +n1h

2

2 dx =

( nh + n ) nh +

n1h dx.

nh nh + n1h dx 12 nh + n1h 2 + n nh + n1h .

nh nh + n1h dx = nh 2 n1h 2 t .


71/195

nh

20,

n

1

h

2

0, t 12

nh +

n

1

h

2

0, + n

0, nh +

n

1

h 0, .

(1.114) ,

(1.113)

nh2 n1h

2

t n 0, nh 0, + n1h 0, .

nh 0, n1h 0, + t n 0, , n .

n2

t n2 0, = t u (tn ) u t (tn12 ) 0,

= tu(tn ) u(tn1)

t u t (tn12 ) 0,

= u(tn ) u(tn1) t u t (tn12 ) 0, .

u(tn ) = u(tn12 ) + t

2 ut (tn12 ) +

t2

8 utt (tn12 ) +

12 tntn 12 (tn s)2u ttt (s) ds,

tn1

u(tn1) = u(tn12 ) t

2 ut (tn12 ) +

t2

8 utt (tn12 ) +

12 tn 1t n 12 (tn1 s)2u ttt (s) ds.

u(tn ) u(tn1) t u t (tn12 ) = 12 tntn 12 (tn s)2u ttt (s) ds 12

tn 1

tn 12

(tn1 s)2u ttt (s) ds.


72/195

u(tn)

u(t

n1)

t u

t(t

n12)

0,

1

2 tntn 12 (tn s)2 u ttt (s) 0, ds + 12 tn 1

tn 12

(tn1 s)2 u ttt (s) 0, ds

t2

8 tntn 12 u ttt (s) 0, ds + t2

8 tn 12t1 u ttt (s) 0, ds=

t2

8 tntn 1 u ttt (s) 0, ds.

t n2 0, t2

8 tntn 1 u ttt (s) 0, ds. n3 0,

12

u(tn ) = 1

2u(tn12 ) +

t4

ut (tn12 ) + 12 tntn 12 (tn s)u tt (s) ds,

12

u(tn1) = 1

2u(tn12 )

t4

ut (tn12 ) + 12

tn 1tn 12

(tn s)u tt (s) ds.

u(tn12 ) 12

(u(tn ) + u(tn1)) = 12 tntn 12 (tn s)u tt (s) ds 12

tn 1

tn 12

(tn1 s)u tt (s) ds.

u(tn12 ) 12

(u(tn ) + u(tn1))0,

1

2 tntn 12 (tn s) u tt (s) 0, ds + 12 tn 12

tn 1 |tn1 s| u tt (s) 0, ds

t

4 tntn 1 u tt (s) 0, ds.


73/195

t n3 0, t2

4

tn

t n 1 u tt (s) 0, ds.

n1 0,

n1 := ( R h I ) u (tn ) = ( R h I ) u(tn ) u(tn1)

t ,

R h (1.18)

X h M h M h . 1.4.9 c > 0

t n1 0, ch u(tn ) u(tn1) H 2 , ()

= ch tnt n 1 u t (s) ds H 2 , () .

t n1 0, ch tntn 1 u t (s) H 2 , () ds. (1.115)

nh 0, n1h 0, + t n 0,

n2h 0, + t n1 0, + n 0,

n3h 0, + t n2 0, + n1 0, + n 0,

0h 0, + tn

i=1

i 0, ,

0h = u0h uh (0) = 0 . (1.117) , (1.118) (1.119) ,

t n2 0, t2

8 tntn 1 u ttt (s) 0, ds,


74/195

t n3 0, t2

4 tnt n 1 u tt (s) 0, ds, t n1 0, ch

tn

tn 1u t (s) H 2 , () ds.

i = i1 + i2 + i3,

nh 0, = unh uh (tn ) 0,

ch tnt0 u t (s) H 2 , () ds + t2

tn

t 0u ttt (s) 0, ds +

tn

t0 u tt (s) 0, ds

u tt (s) = d2

dt 2 u(s) = uttt (s) f tt (s), u tt (s) uttt (s) f tt (s).

unh uh (tn ) 0, ch tn0 u t (s) H 2 , () ds +2 t2

tn

0u ttt (s) 0, ds +

tn

0f tt (s) 0, ds .

u(tn ) unh 0, : {T h}

, (i) (ii ) 1.4.6

1

w , 1 ,

c > 0 h n

1 :

u(tn ) unh 0, c h |u(tn )|H 1 () + |u(tn )|H 2 , () + tn0 u t (s) H 2 , () ds+ 2 t2 tn0 u ttt (s) 0, ds + tn0 f tt (s) 0, ds .

u(tn ) unh 0, u(t) uh (tn ) 0, + uh (tn ) unh 0, .


75/195

(1.108) (1.23) (1.123) .

p nh ,

(1.5.10)

n := n1 + n2 + n3

f H 1(0, T ; L2()) g + f (0) H 10 ().

nh2 n 2,

nh := p nh ph (tn ).

p nh .q h dx + unh div q h dx = 0 , q h X h , n 1 vh div p nh + p n

1h

2 dx = (f (tn12 ) u nh ) vh dx, vh M h , n 1.

(1.125)(i)

ph (tn ). q h dx + uh (tn ) divq h dx = 0 , q h X h ,

nh . q h dx + nh divq h dx = 0 , q h X h , nh := p nh ph (tn ) nh := unh uh (tn )

(1.126) n n 1,

nh . q h dx + nh divq h dx = 0 , q h X h , q h = nh +

n1h (1.127) .

nh20, n1h

20, = t div nh + n1h nh dx.


76/195

div nh +n1h .

(1.107) (1.110) ,

vh div nh + n1h

2 dx = nh + n vh dx,

vh M h , vh = 1K , K T h

div nh +

n1h2

= P 0h nh +

n = nh + P 0h

n ,

nh = nh n

1h

t M h . (1.128) (1.129) n

h

2

0, n1

h

2

0, =

2 t n

h

2

2 t

P 0h

n nh

dx

t P 0h n20, + t

nh

20, 2 t nh

20,

t n 20, .

nh20,

n1h

20,

t n 20, ,

(1.130) t

nh2 n 2.

c > 0 h t

nh2 ch2 tn0 u t (s) 2H 2 , () ds + c t4 tn0 u ttt (s) 20, ds + tn0 u tt (s) 20, ds .

(1.122) ,

t j1 0, ch tjt j 1 u t (s) H 2 , () ds.


77/195

t

j = n

j =1

j1

2

ch2

t

j = n

j =1 tj

t j 1 u t (s) H 2 , () ds

2

ch2 j = n

j =1 tjt j 1 u t (s) 2H 2 , () ds= h2 tnt0 u t (s) 2H 2 , () ds .

(1.120) ,

t j2 0, t2

8 tj

t j 1u ttt (s) 0, ds,

t j = n

j =1

j22

c t 4 tnt0 u ttt (s) 20, ds. (1.121) ,

t j = n

j =1 j3 2 c t4

tn

t0 u tt (s) 20, ds.

1.5.20.

1h2

0h2

t 12

2h2

1h2

t 22

nh2 n1h 2 t n 2

0h (1.89) u0h =

uh (0) ,

nh2 t

j = n

j =1

j 2 .


78/195

n = n1 + n2 + n3

n

h2

3 t j = n

j =1

j1

2

+ 3 t

j = n

j =1

j2

2

+ 3 t

j = n

j =1

j2

2

.

(1.135) , (1.134) (1.133)

{T h} , (i) (ii ) 1.4.6

1

w , 1 ,

c > 0 h t n

1 :

p(tn ) p nh 0, h |u(tn )|H 2 , () + u t L 2 (0 ,t n ;H 2 , ()) + tn0 u t (s) 2H 2 , () ds

+ t2 tn0 u ttt (s) 20, ds + tn0 u tt (s) 20, ds . 1.4.6 (1.139) .


79/195

: ( p nh , unh )n N X h M h

p nh . q h dx + unh divq h dx = 0 , q h X h , n 0 vh div p nh dx = (f (tn ) u nh ) vh dx, vh M h , n 1

u0h (c.i. ),

X h : = q h H (div , ); K T h : q h/K RT 0(K ) ,M h : = vh L2(); vh/K P 0 , K T h ,

M h v(1)h , . . . , v

(L)h

K T h L


80/195

X h

q (1)h , . . . ,q (J )h E

q (E )h := |E |2|T | (

x P ) x T ,0

E T + T , P , E |E | , E |T | ,

T.

E E

T +

q (E )h E =0 ( E ) \E,1 E ;

E ; q (E )h H (div, ) ; ( q (E )h : E E ) RT 0 (T h ) ,

divq (E )h = |E |2|T |

T 0


81/195

p nh =J

j =1

j (tn )q ( j )h u

nh =

L

l=1

l(tn )v(l)h .

J = card (E ), L = card (T h ) . (1.140)

J j =1 j (tn ) q

( j )h . q

( j )h dx +

Ll=1 l(tn )v

(l)h div q

( j )h dx = 0 , j = 1 , 2,...,J

v(l )h ( J j =1 j (tn )divq ( j )h ) dx = (f (tn ) Ll=1 l (tn ) l (tn 1 ) t v(l)h ) v(l )h dx,l = 1 , 2, ..., L.

J j =1 (

q ( j )h .q

( j )h dx) j (tn ) +

Ll=1 (

v(l)h div q

( j )h dx) k(tn ) = 0 ,

j = 1 , 2, ..., J,

t J j =1 ( v(l )h divq ( j )h dx) j (tn ) + Ll=1 ( v(l)h v(l )h dx) l(tn ) = t

f (tn )v

(l )h dx Ll=1 (

v

(l)h v

(l )h dx) l(tn1), l = 1 , 2, ..., L.

a ll = v(l)h v(l )h dx , b jj = q ( j )h q ( j )h dx , c j l = (div q ( j )h )v(l )h dx j, j = 1 , 2, ...,J, ; l, l = 1 , 2,...,L.


82/195


83/195

(tn ) =

1(tn )

L (tn )

RL , (tn ) =

1(tn )

J (tn )

RJ , F (tn ) =

K 1 (f (tn )dx

K L f (tn )dx RL .

1.141

tC (t) A (tn ) = tF (tn ) A (tn1),

B (tn ) + C (tn ) = 0 .

(tn ) = B 1C (tn ). (1.142)

tC B 1C (tn ) A (tn ) = tF (tn ) A (tn1),

(tn ) = B 1

C (tn ). G :=

C B 1C A A + tG

(A + tG ) (tn ) = F (tn ) ,

(0) = 0 (i.c. )

F (tn ) = tF (tn ) + A (tn1) .

u : [0, T ] R : (x, t ) exp( t

10) r

23 sin (

23

),


84/195

(r, ) r = x21 + x22 0, 32 sin =

x2

r

T = 1 pnh,x pnh,y

T = 1

pnh,x pnh,y

t = 0 .1

n

n = 4

t [0, T ] , u (t) H 2, () > 1 = 1 32 = 13 .


85/195

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

= 0 .375

:= 11

= 1 .6.

n = 4


86/195


87/195

0.5 1 1.5 2 2.5 3 3.5 4 4.56

5.5

5

4.5

4

3.5

3

2.5

2

1.5

log(n)

| | u

u

h | |

0 ,

,

t = T

raffinuniforme

1

1

||u (tn ) unh ||0 ,


88/195

0.5 1 1.5 2 2.5 3 3.5 4 4.55

4.5

4

3.5

3

2.5

2

1.5

log(n)

| | p

p

h | |

0 ,

,

t = T

raffinuniforme

1

1

1

2 / 3

|| p (tn ) pnn ||0 ,


89/195

u (t) p (t) (t) := u (t) t

(t) 0

K

p (t)

u (t)

u t (s) H 1 s [0, T ].

P 1,

D


90/195

1

1

R2

Q := ]0, T [

T > 0

Q f = ( f 1, f 2) L2(0, T ; (L2())2) u = ( u1, u 2) H 1(0, T ; (H 10 ())2)

p L2(0, T ; L20())

u t (x, t ) u (x, t ) +grad p(x, t ) = f (x, t ) Q,

divu (x, t ) = 0

Q,

u (x, t ) = 0 := ]0, T [,

u (x, 0) = u 0(x) x ,

= grad u

u t

div(

p ) = f Q,

div u = 0 Q,

u = 0 ,

u (0) = u 0 ,


91/195

= 1 0

0 1

div (div )i = 2 j =1 ijx j (i = 1 , 2).

V := v (H 1())2; div v = 0 .

([ ] ) f L2(0, T ; L2()2) u 0 L2()2

u L2(0, T ; V ) C ([0, T ];L2()2) u (0) = u 0 u L2(0, T ; V )

ddt

(u,v ) +

u : v =

f v, v V. (u, p )

f L2(0, T ; L2()2) u 0 V

u H 1(0, T ; L2()2).

p L2(0, T ; L20())

ddt

(u,v ) + u : v p divv = f v, v H 10 ()2. V w1, w2, . . . , wm , . . .

V u m (t) = mi=1 gim (t) wi


92/195

du m

dt (t)

w j dx +

u m (t) : w j dx =

f (t)

w j dx,

j = 1 ,...,m

u m (0) = u 0m u m (t) w1, ..., wm

u m (t)2

0,+ u m (t) : u m (t) dx = f (t) u m (t) dx.

d

dt u m (t) 2H 10 () 2 = 2

u m (t) : u m (t) dx.

2 u m (t)2

0,+

ddt u m (t)

2H 10 () 2 f (t)

2

0,+ u m (t)

2

0,,

u m (t)2

0,+

ddt u m (t)

2H 10 () 2 f (t )

2

0,.

T

0 u m (t)

2

0, dt + u m (T ) 2H 10 () 2 u m (0) 2H 10 () 2 + T

0 f (t)

2

0, dt.

u m (0) u (0) H 10 ()2, u m (0) H 10 () 2 u (0) H 10 () 2 .

T 0 u m (t) 20, dt u (0) 2H 10 () 2 + T 0 f (t) 20, dt.

u m (t)m

1

L2(0, T ; (L2())2)

v L2(0, T ; (L2())2) u m

u mw v L2(0, T ; (L2())2)

D(]0, T [) J V

L2(0, T ; V ) Rg T 0 g (t), J V,V (t) dt,


93/195

L2(0, T ; V )

T

0g (t), J V,V (t) dt

T

0g (t) V

J V

|(t)

| dt

J V T 0 g (t) 2V dt12 T 0 | (t) |2 dt

12

= J V g L 2 (0 ,T ;V ) L 2 (0 ,T )

= C g L 2 (0 ,T ;V ) . u m

w u L2(0, T ; V )

T 0 u m (t), J V,V (t) dt T 0 u (t), J V,V (t) dt. J (L2()2)

L2(0, T ; L2()2) R h

T

0

h(t), J L 2 () 2 ,(L2 () 2 )

(t) dt,

L2(0, T ; (L2())2). u mw v L2(0, T ; (L2())2),

T 0 um (t) , J L 2 () 2 ,(L2 () 2 ) (t) dt T 0 v (t), J L 2 () 2 ,(L 2 () 2 ) (t) dt J (L2()2) J |V V ,

T

0

u m (t), J (t) dt = T

0

um (t) (t) dt,J

= T 0 um (t) (t) dt,J = T 0 u m (t), J (t) dt T 0 u (t), J (t) dt .


94/195

T

0u (t), J (t) dt =

T

0v (t), J (t) dt, J L2()2 .

T 0 u (t) (t) dt = T 0 v (t) (t) dt, D (]0, T [). (u ) = v v L2(0, T ; (L2())

2)

(u ) L2(0, T ; (L2())2).

du

dt L2

(0, T

;L2

()2

).

p

q = f dudt

+ u.

u H 1(0, T ; H 10 ()2),

u L2(0, T ; H 1() 2).

q L2(0, T ; (H 1())2)

: L20() H 1()2

L20() V V ) [ ] q (t) V ,

t ]0, T [ p(t) L

20()

q (t) = p(t)

1 : V L20() : L20() V

p(t) = 1

q (t) .

q L 2(0, T ; (H 1())2), p L 2(0, T ; L20()) q = p


95/195

X := (, q ) (L2())22 L20() ; div( q ) L2()2 Y := ( L2())2

(, p ) L2(0, T ; X ) u H 1(0, T ; Y )

(t) : dx +

div(

q )

u (t) dx = 0 ,

(, q ) X,

t I,

div( (t) p(t) ) v dx = ( f (t) u t (t)) v dx, v Y, t I ,u (0) = u 0.

(2.12)

(, p ) L2(0, T ; X ). = xu, u L2(0, T ; (H 10 ())2)

= xu L2(0, T ; L2()22).

(2.2)(i)

div( p ) = u p = f + dudt L2(0, T ; L2()

2),

(, p ) L2(0, T ; X ). (2.12) (2.12)(ii ) (2.12)(i)

u L2(0, T ; V ) div u (t) = 0 , t [0, T ],

xu : q dx = 0 q L2(), t [0, T ].


96/195

(t) : dx = xu (t) : ( q ) dx (, q ) X, t [0, T ]

= ( q ) n,u (t) H 12 () 2 ,H 12 () 2 u (t) : div( q ) dx ,= u (t) div( q ) dx

u (t) (H 10 ())2 , t [0, T ].

(t) : dx + u (t ) div( q ) u (t) dx = 0 , (, q ) X, t [0, T ]. (u, p ) (2.2) ((, p ),u )

(2.12) u 0 V

(2.2) (2.12)

((1, p1),u 1) , ((2, p2),u 2) (2.12)

((, p ),u ) := (( 1 2, p1 p2),u 1 u 2)

(t) : dx + div( q ) u (t) dx = 0 (, q ) X, t I , div( (t) p(t) ) v dx = u t (t) v dx v Y, t I,

u (0) = 0 .

(, q ) = ( (t), p(t)) (2.13)(i) ((t), p(t)) X t [0, T ]

|(t)|2 dx + div( (t) p(t) ) u (t) dx = 0 , t I. v = u (t) (2.13)(ii ) , t [0, T ] : u (t)


97/195

Y = L2()2 t [0, T ]

div( (t) p(t) ) u (t) dx = ut (t) u (t) dx,

= 1

2 ddt |u (t)|2 dx. (2.15) (2.14)

|(t)|2 dx + 12 ddt |u (t)|2 dx = 0

ddt |u (t)|2 dx 0 |u ()|2 dx

|u ()|2 dx H 1 ([0, T ]) C ([0, T ]) , u (0) = 0 u (t) = 0 t [0, T ] (2.16) = 0

(2.13)(ii ) ,

x p(t) v dx = 0 v L2() 2 . (0, p(t)) X, p(t) H 1() L20(). v = x p(t),

x p(t) = 0 p(t) = t [0, T ] p(t) L20(),

p = 0 .

R2 := N j =1 j , j j = 1 , 2,...,N


98/195

f L2

(0, T ; L2

()2

)

u 0 H

1

()2

divu 0 = 0 ,

u

L2(0, T ; L2()2).

u + p = f dudt L2(0, T ; L2()2),

divu (t) = 0 ,

u (t) = 0

.

u L2(0, T ; H 2, ()2) p L2(0, T ; H 1, () L20()) ]10(), 1[

0() = inf R + ; z = + i sin2 z = z 2 sin2 , z = 1 .

f H 1(0, T ; L2()2), f (0) + u 0 H 1 ()2 div f (0) = 0 .

( w, ) L2 (0, T ; V ) L2 (0, T ; L20 ()) ([ ] )

d wdt

(t) w (t) + (t) = f (t) , t ]0, T [

div w(t) = 0 , t ]0, T [

w(0) = f (0) + u 0 .

([ ] ) V = v H 1 ()2 ; divv = 0 . w L2(0, T ; L2()2) ( w, ) L2 (0, T ; H 2, ()2) L2 (0, T ; H 1, () L20()) .

v (t) = u 0 + t0 w (s) ds q (t) = t0 (s) ds.


99/195

dv

dt (t) v (t) +

q (t) = f (t) , t ]0, T [

divv (t) = 0 , t ]0, T [

v (0) = u 0 .

u = v du

dt = w L2 0, T ; H 2, ()2 . p = q

dp

dt =

L2

(0, T

;H 1,

() L2

0()).

f H 1(0, T ; L2()2), div f (0) = 0 ,

u 0 V f (0) + u 0 H 1 ()2 .

(u, p ) L2 (0, T ; V ) L2 (0, T ; L20 ())

H 1(0, T ; H 2, () 2) H 1(0, T ; H 1, ())

]10(), 1[ 0() = inf { R + ; z = + i sin2 z = z 2 sin2 ,z = 1}.

(2.12),

(T h )h X h Y h X Y :

X h : = ( h , q h ) X ; h(i,) RT 0(K ) i = 1 , 2 q h |K 0(K ), K T h ,Y h : = v h Y ; v h |K ( 0(K ))2 , K T h .


100/195

0 K RT 0(K )

K

RT 0(K ) = {v : K R ; a,b,c R : v(x) = ( a, b) + c(x1, x 2), x = ( x1, x 2) K } u 0,h = hu 0 h (L2())

2 K T h ( 0(K ))2

(h , ph ) L2(0, T ; X h ),

u h L2(0, T ; Y h )

h (t) : h dx +

div( h

q h )

u h (t) dx = 0 ,

( h , q h ) X h ,

t I,

div( h (t) ph (t) ) v h dx = ( f (t) u h,t (t)) v h dx, v h Y h , t I ,u h (0) = u 0,h .

( [ ]) .

u +grad p = f ,

div u = 0 ,

u = 0 .

(, p ) X u Y

: dx +

div(

q )

u dx = 0 ,

(, q ) X,

div ( p ) v dx = f v dx v Y.

(u, p ) (, p ) X, = u ((, p ) ,u )

((h , ph ) ,u h ) X h Y h


101/195

h : h dx +

div ( h

q h )

u h dx = 0 ,

( h , q h ) X h ,

div ( h ph ) v h dx = f v h dx , v h Y h .

f (L2())

2 T

T : Y X Y : f T f = T 1 f , T 2 f = (( , p ) ,u ) ((, p ) ,u ) ( ) .

T h

T h : Y X h Y h : f T h f = T h, 1 f , T h, 2 f = (( h , ph ) ,u h ) ((h , ph ) ,u h ) ( )

{T h} , (i) (ii ) 1.4.6 ]10(), 1[ T f =

((, p ) ,u ) T h f = (( h , ph ) ,u h ) (, p ) (H 1, ())4

(H 1, () L20()) , C > 0 h

h 0, Ch |u |H 2 , () 2 + | p|H 1 , () ,

p ph 0, Ch |u |H 1 , () 2 + | p|H 1 , () , u u h 0, Ch |u |H 2 , () 2 + | p|H 1 , () + |u |H 1 () 2 .

(2.19)

((h , ph ) ,u h ) L2(0, T ; X h ) L2(0, T ; Y h )


102/195

g Y. T h, 1 T h, 2 ( ) T h, 1 : Y

X h T h, 2 : Y

Y h

g T h, 1g = ( h , ph ) g T h, 2g = u h ((h , ph ) ,u h )

h : h dx + ( h q h ) u h dx = 0 , ( h , q h ) X h , ( h ph ) v h dx = g v h dx , v h Y h .

g

(L2())2 := Y.

T h ,

(h (t), ph (t)) = T h, 1 f (t) d u hdt (t)

u h (t) = T h, 2 f (t) du hdt (t) .

u h (t) + T h, 2du hdt

(t) = T h, 2 f (t) u h (0) = u 0,h ,

T h, 2 Y h

f h Y h f h :

h : h dx + ( h q h ) u h dx = 0 , ( h , q h ) X h , ( h ph ) v h dx = f h v h dx , v h Y h .

T h, 2 f h = u h

T h, 1 f h = ( h , ph )

T h, 2 f h f h dx = f h u h dx = ( h ph ) u h dx= h : h dx = |h |2 0.

(T h, 2 f h ) f h dx = 0 h = 0 . h = 0 , ( h ph ) H (, ) ph H 1() ph (L2())

2


103/195

ph |K P 0(K ) ph = 0 ph = ph L20() ph = 0 . h = 0 f h = 0

(2.29)(ii ) . f h = 0

|h |2 > 0 T h, 2 f h f h dx > 0 T h, 2|Y h : Y h Y h

u h (t) = exp( t A h )u 0,h

t

0exp(( t

s)Ah )Ah T h, 2 f (s) ds

u h (t) = u 0,h , Ah = T h, 2|Y h 1 .

u h (2.19) (h , ph ).

((, p ),u ) (2.12) ((h , ph ),u h ) (2.19) ( (t) p(t) ) (L2())

2 t I,

(, p ) L2(0, T ; X )

( (t) p(t) ) ( (t) p(t) ) W 1,q()4 q > 1

t I

h ((t), p(t )) :=1

1h ( (t) p(t) ) + h ( p(t)) , h ( p(t)) , t I,

1) K T h : 1 [ 1h ( (t) p(t) ) + h ( p(t)) ]|K = 1 1K ( (t) p(t) )|K + h p(t)|K RT 0(K )2


104/195

1h [ ] 1K

2) h L20() {q h L20() q h |K P 0(K ), K T h} (h q )|K := K q :=

1

|K | K qdx, K T h . u h (0) = P hu (0)

(T h )h , (i) (ii ) 1.4.6

(0) h (0) 0, (0) h (0) 0, . (((0) , p(0)) ,u (0)) (2.12) t = 0 ,

(h (t), ph (t)) := h ((t), p(t))

(2.19)(i) t = 0 .

h (0) : h dx =

( h

q h )

u h (0) dx

( h , q h ) X h ,

= ( h q h ) P hu (0) dx,=

K

( h q h ) K P hu (0) dx,=

K

( h q h )

K

u (0) dx =

( h q h ) u (0) dx.

(2.12)(i) t = 0

( h q h ) u (0) dx = (0) : h dx, ( h , q h ) X h X, ( h , q h ) X h ,

((0) h (0)) : h dx = 0 .


105/195

(0)

h (0) 2

0 ,

=

((0)

h (0)) : ((0)

h (0)) dx,

= ((0) h (t)) : ((0) h (0)) dx + (h (t) h (0)) : ((0) h (0)) dx (2.31) h = h (0) h (0) RT 0(K )2

(0) h (0) 20, = ((0) h (0)) : ((0) h (0)) dx

(0)

h (0) 0, (0)

h (0) 0, .

(0) h (0) 0, (0) h (0) 0, .

(0) (2.18)

u C ([0, T ]; H 2, () 2)

C ([0, T ]; H 1, ()22).

(u (t), p(t)) (2.1) , t

(h (t) , ph (t)) ; uh (t) = T h ( u (t) + grad p (t)) , (h (t) , ph (t)) ; uh (t)

h (t) : h dx + div( h q h ) uh (t) dx = 0 , , ( h , q h ) X h , div ( h (t) ph (t) ) v h dx + ( u (t) + grad p(t)) v h dx = 0 , t I, v h Y h .

(2.1)

(, p) ; u = T ( u + grad p),


106/195


107/195

u (t) uh (t) 0, c h |u (t)|H 1 () 2 + |u (t)|H 2 , () 2 + | p(t)|H 1 , ()

p(t) ph (t) 0, c h(|u (t)|H 2 , () 2 + | p(t)|H 1 , () ). 2.4.5

(t) h (t) 0, (2.19)

(T h )h , (i) (ii ) 1.4.6 ]10(), 1[

c > 0 h t I :

(t) h (t) 0,

c h suptT |u (t)|H 2 , () 2 + | p(t)|H 1 , () +

dudt L 2 (0 ,T ;H 2 , () 2 )

+dpdt L 2 (0 ,T ;H 1 , ())

.

(2.19)

(2.34)

h (t) : h dx + div( h q h ) u h (t) dx = 0 ( h , q h ) X h , t I, div( h (t) ph (t) ) v h dx = ( f (t) u h,t (t)) v h dx v h Y h , t I,

u h (0) = P hu (0) ,

t I

h (t) : h dx + div( h q h ) uh (t) dx = 0 ( h , q h ) X h , div( h (t) ph (t) ) v h dx = ( f (t) u t (t)) v h dx v h Y h .


108/195

(2.40) (2.39),

h (t) : h dx +

div( h

q h )

h (t) dx = 0 ,

t I, ( h , q h ) X h ,

div( h (t) r h (t ) ) v h dx = ddt (u h (t) u (t)) v h dx t I, v h Y h ,

h = h h , h = u h uh h = u uh rh = ph ph .

ht (t) : h dx + div(

h q h ) h

t (t) dx = 0 , ( h , q h ) X h , t I

v h = 2 ht (ii ) ( h , q h ) = 2 (h , r h )

2 ht (t) : h dx + 2 div( h r h ) ht (t) dx = 02 div ( h (t) r h (t) ) ht (t)dx + 2 ht (t)

2dx = 2 ht (t) ht (t) dx.

2 ht (t) : h dx + 2 ht (t)2

dx = 2 ht (t) ht (t) dx,

ddt |h (t)|2 dx + 2 ht (t)

2

dx = 2 ht (t) ht (t) dx

2 h

t 0, h

t 0,

ht

2

0,+

ht

2

0,

.

ddt |h (t)|2 dx + d hdt

2

0,

d hdt

2

0,.


109/195

ddt

|h (t)|

2 dx 1

d hdt

2

0,.

t,

|h (t)|2 dx |h (0)|2 dx 1 t0 d hdt 20,(s) ds,

|h (t)|2 dx |h (0)|2 dx + 1 t0 d hdt 20,(s) ds.

h (0) = h (0) h (0) 0, h (0) (0) 0, + (0) h (0) 0, . h ((t), p(t))

h (0) (0) 0, (0) h (0) 0,

= (0) 1

1h ( (0) p(0) ) + h ( p(0)) 0,

=1 ( (0) p(0)) 1h ( (0) p(0) ) +

1 ( p(0) h ( p(0)) 0,

1

( (0) p(0) ) 1h ( (0) p(0) ) 0, +

2 p(0) h ( p(0)) 0, .

h (0) (0) 0, c h ( |u (0)|H 2 , () 2 + | p(0)|H 1 , () . t = 0

h (0) 0, c h |u (0)|H 2 , () 2 + | p(0)|H 1 , () .

ht 0,

=dudt

d uhdt

0,

.


110/195

h (t)20, h (0)

20, +

1

t

0

dudt

d uhdt

2

0,ds.

(2.49) , (2.36)

h (t) 20,

c h2 |u (0)|H 2 , () + | p(0)|H 1 , ()2 + t0 ( ut 2H 2 , () 2 + ut 2H 1 () 2 + pt 2H 1 , ()) ds .

h (t) h (t) 0,

c h |u (0)|H 2 , () + | p(0)|H 1 , () +dudt L 2 (0 ,T ;H 2 , () 2 )

+dpdt L2 (0 ,T ;H 1 , ())

.

(t)

h (t) 0,

(t)

h (t) 0, + h (t)

h (t) 0,

c h suptT |u (t)|H 2 , () + | p(t)|H 1 , () +

dudt L 2 (0 ,T ;H 2 , () 2 )

+dpdt L 2 (0 ,T ;H 1 , ())

.

{T h} c > 0 h t I :

u (t) u h (t) 0, c inf v h Y h u (t) v h 0, + (t) h (t) 0, .

{T h} , (i) (ii ) 1.4.6 ]10(), 1[ .


111/195

c > 0 h t I :

u (t)

u h (t) 0,

c h suptT |u (t)|H 2 , () + | p(t)|H 1 , () +

dudt L2 (0 ,T ;H 2 , () 2 )

+dpdt L 2 (0 ,T ;H 1 , ())

.

u (t) u h (t) 0, c inf v h Y h u (t) v h 0, + (t) h (t) 0, . v h = P hu (t) Y h , t I , P hu (t)

K T h u (t) K

u (t) u h (t) 0, c u (t) P hu (t) 0, + (t) h (t) 0, . c > 0 h t I :

u (t) P hu (t) c h |u (t )|H 1 () 2 . (2.38) ,

u (t) u h (t) 0,

c h suptT u (t) H 2 , () 2 + | p(t)|H 1 , () +

dudt L2 (0 ,T ;H 2 , () 2 )

+dpdt L 2 (0 ,T ;H 1 , ())

.

{T h

} ,

(i) (ii ) 1.4.6 ]10(), 1[ C > 0 h t I :

p() ph () L2 (0 ,T ;L2 ()) C h u () L 2 (0 ,T ;H 2 , () 2 ) + p() L 2 (0 ,T ;H 1 , ())+ C h u (0) H 2 , () 2 + p(0) H 1 , () +

dudt

() L 2 (0 ,T ;H 2 , () 2 ) +dpdt

() L 2 (0 ,T ;H 1 , ())


112/195

(2.41) ,

h (t) : h dx +

div( h q h ) h (t) dx = 0 , t I, ( h , q h ) X h ,

div( h (t) r h (t ) ) v h dx = ddt (u h (t) u (t)) v h dx t I, v h Y h ,

h = h h , h = u h uh h = u uh rh = ph ph . v H 10 ()2 t I

r h (t)

v dx =

div rh (t)

v dx

= r h (t) : v dx= ( h (t) r h (t) ) : v dx h (t) : v dx.

C > 0

h (t) : v dx

C

h (t) 0,

v H 1 () 2 .

v H 10 ()2

( h (t) r h (t) ) : v dx = div( h (t) r h (t) ) v dx= div( h (t) r h (t) ) P hv dx=

(u t (t)

u h,t (t))

P hv dx

(2.54)(ii )

C > 0

( h (t) r h (t) ) : v dx u t (t) u h,t (t) 0, P hv 0, C u t (t) u h,t (t) 0, v H 10 () 2 .


113/195

(2.55) (2.56) (2.57) ,

r h (t) H 1 () 2

C u t (t)

u h,t (t) 0, + h (t) 0, .

C > 0 w L20()

w L 20 () C w H 1 () 2 = supv H 10 () 2 w v dxv H 1 () 2 . (2.59) ,

r h (t) L20 () C u t (t) u h,t (t) 0, + h (t) 0, , h = u h uh h = u uh

r h (t) L20 () C h,t (t) 0, + h,t (t) 0, + h (t) 0, . (2.44)

ddt |h (t)|2 dx + d hdt (t)

2

0,

d hdt

(t)2

0,.

(2.63) t I,

h (t) 20, + t0 d hdt (t)2

0,

dt t0 d hdt (t) 20, dt + h (0) 20, . (2.54) h h

:

h (t) : h dx +

div( h q h ) h (t) dx = 0 t I, ( h , q h ) X h ,

div( h (t) r h (t) ) v h dx = h,t (t) v h dx h,t (t) v h dx t I, v h Y h . v h = h (t), h = h (t) q h = r h (t).

|h (t) |2 dx + div( h (t) r h (t) ) h (t) dx = 0 , div( h (t) r h (t) ) h (t) dx = h,t (t) h (t) dx h,t (t) h (t) dx.


114/195

|

h (t)

|2 dx +

h,t (t)

h (t) dx =

h,t (t)

h (t) dx,

h (t) 20, + 12

ddt

h (t)2

0,= h,t (t) h (t) dx h (t) 0, h,t (t) 0,

1

2 h(t)

2

0,+

h,t(t) 2

0,.

C > 0

T 0 h (t) 20, dt + h (t) 20, C T 0 h,t (t) 20, dt + h (0) 20, . (2.62) ,

T 0 r h (t) 2L 20 () dt C T 0 h,t (t) 20, + h,t (t) 20, + h (t) 20, dt. (2.64) , (2.65)

T 0 r h (t) 2L 20 () dt C T 0 h,t (t) 20, dt + h (0) 20, + h (0) 20, .

h (0)0, u h (0) u (0) 0, + u (0) uh (0) 0,

=

P hu (0)

u (0) 0, + u (0)

uh (0)

0,.

(2.53) , (2.36) ,

h (0)0,

= u h (0) uh (0) 0, C h u (0) H 2 , () 2 + p(0) H 1 , () .

h (0) = h (0) h (0) 0, C h u (0) H 2 , () 2 + p(0) H 1 , () .


115/195

(2.36) ,

h,t (t) = u t (t) uh,t (t) 0, C h

dudt (t) H 2 , () 2 +

dpdt (t) H 1 , () .

p(t) ph (t) 0, p(t) ph (t) 0, + ph (t) ph (t) 0, , r h (t) = ph (t) ph (t),

p(t) ph (t) 20, 2 p(t) ph (t) 20, + 2 r h (t) 20, .

T 0 p(t) ph (t) 20, dt 2 T 0 p(t) ph (t) 20, dt + 2 T 0 r h (t) 20, dt. (2.37), (2.67) , (2.68) , (2.69) (2.70)

T 0 p(t) ph (t) 20, dt C h 2 T 0 u (t) 2H 2 , () 2 dt + T 0 p(t) 2H 1 , () dt+ C h

2u (0)

2H 2 , () 2 + p(0)

2H 1 , () +

T

0

dudt (t)

2

H 2 , () 2dt +

T

0

dpdt (t)

2

H 1 , ()dt .

p(t) ph (t) L 2 (0 ,T ;L 2 ()) C h u (t) L 2 (0 ,T ;H 2 , () 2 ) + p(t) L 2 (0 ,T ;H 1 , ())+ C h u (0) H 2 , () 2 + p(0) H 1 , () +

dudt

(t)L2 (0 ,T ;H 2 , () 2 )

+dpdt

(t)L 2 (0 ,T ;H 1 , ())

.


116/195

[0, T ] N [tn1, t n ] n

0 = t 0 tn < tN = T , k = tn tn1 u nh

tn = nk. u ht tn ,

u nh = (u nh u n1h )

k

.

nh : h dx +

div ( h q h ) u nh dx = 0 , ( h , q h ) X h , n 0

div ( nh pnh ) v h dx + ( f (tn ) u nh ) v h dx = 0 , v h Y h , n 1u 0h = u 0,h .

( ) ((nh , pnh ) ,u nh ) X h Y h .

F (v h ) := ( f (tn ) + 1ku n1h ) v h dx = 0 , v h Y h .

nh : h dx + div ( h q h ) u nh dx = 0 , ( h , q h ) X h , div ( nh pnh ) v h dx 1k u nh v h dx = F (v h ), v h Y h .


117/195

((nh , pnh ) ,u nh ) X h Y h , X h Y h :

( h , q h ) nh : h dx + div ( h q h ) u nh dxv h div ( nh pnh ) v h dx 1k u nh v h dx

.

X h Y h ((nh , pnh ) ,u nh ) X h Y h

((nh , pnh ) ,u nh )

nh : h dx + div (

h q h ) u nh dx = 0 ,

div ( nh pnh ) v h dx 1k u nh v h dx = 0 ,

(( h , q h ) ,v h ) X h Y h (2.74) : h = nh , q h = p

nh v h = u

nh .

|nh |2 dx + div ( nh pnh ) u nh dx = 0

div ( nh pnh ) u nh dx = 1k |u nh |2 dx,

|nh |2 dx + 1k |u nh |2 dx = 0 . u nh = 0 nh = 0 . pnh = 0 .

pnh |K = cte,

nh = 0 ,

pnh H (div, )

pnh = cte

, pnh L20()

(2.72),


118/195

k 12 ,

u N h 0, 2 exp(2) u 0h 0, + N

n =1k f (tn )

2

0,

(2.72),

nh : h dx + div ( h q h ) u nh dx = 0 , ( h , q h ) X h , div ( nh pnh ) v h dx + ( f (tn ) u nh ) v h dx = 0 , v h Y h .

v h = u nh (2.77, ii ),

div ( nh pnh ) u nh dx = ( f (tn ) u nh ) u nh dx= f (t n ) u nh dx + u nh u nh dx= f (t n ) u nh dx + u nh u n1hk u nh dx= f (t n ) u

nh dx + 1k |u

nh |2 dx 1k u

nh u n1h dx.

h = nh q h = pnh (2.77, i )

|nh |2 + 1k |u nh |2 dx 1k u nh u n1h dx = f (tn ) u nh dx,

|nh |

2

+ 1k |u

nh |

2

dx = 1

k unh u

n

1

h dx + f (tn ) u

nh dx

f (tn ) u nh dx + 12k |u nh |2 dx + 12k u n1h 2 dx.

12k |u nh |2 dx 12k u n1h 2 dx + f (tn ) u nh dx,


119/195

u nh20, u n1h

20, + 2 k

f (tn )0, u nh 0, .

n = 1 N,

N

n =1

u nh20,

N

n =1

u n1h20, + 2 k

f (tn )0, u nh 0, .

u N h20, u 0h

20, + k

N

n =1

f (tn )2

0,+ k

N

n =1

u nh2 .

u N h2

0,

(1 k) u N h20, u 0h

20, + k

N

n =1

f (tn )0,

+ kN 1

n =1

u nh .

k 12

u N h20, 2 u 0h

20, + k

N

n =1

f (tn )2

0,+ k

N 1

n=1

u nh2 .

u N h20, exp 2k

N 1

n =1

1 2 u 0h20, + 2 k

N

n =1

f (t n )2

0,.

u N h20, 2exp(2T ) u 0h

20, + k

N

n=1

f (tn )2

0,.

nh ,

C > 0

N n =1

k nh20, C u 0h 0, + k N

n =1

f (tn )2

0,


120/195

(2.72)

h = nh , q h = pnh v h = u nh ,

|nh |2 dx ( f (tn ) u nh ) u nh dx = 0 . k

k |nh |2 dx k ( f (tn ) u nh ) u nh dx = 0 . n = 1 ...N,

N

n =1

k nh20, +

N

n =1 u nh u n1h u nh dx = k f (tn ) u nh dx.

u nh u n1h u nh dx = |u nh |2 dx u n1h u nh dx u nh 20,

12

u nh20,

12

u n1h20,

= 12

u nh 20, 12 u n1h 20, .

(2.79) ,

N

n =1

k nh20, +

12

u N h20,

12

u 0h20,

N

n =1

k f (tn )0, u nh 0, .


121/195

(2.80)

N

n =1

k f (t

n ) 0, u n

h 0, =

N

n =1

k12 f

(t

n ) 0, k

12 u n

h 0,

N

n =1

k f (tn )2

0,

12 N

n =1

k u nh20,

12

N

n =1

k f (tn )2

0,

12

Cte u 0h20, + k

N

n=1

f (tn )2

0,

12

(2.76) ,

Cte N

n =1

k f (tn )2

0,

1

2

u 0h20, + k

N

n=1

f (tn )2

0,

1

2

Cte N

n =1

k f (tn )2

0,+ u 0h

20, + k

N

n =1

f (tn )2

0,.

(2.80) .

N

n =1

k nh20, +

12

u N h20, Cte u 0h

20, + k

N

n=1

f (tn )2

0,.

N

n=1

k nh20, Cte u 0h

20, + k

N

n =1

f (tn )2

0,.

pnh 0, .

C > 0,

N

n =1

k pnh20, C u 0h 0, + 0h 0, +

N

n =1

k f (tn )2

0,

C > 0, H (div ; )2,

tr ( ) dx = 0


122/195

0, C D 0, + div ( ) 0, (nh pnh ) ,

(2.81) . (2.72)(i) :

h = q h = 0 = 1 0

0 1.

(2.72)

tr (nh

) dx = 0

pnh L20(),

tr (nh pnh ) dx = 0 . (2.82) ,

nh pnh 0, C (nh pnh )D 0, + div (nh pnh ) 0, .

(nh pnh )D = ( nh pnh ) 12

tr (nh pnh )

= nh, 11 pnh nh, 12

nh, 21 nh, 22 pnh

12

(nh, 11 + nh, 22 2 pnh )

=12

nh, 11 12 nh, 22 nh, 12

nh, 21 12 nh, 22 12 nh, 11.

. F x (nh pnh )D (x)

2F

= 1

2nh, 11(x)

2 + 12

nh, 22(x)2 + nh, 12(x)

2 + nh, 21(x)2

nh, 11(x) nh, 22(x) nh, 11(x)

2 + nh, 22(x)2 + nh, 12(x)

2 + nh, 21(x)2 = nh (x)

2F .


123/195

,

(nh

pnh )

D 20,

nh

20, .

(2.83) (2.85)

nh pnh 0, C nh 0, + div (nh pnh ) 0, .

nh pnh 0, pnh 0, nh 0, pnh 0, nh 0, .

pnh 0, C nh 0, + div (nh pnh ) 0, . div (nh pnh ) 0, . (2.72)(ii ) v h Y h :

div ( nh pnh ) v h dx = ( f (tn ) u nh ) v h dx,

div ( nh pnh ) = P 0h (u nh f (tn )) , P 0h L

2()2 Y h

div (nh pnh ) 0, u nh 0, + f (tn ) 0, . (2.88) (2.87)

pnh 0, C nh 0, + u nh 0, + f (t n ) 0, ,

N

n =1

k pnh20, 3C

N

n =1

k nh20, +

N

n =1

k u nh20, +

N

n =1

k f (tn )2

0,.

2.5.3

N n=1

k nh20, C u 0h 0, + k N

n =1

f (tn )2

0,.


124/195

(2.89) , N n =1 k unh

20,

(2.72) ,

nh : h dx + div ( h q h ) u nh dx = 0 ( h , q h ) X h , (2.91) , h = nh q h = pnh ,

nh : nh dx + div ( nh pnh ) u nh dx = 0 . v h = u nh (2.72)(ii ) .

div (nh p

nh )

u

nh dx + (

f (tn )

u

nh )

u

nh dx = 0 .

nh : nh dx = f (tn ) u nh dx u nh 20, .

nh : nh dx + u nh 20, f (tn ) 0, u nh 0, .

nh : nh dx = k nh 20, k n1h : nh dx 2k nh 20, 2k n1h 20, . 2k

nh20, n1h

20, + 2 k u

nh

20, 2k f (tn ) 0, u nh 0,

k f (tn )

2

0,+ u nh

20, .

nh20, n1h

20, + k u

nh

20, k f (tn )

2

0,.

n = 1 , . . ,N,

N h20, 0h

20, +

N

n =1

k u nh20,

N

n =1

k f (tn )2

0,.


125/195

N

n =1

k u nh20,

N

n =1

k f (tn )2

0,+ 0h

20, .

(2.95) , (2.90) (2.89)

N

n =1

k pnh20, C u 0h

20, +

0h

20, +

N

n =1

k f (tn )2

0,.

u u nh

(L2)2 , tn :

(( h (tn ), ph (tn )); uh (tn ) X h Y h

h (tn ) : h dx + div ( h q h ) uh (tn ) dx = 0 , ( h , q h ) X h ,

div ( h (tn ) ph (tn ) ) v h dx +

(

f (tn ) u t (tn )) v h dx = 0 , v h Y h .

c > 0 h n

u nh uh (tn ) 0, ch tn0 u t (s) H 2 , () 2 + pt (s) H 1 , () ds + k tn0 u tt (s) 0, ds.

nh = nh

h (tn ), r nh = p

nh

ph (tn ) nh = u

nh

uh (tn ).

(2.72) (2.97) ,

nh : h dx + div ( h q h ) nh dx = 0 ( h , q h ) X h , div ( nh r nh ) v h dx = (u t (tn ) u nh ) v h dx, v h Y h .


126/195

h = nh , q h = rnh v h =

nh .

(2.99)

|nh |2 dx + div ( nh r nh ) nh dx = 0 , div ( nh r nh ) nh dx = (u t (tn ) u nh ) nh dx.

|nh |2 dx + ( uh (tn ) u t (tn )) nh dx = nh nh dx.

nh 2 dx nh n1h dx = k |nh |2 dx k ( uh (tn ) u t (tn )) nh dx k ( uh u t (tn )) nh dx k nh 0, uh (tn ) u t (tn ) 0, .

nh2

0, nh n1h dx + k nh 0, uh (tn ) u t (t n ) 0, .

nh0, n1h 0, + k uh (tn ) u t (tn ) 0, .

n := uh (t n )

u t (tn )

n := n1 + n2 n1 = uh (tn ) u (tn ) n2 = u (tn ) u t (tn ) . n1 0, T T h

(2.23) (2.24)

n1 = uh (tn ) u (tn ) = T h, 2 f (tn ) u t (tn ) T 2 f (tn ) u t (tn ) .


127/195

n1 0,

= (T h, 2

T 2) f (t

n)

u

t(t

n)

0,

=1k

(T h, 2 T 2) tntn 1 f t (s) ds tntn 1u tt (s) ds 0,

1k

(T h, 2 T 2) tnt n 1 f t (s) u tt (s) ds 0,

1k

tn

tn

1

(T h, 2 T 2) f t (s) u tt (s) 0, ds.

n1 0, 1k tntn 1 u t,h (s) u t (s) 0, ds.

c > 0 h

n1 0, ch1k

tn

tn 1u t (s) H 2 , () 2 + pt (s) H 1 , () ds.

kN

n =1

n1 0, ch1k tnt0 u t (s) H 2 , () 2 + pt (s) H 1 , () ds.

n2 = u h (tn ) u t (tn )

= 1

k (u (tn ) u (tn1) ku t (tn ))

= 1

k tntn 1 (tn1 s) u tt (s) ds.

n2 0, tntn 1 u tt (s) 0, ds.


128/195

kN

n =1

n2 0, k

tn

0u tt (s) 0, ds.

(2.103) (2.102) ,

nh0, 0h 0, + ch tnt0 u t (s) H 2 , () 2 + pt (s) H 1 , () ds + k tnt0 u tt (s) 0, .

u 0h = uh (t0), 0h = 0 .

nh0, ch tnt0 u t (s) H 2 , () 2 + pt (s) H 1 , () ds + k tnt0 u tt (s) 0, .

u nh u (tn ) 0, O(h). (T h )h ,

(i) (ii ) 1.4.6 ]10(), 1[ c > 0 h t I :

u nh u (tn ) 0, c h |u (t)|H 1 () 2 + |u (t)|H 2 , () 2 + | p(t)|H 1 , ()+ ch tnt0 u t (s) H 2 , () 2 + pt (s) H 1 , () ds + k tnt0 u tt (s) 0, ds.

nh 0, .

nh20,

1 n

20, .

(2.99) ,

nh : h dx + div ( h q h ) nh dx = 0 , ( h , q h ) X h ,


129/195

h = nh q h = r nh ,

nh : nh dx =

div ( nh

r nh )

nh dx

= n nh nh dx (2.99, ii )

= n nh dx nh 20, n 0, nh 0, nh

2

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Korikache R., Methode d'elements finis mixtes.pdf