e methodos ton peperasmenon stoikheion g - marianna pepona

7/23/2019 E Methodos Ton Peperasmenon Stoikheion g - Marianna Pepona

1/93

Stokes


2/93


3/93

Sobolev

Holder

Sobolev

Sobolev

Sobolev

Sobolev

Poincare

Friedrich

H1

Lax-Milgram

Neumann

Ritz-Galerkin


4/93

Bramble-Hilbert

Clement

Stokes

Cea

Saddle Point

Inf-Sup

Fortin

Stokes

Inf-Sup


5/93

Sobolev

Sobolev

Ritz-Galerkin

3.4

L2

Sobolev

H1

saddle point

Stokes

saddle point

Freefem

Stokes


6/93


7/93

Sobolev

Holder

U Rn

U

0< 1

u : U R

Lipschitz

x, yU C

|u(x) u(y)| C|x y|.

u : U R

Holder

0< 1 x, yU C

|u(x) u(y)| C|x y|.

i

u : U R

uC(U):=supxU|u(x)|, U U

ii

Holder

u: U R

[u]C0,(U) := supx,yU,x=y

|u(x) u(y)||x y|

,

Holder

u

uC0,(U):=uC(U)+ [u]C0,(U).


8/93

SOBOLEV

HolderCk,(U)

uCk(U)

uCk,(U):=||k

DuC(U)+||=k

[Du]C0,(U)

Ck,(U)

u

k

k

Holder

Ck,(U)

Banach

Sobolev

Holder

Sobolev Wk,p(U)

U Rn Cc (U)

:U R

U

(support)

Cc (U)

u, vL1loc(U) L1loc(U)

{u: U

R

|v

Lp(V)

V

U

}

v

u

Du= v

U

uD dx= (1)||

U

v dx

Cc (U)

u


9/93

SOBOLEV

v, v

L1loc(U)

U

uD dx= (1)||

U

v dx= (1)||

U

v dx

Cc (U)

U

(v v) dx= 0

Cc (U) v v= 0

Green

1p k

Sobolev Wk,p(U)

u: U R

Du

|| k Lp(U) p= 2

Hk(U) =Wk,2(U) (k= 0, 1, . . . ).

H

Hk(U)

Hilbert

H0(U) = L2(U)

L2(U)

{u: U R| uL2(U)


10/93

SOBOLEV

u

Wk,p(U)

uWk,p(U):=

(

||k

U|Du|p dx)1/p (1p


11/93

SOBOLEV

u, vWk,p(U) || k

(i)DuWk||,p(U)

D(Du) =D(Du) =D+u

,

|| + || k (ii)

, R

u+v Wk,p(U)

D(u+v) = Du+

Dv, || k (iii)

V

U

uWk,p(V)

(iv)

Cc (U) uWk,p(U)

Leibniz

D(u) =

DDu,

= !

!()!

k = 1, . . .

1 p

SobolevWk,p(U)

Banach

Sobo-

lev

Sobolev

Sobolev

k

1 p

U={xU|dist(x,U)> } U

C(Rn)

(x) := Cexp 1|x|21 , |x|< 1,

0 , |x| 1,

C >0

Rn

dx= 1

>0

(x) :=

1n

( x

)

f :U R

f := f U

f(x) =

U

(x y)f(y)dy=

B(0,)

(y)f(x y)dy,

xU


12/93

SOBOLEV

Sobolev

uWk,p(U)

1

p 0, (ii) u u Wk,ploc(U) 0.

Wk,p(U) Wk,ploc(U)

Sobolev

U

U

u Wk,p(U) 1 p


13/93

SOBOLEV

1

p

U

U C1

V

U V

E:W1,p(U)W1,p(Rn)

uW1,p(U) (i) Eu = u

U

(ii) Eu

V

(iii)EuW1,p(Rn) CuW1,p(U) C

p U

V

Eu

u

R

n

U

u W1,p(U)

C1

u C(U)

u

U

uW1,p(U)

U

1 p


14/93

SOBOLEV

Sobolev

Sobolev

Sobolev W1,p(U)

u

SobolevW1,p(U)

u

u

p

(i)

1p < n (ii) p= n (iii) n < p

1p < n

uLq(Rn)CDuLp(Rn),

C > 0

1 q 0

u(x) :=u(x), x Rn.

u

uLq(Rn)CDuLp(Rn).

Rn

|u|qdx=Rn

|u(x)|qdx= 1n

Rn

|u(y)|qdy,

Rn

|Du|pdx= pRn

|Du(x)|pdx= p

n

Rn

|Du(y)|pdy,

1

n/quLq(Rn)C

n/pDuLp(Rn),

uLq(Rn)C 1np+

nq DuLp(Rn).


15/93

SOBOLEV

1

np

+ nq

= 0

0

1 n

p+ n

q = 0

1q

= 1p 1

n

q= npnp

1p < n

Sobolev

p

p := np

n p.

q= p

1p < n

C

p

n

uLp(Rn)CDuLp(Rn),

uC1c (Rn)

Gagliardo-Nirenberg-Sobolev

SobolevW1,p(U)

W1,p0 (U) L

p

U Rn

U

R

n

U

C1

1p < n

uW1,p(U) uLp(U)

uLp(U)CuW1,p(U),

C

p, n

U

U

R

n

uW1,p0 (U) 1p np

uCk[np ]1,(U)

=

[ n

p] + 1 n

p , n

p Z,

0<


18/93

SOBOLEV

U

Rn U C1 1 p < n

W1,p(U)Lq(U),

1q < p

p > p

p

p n

W1,p(U) Lp(U)

1 p

W

1,p

0 (U)Lp

(U)

U

C1

Poincare Friedrich

Poincare

U

B(x, r)

x

r

U

R

n

U

C1

1 p C

n, p

U

u (u)ULp(U)CDuLp(U),

uW1,p(U)

u

U

(u)U =

Uudy u B(x, r)

(u)x,r =

B(x,r)udy

1p

C

n

p

u (u)x,rLp(B(x,r))C rDuLp(B(x,r)),

B(x, r)Rn

uW1,p(B0(x, r))


19/93

SOBOLEV

Friedrich

Lp

Lp

Friedrich

Sobolev

U

R

n

d

u : U R

SobolevWk,p0 (U)

uLp(U)dk||=k

DupLp(U)

1/p

,

(1, . . . , n) ||= 1+ +n

Du

||u1x1...

nxn

Sobolev

u: U R

VU

(i)

i

h

Dhiu(x) =u(x+hei) u(x)

h , i= 1, . . . , n ,

xV

h R, 0


20/93

SOBOLEV

SobolevW1,

Lipschitz

U

U

C1

u: U R

Lipschitz

uW1,(U)

U

u

W1,loc (U) u Lipschitz

U 1 p


21/93

H1

Rademacher u Lipschitz

U

u

U

H1

U

U

H1(U) H

1(U)

u

H1(U)

H1(U) ={u H1(U)|u = 0 } = U H1U(U)

H10(U) H10 (U) ={uH1(U)|u= 0 U}

X

f : X R

X

X

H1(U)

H10(U)

f

H1(U)

f

H10(U)

fH1(U)

fH1(U) = sup{< f,u >| uH10 (U), uH10 (U)1}.

(i)

f H1(U)

f0, f1, . . . , f n

L2(U)

< f, v >= U f0v+

n

i=1 fivxidx,vH10(U).

(ii) fH1(U) =inf{(

U

ni=0 |fi|2dx)1/2| f

f0, . . . , f n L2(U)}


22/93

SOBOLEV


23/93

u= u(x)

x= (x1, . . . , xn)

U

R

n, n

2

F(x, u , ux1, ux2, . . . , ux1x2, . . . ) = 0 ,

U

U

R

n

U

U

u

F

u

u


24/93

F

u F u

u

F

n

n

i,j=1ij(x)uxixj +

n

i=1bi(x)uxi+ c(x)u= f(x),

x= (x1, . . . , xn)

ij bi c x

uxixj =uxjxi

(i)

x

A(x) := (ij(x))

(ii)

x

A(x)

n

1

(iii) x A(x)

(A(x), b(x))

n

(iv)

Lu= f,

L

Poisson

u= f.

utt+Lu= f,

ut+Lu= f,


25/93

L

u= 1

c2utt,

u= 1

ut.

L

Lu=0u+c0u.

Lu= f uU,

u= 0 uU,

U

R

n

u: U R

u= u(x) f :U R

L

Lu=n

i,j=1

(ij(x)uxi)xj +n

i=1

bi(x)uxi+ c(x)u


26/93

Lu= ni,j=1

ij(x)uxixj +n

i=1

bi(x)uxi+ c(x)u,

ij, bi, c

i, j = 1, . . . , n

Lu= f

L

L

u= 0

U

Dirichlet

ij

i, j =

1, . . . , n

C1

L

L

Lu=n

i,j=1

ij(x)uxixj +n

i=1

bi(x)uxi+ c(x)u,

bi :=bi nj=1 ijxj i = 1, . . . , n

Dirichlet

Lu= f uU,

u= g uU,

u0 g

U

Lu0

Lw= f1 wU,

w= 0 wU,

w:= u u0 f1:=f Lu0

ij =ji

i, j= 1, . . . , n

L

>0

ni,j=1

ij(x)ij||2

xU Rn

xU

A(x) = (ij (x))


27/93

u

U A := D2u =

ni,j=1

ijuxixj

u

U

(ij)

F :=ADu

F Du0,

u

b

Du= ni=1 biuxi u

U cu

u

u

L

ij, bi, c

L(U)

i, j = 1, . . . , n

fL2(U)

V

B( , )

V

VV

R

B(1u1+2u2, v) =1B(u1, v) +2B(u2, v),

B(u, 1v1+2v2) =1B(u, v1) +2B(u, v2),

u,v,u1, u2, v1, v2V 1, 2, 1, 2R

(i)

B(, )

u, vV

B(u, v) =B(v, u).

(ii)

B(, )

u, vV

|B(u, v)| c1uVvV,


28/93

c1

0

(iii) B(, ) V

vV

B(v, v)c2v2V,

c2>0

(i)

B[, ]

L

B[u, v] :=

U

ni,j=1

ij

uxivxj +

ni=1

bi

uxiv+cuv dx

u, vH10 (U)

(ii)

u H10 (U)

B[u, v] = (f, v)

v H10 (U) (, )

L2(U)

Lu= f0 ni=1 fixi uU,u= 0 u U,

L

fi L2(U)

i= 0, . . . , n

1.6

f=f0

ni=1 f

ixi

H1(U)

H10 (U)

u H10 (U)

B[u, v] =< f, v >

vH10 (U) < f,v >=

U

f0v+n

i=1 fivxi dx

H1(U)

H10 (U)


29/93

HilbertV

(, )V V B(, )

V

F = (f, v)

V

B

V

F

v V

|F(v)| c3vV,

c30

u V B(u, v) = F(v) B(u, v) = (f, v)

vV

u V

J(u) = minvVJ(v) J

V

J(v) = 12

B(v, v)F(v) J(v) = 1

2B(v, v) (f, v)

u V

u

wV V

u+wV

J(u+w) = 1

2B(u+w, u+w) (f, u+w)

= 1

2B(u, u) (f, u) +1

2B(w, w) (f, w) +B(u, w)

= J(u) +1

2B(w, w)

J(u),

B(w, w)c2w2V0 J(u+w)J(u) , wV

v=u+w

J(u) =minvVJ(v),


30/93

u

u V

u

vV

t R

V

u+tv V

u

J(u)J(u + tv)

t R

g : R R

g(t) =J(u+tv)

g(0)g(t) t R 0

g

g(0) = 0

g(0)

g(t) = J(u+tv) =1

2B(u+tv, u+tv) (f, u+tv)

= 1

2B(u, u) (f, u) +t[B(u, v) (f, v)] +1

2t2B(v, v)

= J(u) +t[B(u, v) (f, v)] +12

t2B(v, v)

g(0) = limt0

g(t)g(0)t

= limt0t(B(u,v)(f,v))+ 1

2t2B(u,v)

t = 0

g(0) = B(u, v)

(f, v) = 0

B(u, v) = (f, v)

u

J

v

u

u

V

B(u, v u)< f, v u >,vV.

L

C2(U) C0(U)


31/93

C2(U)

C0(U)

Dirichlet

Lu = f

Euler

Euler-Lagrange

v

w

C1

i

i

U

Green

U

viwdx=

U

wivdx+

U

vwids.

w:= ijju

U

vi(ijj u)dx=

U

ijivj udx,

v= 0

U

B(u, v) =

U

ni,j=1

ij

iujv+

ni=1

bi

iuv+cuv

dx,

(f, v) =

U

fvdx.

i, j

vC1(U) C(U) v= 0

U

B(u, v) (f, v) =U

v

n

i,j=1i(

ijju) +n

i=1biiu+cu f

dx

=

U

v(Lu f)dx= 0,

Lu= f

u

C2(U) C0(U)


32/93

2.2.1

Lu= f uU,

u= g uU,

Lw= f1 wU,w= 0 wU,

w = uu0 f1 = f Lu0 u0

C2(U)

C0(U)

H1(U)

g

U U

w H10 (U) B(w, v) = (f Lu0, v) v H10 (U)

B(u0, v) = (Lu0, v)

u H1(U)

B(u, v) = (f, v)

u u0H10 (U) vH10 (U)

u= f uU,u= 0 uU.

uH10 (U)

B(u, v) = (f, v)

vH10 (U)

B

Green

B(u, v) =

U

(u)v dx=

U

(2u)v dx

=

U

u v dx= (u, v)

F(v) = (f, v)

(f, v) =

U

fv dx.

B

F

B

F

|B(u, v)| | u| L2(U) | v| L2(U) uH10 (U)vH10 (U),


33/93

|(f, v)| fL2(U)vL2(U)cvH10 (U).

H10 B

Poincare

U

u2 dxc

U

| u|2 dx,uH10 (U),

c

B(u, u) =

U| u|2 dx 1

c+ 1

U

(u2 + | u|2) dx=

1

c+ 1u2H1(U).

Lax-Milgram

2.3.2

H

Hilbert

(, )

H

Lax-Milgram V

HilbertH

B :HH R

f

V

J

J(v) =1

2B(v, v)< f, v > 1

2c2v2fv= 1

2c2(c2vf)2f

2

2c2 f

2

2c2.


34/93

c := inf

{J(v)

|v

V

}

{vn

}n=1

limn J(vn) =infvVJ(v) =c

c2vn vm2 B(vn vm, vn vm)= 2B(vn, vn) + 2B(vm, vm) B(vn+vm, vn+vm)= 4J(vn) + 4J(vm) 8J(vm+vn

2 )

4J(vn) + 4J(vm) 8c,

V

12

(vn+vm)V

J(vn), J(vm)c vn vm 0 n, m

{vn}

Cauchy

H

u= limn vn V u

V

J

J(u) = limn J(vn) = infvVJ(v)

V

u1, u2

u1, u2, u1, u2, . . .

limn J(vn) =c

Cauchy

u1=u2

H

B

vB :=

B(v, v).

HilbertH

V =H

Lax-Milgram

B :H H R

, > 0

|B[u, v]| u v, u , vH

u2 B [u, u], uH.


35/93

f :H

R

H uH B[u, v] =< f, v >

vH

Riesz u H

uH

< u, v >= (u, v),vH.

v B[u, v] u H

H

wH B[u, v] = (w, v),vH.

A

Au= w

B[u, v] = (Au,v),

u, v

H.

u = A1w

A

A: HH

1, 2 R u1, u2 H

vH

(A(1u1+2u2), v) = B[1u1+2u2, v]

= 1B[u1, v] +2B[u2, v]

= 1(Au1, v) +2(Au2, v)

= (1Au1+2Au2, v).

A

uH

Au2 = (Au, Au) =B [u,Au]uAu.

uH

Au u

A


36/93

A

1

1

R(A)

H

u2 B [u, u] = (Au,u) Auu

u Au

A

{yj}j=1 R(A) y H

xj H Axj = yj yjykH =A(xjxk)H xjxkH {xj}j=1

Cauchy

{xj} xj xH

y =Ax

R(A)

A

yj =AxjAx = y R(A)

A

R(A) =H

R(A)H

R(A)

wH

wR(A)

w2 B[w, w] = (Aw,w) = 0

A

Riesz

< f, v >= (w, v)

v H

w

H

A

1

1

R(A) =H uH Au= w

B[u, v] = (Au,v) = (w, v) =< f,v >,vH.

uH

u1, u2H

B[u1, v] =< f, v > B[u2, v] =< f, v >

B[u1, v] = B[u2, v] B[u1, v] B[u2, v] = 0 B[u1 u2, v] = 0

vH

v =u1 u2 u1 u22 B [u1u2, u1

u2] = 0

u1

u2

= 0

u1=u2

B[, ]

, > 0

0


37/93

(i)|B[u, v]| uH10 (U)vH10 (U)

(ii) u2H10 (U)B [u, u] +u2L2(U)

u, vH10(U)

B[, ]

|B[u, v]| n

i,j=1

ijL(U)

U

|Du||Dv|dx

+n

i=1

biL(U)

U

|Du||v|dx+ cL(U)

U

|u||v|dx

uH10 (U)vH10 (U),

L

ij

(x)

x U

x U

n

i=1

|uxi|2 n

i,j=1

ij(x)uxiuxj .

U

B

U |Du|2

dx Un

i,j=1

ij

uxiuxjdx

= B[u, u]

U

ni=1

biuxiu+cu2dx

B[u, u] +n

i=1

biL(U)

U

|Du||u|dx

+ cL(U)

U

u2dx.


38/93

Cauchy

aba2 + b2

4, >0,

U

|Du||u|dx

U

|Du|2dx+ 14

U

u2dx, >0.

>0

n

i=1

biL(U)0

0


39/93

= 0

B[

,

]

Lax-Milgram

B

0

f L2(U) u H10 (U)

Lu+u= f, uU,u= 0, u U.

u

H10 (U)

Lax-Milgram

B

B[u, v] :=B [u, v] +(u, v).

|B[u, v]| = |B[u, v] +(u, v)| |B[u, v]| +|(u, v)| uH10 (U)vH10 (U)+uL2(U)vL2(U) c1uH10 (U)vH10 (U),

u2H10 (U) B[u, u] +u2L2(U)

= B[u, u] (u, u) +u2L2(U)

= B[u, u] u2L2(U)+u2L2(U)= B[u, u] + ( )u2L2(U),

c2u2H10 (U)u2H10 (U)

+ ( )u2L2(U)B[u, u],

B


40/93

v

< f, v >:= (f, v)

L2(U) H10 (U)

vH10 (U) |< f, v >|=|(f, v)| fL2(U)vL2(U)c3vH10 (U).

Lax-Milgram

uH10(U)

B[u, v] =< f, v > vH10 (U)

fi L2(U)

i= 0, 1, . . . , n

u

Lu+u= f0 ni=1 fixi , uU,

u= 0, u U.

2.2.3

B

f

Lax-Milgram

uH10 (U)

(i)

L

L

Lv:=n

i,j=1

(aijvxj)xin

i=1

bivxi+ (c n

i=1

bi,xi)v,

bi C1(U)

i= 1, . . . , n

(ii)

B :H H R B[v, u] =B [u, v],

u, vH10(U)

(iii)

v H10 (U)

Lv=f, vU,v= 0, vU,

B[v, u] = (f, u)

uH10 (U)


41/93

(i)

()

f L2(U)

u

Lu= f, uU,

u= 0, uU,

()

u= 0

Lu= 0, uU,

u= 0, uU.

(ii)

()

N H10 (U)

N H10 (U)

Lv= 0, vU,v= 0, v U.

(iii)

()

(f, v) = 0 vN

(i)

R

Lu= u+f, uU,

u= 0, uU

fL2(U)

/

(ii)

={k}k=1 k

k+

L


42/93

Lu= u, uU,

u= 0, uU,

u= 0

L

w

/

C

uL2(U)CfL2(U),

fL2(U)

uH10(U) Lu= u+f, uU,

u= 0, uU.

C

U

L

Neu-

mann

Dirichlet

Neumann

U

U

C1

f L2(U)

g L2(U)

u H1(U)

J(u) = minvH1(U)J(v) J H1(U)

J(v) :=1

2B[v, v] (f, v)U (g, v)U,


43/93

NEUMANN

C2(U)C1(U)

Lu= f, uU,n

i,k=1 iikku= g, uU.

L

Lu :=ni,k=1 i(ikku) + 0u 0(x) > 0 xU

ik i, k =1, . . . , n

:=(x)

U

Lax-

Milgram

B

B[u, v] := U n

i,k=1

ikiukv+0uv dx

< l, v >:=

U

fvdx+

U

gvds

fL2(U)

gL2(U)

vH1(U)

B[v, v]

|a|=1

U |Dv

|2dx+

v

2H0(U)=

v

2H1(U),

B

H1(U)

< l, v >

Lax-Milgram

u

B[u, v] = (f, v)U+ (g, v)U,vH1(U).


44/93

u

C2(U)C1(U) vH10 (U)

T v= 0

B[u, v] = (f, v),vH10 (U),

U

Lu= f.

Green

v H1(U)

U

vi(ikku)dx=

U

ivikkudx+

U

vikkuids.

B[u, v] (f, v)U (g, v)U=

U

v(Lu f)dx +

U

v

n

i,k=1

iikku g

ds.

v0 := iikku g

Uv20ds >0 C

1(U)

C0(U)

vC1(U)

U

v0vds >0

Neumann Helmholtz

u+0(x)u= f, uU

u

:= u= g, u U,

u


45/93

NEUMANN

Neumann

Pois-

sonu= f , uU

u

=g, u U.

Gauss

U div(w)dx= U wds

f

g

U

f dx =

U

(u)dx

=

U

div(u)dx

=

U

u ds

= U

u

ds

=

U

gds

U

f dx+

U

g ds= 0.

Lu= f, uU,u= g, u U1,

u

=h, u U2,

U := U1 U2


46/93

u

Lu = f

U

Hs

B[, ] V

V Hm(U)

m 1

B[u, v] = (f, v),vV

Hs

c = c(U, B , s)

f Hs2m(U)

u Hs(U)

uHs(U)cfHs2m(U).

U Rn

u H10 (U)

Lu= f

U

L

u

U

H2

L ij C1(U) bi, c L(U) i, j = 1, . . . , n

fL2(U)

uH1(U)

Lu= f

U

uH2loc(U) VU

u

H2(V)

CfL2(U)+ uL2(U) ,

C

V

U

L

ij, bi, c Cm+1(U) i, j = 1, . . . , n m

f Hm(U)


47/93

u

H1(U)

Lu = f U u Hm+2loc (U)

VU

uHm+2(V)CfHm(U)+ uL2(U) ,

C

m

U, V

L

ij , bi, cC(U) i, j= 1, . . . , n fC(U)

uH1(U)

Lu= f

U

uC(U)

u

U

U

ij

C1

(U)

bi

, c L

(U)

i, j = 1, . . . , n

f L2(U)

u H10 (U)

Lu= f, uU,

u= 0, uU.

U

C2

uH2(U)

uH2(U)CfL2(U)+ uL2(U)

,

C

U

L

uH10 (U)

uH2(U)CfL2(U).


48/93

ij, bi, c Cm+1(U) i, j = 1, . . . , n m

f Hm(U)

u H10 (U)

Lu= f, uU,

u= 0, uU.

U

Cm+2

u Hm+2(U)

u

Hm+2(U)

C

f

Hm(U)+

u

L2(U) ,

C

m

U

L

u

uHm+2(U)CfHm(U).

ij , bi, cC(U) i, j= 1, . . . , n fC(U)

uH10 (U)

Lu= f, uU,u= 0, uU.

U

C

uC(U)


49/93

Ritz-Galerkin

Ritz-Galerkin

J

Hm(U)

Sh h

h0

u V B[u, v] =< f, v > v V

Ritz-Galerkin

Sh V

uh Sh B[uh, v] =< f, v > vSh uh

u

{1, 2, . . . , N} Sh

N

B[uh, i] =< f, i>, i= 1, 2, . . . , N .

uh i

i= 1, . . . , N

uh =N

j=1

cjj,


50/93

Nj=1

cjB[j, i] =< f, i>, i= 1, 2, . . . , N ,

Ac= b,

A= (aij) =B [j , i] b= (bi) =< f, i> c= (cj)

Ritz-Galerkin

A

B

< f,v >

(i)

B

V

(ii) < f,v >

A

Ritz-Galerkin

zTAz >0,z RN, z= 0.

{j}Nj=1 Ritz-Galerkin uh=

Nj=1 cjj

zTAz =N

i=1zi

N

j=1aijzj

=N

i=1

zi

Nj=1

B[j , i]zj

= B[N

j=1

zjj,N

i=1

zii]

= B[uh, uh]uh2V >0,

uh= 0 z= 0


51/93

RITZ-GALERKIN

Sh

V

uhHm(U)1f.

uh

v= uh

uh2Hm(U)B [uh, uh] =< f, uh> fuhHm(U).

uhHm(U) Cea B V

VHm(U)

u uhV

infvhShu vhV,

u

uh V

ShV

B[u, v] =< f,v >,

v

V ,

B[uh, v] =< f,v >,vSh.

ShV B[u uh, v] = 0,vSh.

vh Sh v = vh uh Sh

B[u uh, vh uh] = 0 u uh2V B[u uh, u uh]

= B[u

uh, u

vh+vh

uh]

= B[u uh, u vh] +B[u uh, vh uh]= B[u uh, u vh] +B[u, vh uh] B[uh, vh uh]= B[u uh, u vh]+< f, vh uh>< f, vh uh>= B[u uh, u vh] u uhVu vhV.

uuhV u uhV u vhV


52/93

Galerkin

2.2.3

B

uhSh J(uh) =minvShJ(v)

Ritz-Galerkin

u= f, uU,u= 0, u,

U

(0, 1) (0, 1)

U

Ritz-

Galerkin

U

Sh v(x, y)

U

(i) v(x, y)

T

U

v(x, y) =ax+by+c.

(ii)

v(x, y)

U= [0, 1] [0, 1]

(iii) v(x, y)

Pj = (xj, yj) j = 1, . . . , n

U

j(x, y) Sh

Pj

1h

dimSh =n = j j = 1, . . . , n

Ritz-Galerkin

nj=1

cj

U

j i dx dy =

U

f i dx dy, i = 1, . . . , n ,


53/93

RITZ-GALERKIN

Ac= b,

A = (aij) =

Uj i dx dy b = (bi) =

U

f i dx dy c= (cj)

V V II

III I

V I V III

IV II

S SE

NW N

W EC

U

A

(aij) C

aCC =

IV III

(C)2 dx dy

= 2

I+II I+IV

2Cdx dy

= 2

I+II I+IV

(xC)

2 + (yC)2

dx dy

= 2

I+II I

(xC)2 dx dy+ 2

I+IV

(yC)2 dx dy

= 2h2

I+II I

dx dy+ 2h2

I+IV

dx dy

= 4,

aCN =

I+IV

C Ndx dy=

I+IV

yC yNdx dy

=

I+IV

(h1)h1 dx dy =1,


54/93

aC NW =

II I+IV

C NWdx dy

=

II I+IV

(xCxN W+yCyN W) dx dy = 0,

aCE=aCS=aCW =aCN=1,

aC SE=aC NW = 0,

i i={C,N,S,W,E,NW,SE}

i

A

C

0 1 01 4 10 1 0

.

U

Sh

(i)

U


55/93

U

U

U

(i)

T ={T1, T2, . . . , T M} U

U=Mi=1Ti

Ti Tj

Ti Tj

i=j

TiTj

Ti Tj Ti Tj

(ii)

T

Th

2h

(iii)

(iv) {Th}

> 0

T Th

T

T hT/,

hT T

(v) {Th}

>0

T Th

T h/

h= maxTThhT

(ii)

vSh Ti i= 1, 2, . . . , M


56/93

v

U vPt

Pt :={v(x, y) =i+jti,j0

cijxiyj},

t

t v

v

Qt

Qt :={v(x, y) =

0i,jt

cijxiyj}

(iii)

vSh

v

Ti

v

(i)

(T, , )

T

R

n

s

T

s


57/93

p

s

s

(ii)

Sobolev

H1(U)

U k1

v : U R

Hk(U)

vCk1(U)

C0(U)

C0

v

Ck

vCk(U)

(i)

(T, , )

T

i, i= 1, 2, 3,

= P1 ={v(x, y) = a+bx+cy| a,b,c R} dim = 3

=

{p(i), i= 1, 2, 3

}

(ii)

(T, , )

T

i, i= 1, 2, 3 i, i= 1, 2, 3

=P2={v(x, y) =a1+ a2x + a3y + a4xy + a5x2 + a6y2 |aiR, i= 1, . . . , 6}

dim = 6

={p(i), p(i), i = 1, 2, 3}


58/93

(iii)

(T, , )

T i, i = 1, 2, 3 i i i i = 1, 2

123 =13

(1+2+3)

= P3 ={v(x, y) =

i+j3i,j0 cijx

iyj | cij R, i , j = 0, 1, 2, 3}

dim = 10

={p(i), p(i), p(i), p(i), p(123), i= 1, 2, 3}

Mk :=Mk(T) :={vL2(U)|v|T Pk T Th},

Mk0 :=Mk H1(U),

Mk0,0 := Mk H10 (U).

(i)

(T, , )

T

i, i= 1, 2, 3, 4

={v C0(U)| v|T P2, v|Ti P1, i = 1, 2, 3, 4, T

Th} = Q1 ={v(x, y) = a1 + a2x+ a3y+a4xy| ai R, i = 1, 2, 3, 4} dim = 4

={p(i), i = 1, 2, 3, 4}


59/93

(ii)

(T, , )

T i, i= 1, 2, 3, 4 i, i= 1, 2, 3, 4

={v C0(U)| v|T P3, v|Ti P2, i = 1, 2, 3, 4 T

Th}={v(x, y) =a1+ a2x + a3y+ a4xy+ a5(x2 1)(y 1) + a6(x2 1)(y+1) + a7(x 1)(y2 1) + a8(x +1)(y2 1)|ai R, i= 1, . . . , 8} dim = 8

={p(i), p(i), i = 1, 2, 3, 4}

serendipity

serendipity

U

Sh Th U Rn

(Tref, ref, )

TjTh Fj :Tref

Tj vSh Tj

v(x) =p(F1j x), pref.

Mk0

(Tref, Pk, k)

Tref :={(, ) R2 |0, 0, 1 0}

Pk k k:={p(i), i= 1, 2, . . . , (k+1)(k+2)2 }


60/93

U

v Ihvm,hcvHt(U), mt, Th ={T1, T2, . . . , T M} U

m1 m,h

vm,h:=

TjTh

v2Hm(Tj),

Ihv

Sh v Hm(U)

vm,h=vHm(U)

Bramble-Hilbert

Bramble-Hilbert


61/93

U

R

2

Lipschitz

U

U

t2

z1, z2, . . . , z s s:=t(t+ 1)/2

U

I : Ht Pt1

t1

c= c(U, z1, . . . , z s)

u IuHt(U)c[u]Ht(U),uHt(U),

[

]Ht(U)

[u]Ht(U):=

||=t

U

|Du|2 dx.

Ht(U)

|v|:= [v]Ht(U)+s

i=1

|v(zi)|.

Ht(U)

||

|v| c1vHt(U) vHt(U)c2|v| vHt(U)

Ht(U)H2(U)C0(U)

|v(zi)| cvHt(U), i= 1, 2, . . . , s .

|v| = [v]Ht(U)+s

i=1

|v(zi)|

vH

t

(U)+cs

vH

t

(U)= (1 +cs)vHt(U).

vHt(U)c|v|,vHt(U)

c

(vk)Ht(U)

vkHt(U) = 1,|vk| 1k

, k= 1, 2, . . . .


62/93

1.5.1

(vk) Ht1(U)

(vk) Ht1(U)

(vk) Cauchy H

t1(U)

[vk]Ht(U)0

vkvl2Ht(U) vkvl2Ht1(U) + ([vk]Ht(U)+ [vl]Ht(U))2 (vk) Cauchy H

t(U)

Ht(U)

(vk) v Ht(U)

vHt(U)= 1,|v|= 0,

[v]Ht(U) = 0

v

Pt1 v(zi) = 0

i= 1, 2, . . . , s v

u IuHt(U) c|u Iu|

= c

[u Iu]Ht(U)+

si=1

|(u Iu)(zi)|

= c[u Iu]Ht(U)= c[u]Ht(U),

Iu= u

D

Iu= 0 ||= t

Bramble-

Hilbert

Bramble-Hilbert U R2 Lip-schitz

t2

L : Ht(U) Y

Y

Pt1kerL kerL L

c = c(U)L 0 L := sup{Lv | v = 1}

Lv c[v]Ht(U),vHt(U).

I : Ht(U) Pt1

IvkerL

Lv=L(v Iv) L v IvHt(U)cL[v]Ht(U),

c


63/93

Th U

Sh = M

t10 (Th) t 2

Th

Ih:Ht(U)Sh

Th

U

t 2

c= c(U,,t)

u Ihum,hchtm[u]Ht(U),uHt(U), 0mt,

Ih t 1

U

T1

Th := hT1 ={

x= hy|

y

T1}

h1

0mt

u IuHm(Th)chtm[u]Ht(Th),

Iu

Pt1

u

c

uHt(Th) v Ht(T1)

v(y) =u(hy).

|| t

v = h||

u

R

2

h2

[v]2Hl(T1) =||=l

T1

|Dv|2 dy =||=l

Th

h2l|Du|2h2 dx= h2l2[u]2Hl(Th).

h1

u2Hm(Th)=lm

[u]2Hl(Th)=lm

h2l+2[v]2Hl(T1)h2m+2v2Hm(T1).


64/93

u

u

Iu

u IuHm(Th) hm+1v IvHm(T1) hm+1v IvHt(T1) hm+1c[v]Ht(T1) chtm[u]Ht(Th),

mt

U

U

U

1 1

F : UU

Fx= x0+Bx,

B v Hm(U) v :=v F Hm(U)

c= c(U , m)

[v]Hm(U)cBm|detB|1/2[v]Hm(U).

F : T1 T2 x Bx+x0 1 1

i

Ti ri

Ti x R2 x 21

y1, z1

T1 x = y1

z1 F(y1), F(z1)

T2

Bx 2r2

B r21

.

T1 T2

B1 r12

B B1 r1r212

.


65/93

Tj Th U

u IhuHm(Tj)chtm[u]Ht(Tj),uHt(Tj).

r= 21/2

= (2+

2)1 2/7

11

F : Tref T T = Tj Th

[u

Ihu]Hm(T)

c

B

m

|detB

|1/2[u

Ihu]Hm(Tref)

cBm|detB|1/2 c[u]Ht(Tref) cBm|detB|1/2 cBt|detB|1/2[u]Ht(T) c B B1m Btm[u]Ht(T).

r/

B

B1 (2 + 2)

B h/4h

[u Ihu]Hl(T)chtl[u]Ht(T).

l 0 m

serendipity

Th

U

c= c(U, )

u

Ihu

Hm(U)

ch2m[u]H2(U),

u

H2(U),

Ihu u

Th

U

serendipity

c = c(U, )

u IhuHm(U)chtm[u]Ht(U),uHt(U), m= 0, 1, t= 2, 3.


66/93

Clement

(Sh)

k

c = c(,k,t)

0mt

vht,hchmtvhm,h,vhSh.

X

Y

(Sh) X

infvhShu vhYchuX,uX,

vhXchvhY,vhSh.

=

Clement

Ih

H2

Clement

H1

Clement

Th

U

xj

j :=xj :={T Th| xjT}

vj M10 vj (xk) = jk

T :={j| xjT}

T

T

Ihv Clement

Ihv:=

j

(Qjv)vjM10 ,


67/93

Qjv=

0, xjD,Qj v, xj /D,

D U Qj : L2(j) P0 L2

Clement

Th

U

Ih :H

1(U)M10

v IhvHm(T)ch1mT vH1(T),vH1(U), m= 0, 1, T Th,

v IhvH0(e)ch1/2T vH1(T),vH1(U), eT, T Th.

u uh chp,

u

uh Sh

p

Sobolev

U

U

Th

U

uhSh=Mk0 k1

u uhH1 chuH2 chfH0.


68/93

U

Dirichlet H2

uH2c1fH0 .

vhSh

u vhH1(U)=u vh1,hc2huH2(U).

Cea

u uhH1 c3c4

infvhShu vhH1,

P1

Th

U

uh

Sh Sh

u uhH1chfH0.

L2

Aubin-Nitsche H Hilbert || (, )

V

Hilbert

V H

ShV

|u uh| Cu uhsupgH

1

|g| infvShg v

,

gH

gV

B(w, g) = (g, w),wV.


69/93

w

Hilbert

|w|= supgH(g, w)|g| ,

supremum

g= 0 u

uh

B(u, v) =< f, v >,vV ,

B(uh, v) =< f, v >,

v

Sh.

v Sh B(u uh, v) = 0 w := u uh

(g, u uh) = B(u uh, g)= B(u uh, g v) c1u uh g v,

B

(g, u

uh)

c1

u

uh

infvSh

g

v

.

|u uh| = supgH(g, u uh)|g| c1u uhsupgH

infvSh

g v|g|

.

u H1(U)

u uhH0cC hu uhH1 .

f L2(U) u H2(U)

u uhH0cC2h2fH0,

c

C


71/93

Stokes

Cea

Hm

SobolevHm(U)

uh

u

Cea

V Hm(U) Sh V

B[u, v] =< f,v >,vV

uh

Sh

vSh Bh[uh, v] =< fh, v > .

Bh

V

V

Strang

c

h


72/93

STOKES

u uhV c

infvhSh

u vhV +supwhSh

|B[vh, wh] Bh[vh, wh]|whV

+ c supwhSh

< f, wh >< fh, wh>

whV

.

vhSh wh = uh vh

c2uh vh2V Bh[uh vh, uh vh]= Bh[uh

vh, wh]

= B[u vh, wh] + (B[vh, wh] Bh[vh, wh]) + (Bh[uh, wh] B[u, wh])= B[u vh, wh] + (B[vh, wh] Bh[vh, wh]) (< f, wh>< fh, wh >) .

uhvhV =whV

B

uhvhVC

u vhV +|B[vh, wh] Bh[vh, wh]|whV +|< fh, wh >< f, wh>|

whV

.

vh Sh

u uhV u vhV + uh vhV.

Sh V

Hm

Sh

h

Bh

V

Sh

Bh

|Bh[u, v]| c1uhvh,uV +Sh, vSh,

Bh[v, v]c2v2h,vSh,

c1 c2 h

Strang

Berger, Scott, Strang

c

h

u uhhc

infvhShu vhh+supwhSh|Bh[u, wh]< fh, wh>|

whh

.


73/93

vhSh

c2uh vh2h Bh[uh vh, uh vh]= Bh[u vh, uh vh] + (< fh, uh vh>Bh[u, uh vh]) .

uhvhh

uh vh = wh

uh

vh

h

c12 c1

u

vh

h+

|Bh[u, wh]< fh, wh>|

whh .

u uhh u vhh+ uh vhh.

Aubin-Nitsche

HilbertV

H

Aubin-Nitsche

ShH

Bh V

Sh

B

V

uh

|u uh| supgH 1|g|(c1u uhhg g,hh+ |Bh[u uh, g] (u uh, g)|+ |Bh[u, g g,h]< f, g g,h >|),

gV g,hSh

Bh[w, ] = (w, g) gH

uh g g,h gH

(u uh, g) = Bh[u, g] Bh[uh, g,h]= Bh[u uh, g g,h] +Bh[uh, g g,h] +Bh[u uh, g,h]= Bh[u uh, g g,h] (Bh[u uh, g] (u uh, g)) (Bh[u, g g,h]< f, g g,h >) .

Bh


74/93

STOKES

Sobolev

m 1

uL2(U)

um,U :=supvHm0 (U)(u, v)

vHm(U) .

Hm(U) L2(U) m,U

Sobolev

Hm0 (U)

Hm(U)

H2(U)H1(U)L2(U)H10 (U)H20 (U). . .

u2,U u1,U uH0(U) uH1(U) uH2(U)

B

Hm

0

uHmc12 fm.

(u, v) umvHm

v= u

B[u, v] = (f, v)

c2u2HmB [u, u] = (f, u) fmuHm.

uHm

Dirichlet

Hm

X

Y

Banach

X

Y

L : X Y

y Y

xly(x) :=< y, Lx >


75/93

X

L : Y X y ly , < Ly, x >:=< y, Lx >

L

V

X

V

V0 :={lX | < l,v >= 0vV}.

V0

V

V

V

:={xX| (x, v) = 0vV}.

L

X

Y

Banach

L : X Y

(i)

L(X)

L

Y

(ii) L(X) = (kerL)0

U

V

Hilbert

B :

UV R L: UV

< Lu,v >:=B [u, v]vV.

f V

uU vV

B[u, v] =< f,v > .

u= L1f

U

V

L

L

L1


76/93

STOKES

U

V

Hilbert

L : U V

B : U V R

(i)

C0

|B[u, v]| CuUvV. (ii)

inf-sup

>0

supvVB[u, v]

v

V

uU,uU.

(iii)

vV

uU

B[u, v]= 0.

(i)

(ii)

L: U {vV | B[u, v] = 0uU}0 V

LvVuU,uU.

infuUsupvVB[u, v]

uUvV >0.

L: UV

L

L

Lu1=Lu2

u1 =u2 Lu1 = Lu2 L

B[u1, v] = B[u2, v] v V supvVB[u1

u2, v] = 0 u1u2= 0 u1u2 = 0

L

f L(U) u= L1f

uUsupvV B[u, v]vV =supvV< f, v >

vV =f.


77/93

L1

L

L

L1

L(U)

L

(iii)

Lax-Milgram

B

V

inf-sup

supvV B[u, v]vV B[u, u]uU uU.

Galerkin

Uh U Vh V

f V

uh Uh

vVh B[uh, v] =< f,v > .

B :UV R

UhU VhV

U

V

Uh Vh

u uh

1 +C

infwhUhu wh.

Uh Vh Babuska

inf-sup

Uh Vh

B[u uh, v] = 0vVh.

wh Uh

B[uh wh, v] =B [u wh, v]vVh.

< l , v >:= B[uwh, v] l Cuwh

Lh :UhVh


78/93

STOKES

B[uh

wh,

]

(Lh)

1

1/

uh wh 1l 1Cu wh.

u uh u wh + uh wh.

Saddle Point

X

M

Hilbert

B :X X R, b: X M R

f X

gM

u

X

J(u) =1

2B[u, u]< f,u >

b[u, ] =< g, >,M

M

J

Lagrange

L(u, ) :=J(u) + (b[u, ]< g, >)

L(, ) J L(u, )

u

saddle point (u, ) XM

vX M

B[u, v] +b[v, ] =< f, v >,b[u, ] =< g, > .

(u, )

saddle point

saddle point

L(u, ) L(u, ) L(v, )(v, )X M.


79/93

SADDLE POINT

Inf-Sup

L: X MX M(u, )(f, g).

A: XX, < Au, v >=B [u, v] vX

B

C : X

M, < Cu, >=b[u, ]

M

C : M X, < C, v >=b[v, ]

vX

b

Au+C= f,Cu= g.

V(g) :={vX| b[v, ] =< g, >M},V :={vX| b[v, ] = 0M}.

b

V

X

(i)

>0

infMsupvXb[v, ]

v.

(ii)

C : V M

Cv v vV. (iii)

C : M V0 X

C M.


80/93

STOKES

(i)

(iii)

(iii)

v V

gV0 w(v, w) C

M

w

b[w, ] = (v, w).

g

g=v

v =g =C

w = v

supMb[v, ]

b[v, ]

=(v, v)

v.

C :V M

C

(ii)

M

= supgM < g, >

g

=supvV

< Cv, >

Cv

= supvV

b[v, ]

CvsupvVb[v, ]

v.

(i)

Brezzi saddle

point

L: X MX M

(i)

B

V

(ii)

b

inf-sup

(ii)

Brezzi

saddle

point

Xh X Mh


81/93

SADDLE POINT

M

(uh, h)

Xh

Mh

vXh Mh B[uh, v] +b[v, h] =< f, v >,

b[uh, ] =< g, > .

Vh :={vXh| b[v, ] = 0Mh}.

Xh

Mh

Babuska-Brezzi

>0

>0

h

(i)

B

Vh

>0

(ii)

hMh

supvXhb[v, h]

v h.

(ii)

Brezzi

Ladyshenskaja-

Babuska-Brezzi

LBB

Xh Mh Babuska-Brezzi

u uh + h c {infvhXhu vh +infhMh h} .

XhX MhM

(C)

Vh V vh Xh

b[vh, h] = 0 hMh b[vh, ] = 0 M

(C)

u uh c infvhXhu vh.


82/93

STOKES

vh

Vh(g) v

Vh

B[uh vh, v] = B[uh, v] B[u, v] +B[u vh, v]= b[v, h] +B[u vh, v] Cu vh v,

b[v, h] (C) v :=

uhvh uhvh2 1Cuhvh u vh uhvh u uh u vh + uh vh

Fortin

Fortin

inf-sup

Fortin

b : X M R inf-sup

Xh Mh h:XXh hMh

b[v hv, h] = 0. h c c h

Xh Mh inf-sup

hMh

h supvXb[v, h]v =supvXb[hv, h]

v

c supvX

b[hv, h]

hv =c supvhXh

b[vh, h]

vh ,

hvXh

Xh Mh

inf-sup

h: XXh

Fortin


83/93

STOKES

b

Xh

Xh Mh inf-sup

c

h

uV(g)

infvhVh(g)u vh c infwhXhu wh.

Fortin

wh

Xh hwh = wh u

V(g)

hu

Vh(g)

uhu=uwhh(uwh) uwh+h(uwh) (1+c)uwh.

Stokes

Stokes

u+grad p= f u

U,

div u= g uU,u= u0 u U,

u : U Rn n = 2 3

p : U R

f

g, u0

= 1

U R2

div u = 0

u0= 0 Stokes

u+grad p= f uU,div u= 0 uU,

u= 0 uU.

Gauss

U

div u dx=

U

u ds=

U

u0 ds.


84/93

STOKES

p

U

p dx= 0.

u

C2(U) C0(U)2 p C1(U) u p

Stokes

Stokes

saddle point

(u, p) XM

v X

qM B[u, v] +b[v, p] = (f, v),

b[u, q] = 0,

X=H10 (U)

2

M=L20(U) :={qL2(U)|

Uq dx= 0}

B[u, v] = Ugrad u: grad v dx,grad u: grad v ni,j=1 uixj vixj ,

b[v, q] = U

div v q dx.

vH10 qH1 Green

b[v, q] =

U

div v q dx

=

U

v grad q dx

U

v q ds

=

Uv grad q dx.

div

grad

b[v, q]

q

M

L2(U)/R

L2


85/93

STOKES

(u, p)

u C2(U) C0(U)2

pC1(U)

saddle point

(u, p)

saddle

point

:= div uL2

= q0+c q0M c

u

H10 v = u q = 1

Udiv u dx = 0

b[u, q] = 0

U

(div u)2 dx =

U

(div u) (div u) dx=

U

(div u) dx

=

U

(div u) (q0+c) dx=

U

(div u) q0 dx+c

U

div u dx

= b[u, q0] +c

U

div u dx= 0.

div u= 0

(grad u, grad v) = (f grad p,v)vH10(U)2.

u C2(U) C0(U)2

u

u= f grad p uU,u= 0 u U,

Inf-Sup

Stokes

(i)

(ii)

Brezzi

V :={vX| (div v,q) = 0qL2(U)}.


86/93

STOKES

B[u, u]1/2 =

grad u

H0(U) = [u]H1(U)

B H10 (i)

(ii)

U

Lipschitz

(i)

grad: L2(U)H1(U)2

H1(U)2

(ii)

c= c(U)

pH0(U)cgrad pH1(U)+ pH1(U)pL2(U),

pH0(U)cgrad pH1(U)pL20(U).

Stokes

Brezzi

pL20(U)

grad pH1(U)c1pH0(U).

vH10 (U)2 vH1(U) = 1

(v, grad p) 12vH1(U)grad pH1(U) 1

2cpH0(U).

b[v, p]vH1(U) = (v, grad p)

1

2cpH0(U),

Brezzi


87/93

e methodos ton peperasmenon stoikheion g - marianna pepona

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