CoursUFE_KhaledSaleh

7/23/2019 CoursUFE_KhaledSaleh

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http://www.ann.jussieu.fr/~saleh/mailto:[email protected]


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C1 C2


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7/29

RN

f( ) T( ) u( )

f( ) u( ) R3 T( )

R3

(u)

u

(u) =12

u + t(u)

=

12

uixj

+ ujxi

1i,jN

,


8/29

(u)

(u)

= 2(u) + ((u)) .

E

= E

2(1 + ), =

E

(1 + )(1 2).

= f ,

= N

j=1

ijxj

1iN

.

u

(2(u) + ((u)) ) = f ,

Ox1x2 f( ), e3 f=f(x1, x2)e3 u e3 u= u(x1, x2)e3

e2

e3

e1

f(x)

u(x)


9/29

(u) = f ,

(u) = 2(u) +

((u))

,

ij =

uixj

+ ujxi

+

3k=1

ukxk

ij ,

ij u1= u2 =u3x3

= 0

=

0 0

ux1

0 0 ux2

ux1

ux2

0

,

( (u)).e3 = 2u

x2

1

+2u

x2

2

u= f ,

u

(u)(x, y) :=2u

x2(x, y) +

2u

y2(x, y).

u

(P)

u= f ,

u= 0 .

u


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u [0, 1]

(P)

u(x) = f(x), x (0, 1),

u(0) =u(1) = 0.

V20 C2 [0, 1] [0, 1]

V20 =

u C2[0, 1], u(0) =u(1) = 0

.

f x u (P)

x u(x) = f(x) u

x

f(x)

u(x)0 1x

(P) (P)

u

(P) u (P) v [0, 1]

10

u(x)v(x)dx= 10

f(x)v(x)dx.

v 1

0

u(x)v(x)dx

u(x)v(x)10

=

10

f(x)v(x)dx,

v v(0) =v(1) = 0 10

u(x)v(x)dx=

10

f(x)v(x)dx.

u (P)

v v(0) =v(1) = 0 10

u(x)v(x)dx=

10

f(x)v(x)dx.


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u C1

C2 (P) u C2 (P) u C1 0

1

u

v

u v

V10 =

u C1 [0, 1], u(0) =u(1) = 0

.

(P)

u V10

(FV) 1

0

u(x)v(x)dx= 1

0

f(x)v(x)dx, v V10.

u (P)

u (FV) u

(FV) u C2 u (P)

10

u(x)v(x)dx=

10

f(x)v(x)dx v V10.

v u(x) = f(x) x (0, 1) u V10

u

u(0) = u(1) = 0

u

u [0, 1]

(N) u(x) + u(x) = f(x), x (0, 1),

u

(0) =, u

(1) =.

x f

x

qv(x) q


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= = 0

u

v

[0, 1]

10

u(x)v(x)dx +

10

u(x)v(x)dx=

10

f(x)v(x)dx.

10

u(x)v(x)dx

u(x)v(x)10

+

10

u(x)v(x)dx=

10

f(x)v(x)dx,

10

u(x)v(x)dx +

10

u(x)v(x)dx=

10

f(x)v(x)dx + v(1) v(0),

u(0) = u(1) =

u

v

C1 [0, 1]

V1 =

u C1 [0, 1]

.

(N)

u V1

(FV)

10

u(x)v(x)dx +

10

u(x)v(x)dx=

10

f(x)v(x)dx + v(1) v(0),

v

V1

u C2 (FV) u (N)

u [0, 1]

u(x) = f(x), x (0, 1),

u(0) =, u(1) = 0.


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u

(P)

u(x, y) = f(x, y), (x, y) ,

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

R2 n(x, y) (x, y) nx ny n dxdy ds

n

nx

ny

ds

C1

x

(x, y) x

(x, y)dxdy =

x

(x, y)(x, y)dxdy+

(x, y)(x, y)nx(x, y)ds.

y

(x, y)

y(x, y)dxdy =

y(x, y)(x, y)dxdy+

(x, y)(x, y)ny(x, y)ds.

u C2 v C1

u(x, y)v(x, y)dxdy=

u(x, y).v(x, y)dxdy+

v(x, y)u(x, y).n(x, y)ds,

u =ux

, uy

u a.b

a

b

R2


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= v = ux

= v = uy

u C2 v(x, y)

u(x, y)v(x, y)dxdy=

f(x, y)v(x, y)dxdy.

u(x, y).v(x, y)dxdy

v(x, y)u(x, y).n(x, y)ds=

f(x, y)v(x, y)dxdy.

v

u(x, y).v(x, y)dxdy=

f(x, y)v(x, y)dxdy.

u v C1

V10 =

u C1 , u= 0

.

u V10

u(x, y).v(x, y)dxdy =

f(x, y)v(x, y)dxdy,

v

V1

0.

R3 f( ) T( ) N D u( )

= f .

3

j=1

ijxj

=fi, i= 1, 2, 3.


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vi

n

j=1ijxj

vid =

fivid i= 1, 2, 3.

3j=1

ijvixj

d

viijnjds

=

fivid i= 1, 2, 3,

3j=1

ijvixj

d

3j=1

viijnjds=

fivid i= 1, 2, 3.

i

: v d

v.(.n)ds =

f.v d,

v = (vi)1i3 v =

vixj

1i,j3

A: B =3

i,j=1

AijBij A B

3 3 .n n= (nj)1j3 (.n)( )ds N (.n)( ) =T( )

: v d

N

v.T ds

D

v.(.n)ds =

f.v d.

D

V

u

V

u D v V D D

V =

u= (u1, u2, u3) C1

, u= 0 D

.

V

u V

(u) : v d =

f.v d +

N

v.T ds, v V .

v

(u) : v d

f.v d +N

v.T ds


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u

2(u) : (v) + ( u)( v)

d =

f.v d +N

v.T ds.


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u V

(FV) a(u, v) = f, v , v V ,

V (u, v)a(u, v) V V v f, v V

V =

u C1 [0, 1], u(0) =u(1) = 0

a f, .

a(u, v) =

10

u(x)v(x)dx, f, v=

10

f(x)v(x)dx.

V Vh

uh Vh

(FV)h a(uh, vh) = f, vh , vh Vh.


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Vh V

Vh (FV)h

(1,...,N) Vh uh

uh(x) =Nj=1

ujj(x).

(FV)h U= (u1,...,uN) RN

a Nj=1

ujj , vh

=f, vh , vh Vh,

Nj=1

a (j , vh) uj =f, vh , vh Vh,

N

j=1

a (j

, i) u

j=f,

i , i {1,...,N},

(1,...,N) Vh

A= (Aij)1i,jN RNN, Aij =a(j , i),

F = (Fi)1iN RN, Fi= f, i ,

AU=F.

A U U=A1F (FV)h A

Vh A

(P)

u= f, ,

u= 0 .


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f [0, 1]

C2([0, 1])

u(x) = x 10 f(t)(1 t)dt

x0 f(t)(x t)dt, x [0, 1].

V =

u C1 [0, 1], u(0) =u(1) = 0

,

Vh V [0, 1]

0 = x0 < x1 < x2 < ... < xN< xN+1 = 1,

[0, 1] [xi, xi+1], i = 0...N [xi, xi+1] hi = xi+1 xi i

h= max0iN

xi+1 xi

xi

xi = ih, h= 1

N+ 1.

Pk k

Pk =

p(x) =

kj=0

ajxj , aj R

.

k+ 1

P1

[0, 1] [xi, xi+1] 0 1

V1h =

v C0([0, 1]), v|[xi,xi+1] P1, 0 i N, v(0) =v(1) = 0

.


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h V1h V1h

N

V1h

v(xi) 1 i N V1h (1,...,N)

[xi, xi+1]

i {1..N} , j {0..N+ 1} , i(xj) = ij ,

ij

(x) =

1 |x| |x| 1,

0 |x|> 1.

i i(x) =

x xi

h

(1,...,N) v V1h

v(x) =Ni=1

v(xi)i(x).

i

x

i(x) = 0 [0, 1]

x0= 0 x1 xi1 xi xi+1

xN+1 = 1

1i

A=

a(j , i)1i,jN

Aij =a(j , i) =

10

j(x)i(x)dx,

j i 1 i i+ 1 j i

Ai1,i= a(i1, i) =

10

i1(x)i(x)dx=

xixi1

1

h

1

hdx=

1

h,

Ai,i= a(i, i) = 10

i(x)2dx= xixi1

1

h2 dx + xi+1xi

1

(h)2 dx= 2

h ,

Ai+1,i= a(i+1, i) =

10

i+1(x)i(x)dx=

xi+1xi

1

h

1

h dx=

1

h.


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A= 1h

2 1 0 0

1 2

0

1 0

2 10 0 1 2

.

X, Y =Ni=1

xiyi X = (x1, . . ,xN)

Y = (y1, . . ,yN) RN

A

A

U RN {0}, AU, U> 0.

A A U= 0 RN

AU = 0 AU, U= 0 U = 0 A

U= (u1, . . ,uN)= 0 Rn

AU

Nj=1

A1juj ,...,Nj=1

ANjuj

= Nj=1

a(j , 1)uj ,...,Nj=1

a(j, N)uj

.

AU, U=Ni=1

Nj=1

a(j , i)ujui= a(Nj=1

ujj,Ni=1

uii) = a(u,u) = 1

0

|u(x)|2dx,

u(x) V1h u(x) =N

i=1 uii(x) (u1, . . ,uN)

u

10

|u(x)|2dx >0

F= (f, i)1iN

f, i=

10

f(x)i(x)dx=

xixi1

f(x)i(x)dx +

xi+1xi

f(x)i(x)dx.

f

[xi, xi+1]


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xi+1xi

(x)dx (xi+1 xi)

xi+ xi+1

2

.

xi+1xi

(x)dx (xi+1 xi)(xi) + (xi+1)

2 .

xi+1xi

(x)dx (xi+1 xi)(xi) + 4

xi+xi+1

2

+ (xi+1)

6 .

1

h2

2 1 0 0

1 2

0

1 0

2 10 0 1 2

u1u2

uN

=

f(x1)f(x2)

f(xN)

,

(u1,...,uN)

uh(x) =

Ni=1

uii(x).

P2

[xi, xi+1] u u

P2

u [a, b] 2 [xi, xi+1]

V2h =

v C0([a, b]), v|[xi,xi+1] P2, 0 i N, v(0) =v(1) = 0

.

xi+ 12

=xi+ xi+1

2 , i {0...N} ,

V2h xi, i {1, . . ,N}

xi+ 12

, i {0...N} V2h 2N+ 1 V2h

i(x) 1 i N

i V2h , i(xj) = ij , i(xj+ 1

2) = 0, j,


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i+ 12

(x) 0 i N

i+ 12

V2h , i+ 12

(xj) = 0, i+ 12

(xj+ 12

) = ij , j.

(x) =

(1 + x)(1 + 2x) 1 x 0

(1 x)(1 2x) 0 x 1

0 |x|> 1

(x) =

1 4x2 x 1/2

0 x >1/2

i(x) =

x xi

h

i+ 12

(x) =

x xi+ 1

2

h

xi1 xi xi+ 12

xi+1

i+ 12

i

P2

(x0, x 12

, x1,...,xN+ 12

, xN+1) (x k2

)0k2N+2

( 12

, 1,...,N+ 12

) ( k2

)1k2N+1

(P)

u= f ,

u= 0 .

R2 u

V =

u C1 , u= 0

,

uh V Vh

Th (Ki)1iN


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h

Ki Kj

Pk

k

Pk=

p(x, y) =

i,j0i+jk

aijxiyj , aij R

.

p(x, y) = a +bx+cy p(x, y) = a + bx + cy+ dxy+ ex2 + fy2

Vh Ki P1

V1h ={u , u|Ki P1 Ki Th, u= 0 }.

i 1 (xi, yi) 0 (xj , yj)


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Aij =a(j , i) =

j(x, y).i(x, y)dxdy,

f, i=

f(x, y)i(x, y)dxdy

uh V u

Vh

V

Vh u

uh v [0, 1] L2 v

||v||L2(0,1) :=

10

|v(x)|2dx

12

,

v1 v2 ||v1 v2||L2(0,1)

H1

||v||H1(0,1) :=

||v||2L2(0,1)+ ||v||2L2(0,1)

12

=

10

|v(x)|2dx +

10

|v(x)|2dx

12

.

V =

u C1 [0, 1], u(0) =u(1) = 0

u ||u||L2(0,1) u ||u||H1(0,1) V

u ||u||L2(0,1) V


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||u uh||H1(0,1)

||u uh||H1(0,1)

u V V =u C1 [0, 1], u(0) =u(1) = 0

Vh V

uh Vh C1 Vh

||u uh||H1(0,1) C (u, Vh)

(u, Vh) = infvhVh||u vh||H1(0,1) u Vh

Vh

(u, Vh)

Vh

h h

Vh P1 Vh = N N =1/h 1 Vh V

u

uh1

uh2

uh3

uV

Vh1

Vh2

Vh3

Vh uh u

Pk limh0

(u, Vh) =

0 0

limh0

||u uh||H1(0,1) = 0.

limh0

(u, Vh) = 0 vh Vh ||u

vh||H1(0,1) 0 h 0

P1 Ihu(x) V1h u

xi


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x0 = a xi xi+1 xN+1 = b

uIhu

Ihu u (0,...,N)

Ihu(x) =Ni=1

u(xi)i(x).

v V Ihv

v V Ihv V1h

limh0

||v Ihv||H1(0,1)= 0.

v C2 K >0 h

||v Ihv||H1(0,1) Kh||v||L2(0,1).

uh u

||u uh||H1(0,1) C infvhVh

||u vh||H1(0,1) C||u Ihu||H1(0,1) 0,

u C2

||u uh||H1(0,1) C Kh||u||L2(0,1) 0.

h 1

H1

||u||L2(0,1) ||u||H1(0,1) L2

L2

||u uh||L2(0,1) C h2||u||L2(0,1),

C h


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u

Ihu u

u(xi) Ihu Ihu

limh0

(u, Vh) = 0


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CoursUFE_KhaledSaleh