Two-Point Boundary Value Problem

DEWeak Form

Discrete FormLinear System

0)1()0(

')''(

uu

fcubuau

1)( )(),(

such that )( Find10

10

HLua

Hu

2

h

hh

SLua

Su

)(),(

such that FindbAU 34

spaces

)(2L

The space of all square integrable funcions defined in the domain

Definition:

dxffL 22 :)(

Definition:

)(',:)( 21 LvvvH

The function and its first derivative are square integrable

Remark: Both spaces are Hilbert spaces. R2 is also a Hilbert space

reals are and ,:2 yx

y

xvvR with inner product 21212

2

1

1 ,),( yyxxvuy

x

y

x

R2 is also a Hilbert space

Triangle inequality

)(2 LTriangle inequality:

)(, 2 Lgf)()()( 222

LLLgfgf

)()()( 111

HHHwvwv

Triangle inequality: )(1 H

)(, 1 Hwv

wvwv

Triangle inequality: 2R2, Rwv

2/12

221

2/122

21

2/12

22

2

11

wwvv

wvwv

Cauchy-Schwarz inequality

)(2 LCauchy-Schwarz inequality:

)(, 2 Lgf )()( 22

,

LL

gfgf

Cauchy-Schwarz inequality: (integral form)

2/11

0

22/11

0

21

0)()()()(

dxxgdxxfdxxgxf

Example: verify CS-inequalityxexgxxf )( ,3)(

Cauchy-Schwarz inequality:2, Rwv wvwv , 2/12

221

2/122

21

2121

wwvv

wwvv

3.09572/13 2 e

Cauchy-Schwarz inequality

Is this true?:

??, gf 11

',' gfgf

Cauchy-Schwarz inequality: (integral form)

2/11

0

22/11

0

21

0)()()()(

dxxgdxxfdxxgxf

Bilinear form

Definition:

A bilinear form on V is a function : V × V → R, which is linear in each argument separately

)or ( 12 HL

Example:

1

0)(')('),( )1 dxxvxuvua

1

0)('')(''),( )2 dxxvxuvua

1

0'''),( )3 dxuvvuvua

the bilinear form a( ・ , ・ ) on V is bounded if there is a constant M such that.

Definition:

1

11 Hw,v , v wM|a(w, v)|

a(w, v). a(w, u) v) u a(w,

a(v,w), a(u,w) v,w) u a(

R , V and u,v,w

),( aThe bilinear form is said to be symmetric if

a(w, v) = a(v,w), ∀v,w ∈ V,

Definition:),( a

Example: prove that a is bounded bilinear form on

1

0)(')('),( dxxvxuvua

1H

Bilinear form

the bilinear form a( ・ , ・ ) on is coercive if there is a constant α > 0 such that.

Definition:

12

1 Hv , v|a(v, v)|

Example: prove that a is coercive on

1

0)(')('),( dxxvxuvua

1H

1H

Linear functional

A linear functional L : V → R is said to be bounded

Definition:

Vv, vc|L(v)|V

Example:

is the smallest constant cL

Remark:V

vL|v|L )(

xxf

, dxxvxfvL

)( where

)()()(0

0

1HV

Lax-Milgram lemma

Lax-Milgram lemma

L

),( a bounded coercive bilinear form on

bounded linear functional on

VV V

there exists a unique vector u ∈ V such that (2) is satisfied

V Hilbert space

Then

VvvLvua

Vu

)(),(

such that Find

L

),( a bilinear form on

linear functional on VV

V

V Hilbert space

whereConsider:

2 )or ( 12 HL

DEWeak Form

0)1()0(

')''(

uu

fcubuau

1)( )(),(

such that )( Find10

10

HLua

Hu

2

dxfL

dxcubuauua

)(

)'''(),(

Example: 2)( xxa

0)( xb

1)( xc

33)( 2 xxxf

0)1()0(

33)'')2(( 2

uu

xxuux

1

)( )(),(

such that )( Find10

10

HLua

Hu

2

1

0

33

1

0)2(

)()(

)''(),(

2 dxL

dxuuua

xx

x

Show that there exist a unique solution for (2)

Lax-Milgram lemma

Example: 2)( xxa

0)( xb

1)( xc

1

)( )(),(

such that )( Find10

10

HLua

Hu

2

Show that there exist a unique solution for (2)

Lax-Milgram lemma

solution: In order to show that there exist a unique solution for (2), we need to satisfy all the conditions of Lax-Milgram lemma

bounded )(

in coercive ),(

bounded ),(

L

a

a proof

Later we will do another proof for a symmetric a

33)( 2 xxxf

0)1()0(

33)'')2(( 2

uu

xxuux

1

0

33

1

0)2(

)()(

)''(),(

2 dxL

dxuuua

xx

x

Poincare’s inequality (HW) 1

010 Hvvv

Show that:10010

'2' Hvvvv

Example: 2)( xxa

0)( xb

1)( xc

44)( 2 xxxf

0)1()0(

44)'')2(( 2

uu

xxuux

1

)( )(),(

such that )( Find10

10

HLua

Hu

2

1

0

44

1

0)2(

)()(

)''(),(

2 dxL

dxuuua

xx

x

Show that there exist a unique solution for (3)

Lax-Milgram lemma

hh

hh

SLua

Su

)(),(

such that Find

3

Thm: A finite dimensional subspace of a Hilbert space is Hilbert

Hilbert V dim-finite V0 V Hilbert 0V

solution: In order to show that there exist a unique solution for (3), we need to satisfy all the conditions of Lax-Milgram lemma

bounded )(

in coercive ),(

bounded ),(

L

a

a

Example: 2)( xxa

0)( xb

1)( xc

44)( 2 xxxf

)( )(),(

such that )( Find10

10

HLua

Hu

2

Stability

Setting ϕ = u in (2) and using (coercive) and (Poincare), we find

Poincare’s inequality

1010

Hvvv

)(),( uLuua

1000),( ufufuf

2

1u

01

fu

continuous dependence of solutions with respect to perturbations of data

A problem that satisfies the three conditions is said to be well posed

1)existence of solutions,2)uniqueness of solutions,3)stability

Solution bounded by the data of the problem

Definintion:

Small change in the data produce small chang in the solution

Stability:

Linear System of Equations

bAU hh

hh

SLua

Su

)(),(

such that Find34

1

1

M

jjjh Uu

),( ijij aa ),( ii fb

(4) has a unique solution iff that the matrix A is invertible ( non-singular )

Remark:Under what condition that(4) has solution

Remark:

A is symmetric and positive definite

Definition:

An nxn matrix A is symmetric and positive definite if

0, 0 VRVAVV nT

Example: show that A is SPD

51

13A

Linear System of Equations

bAU hh

hh

SLua

Su

)(),(

such that Find34

1

1

M

jjjh Uu

),( ijij aa ),( ii fb

(4) has a unique solution iff that the matrix A is invertible ( non-singular )

Remark:Under what condition that(4) has solution

Remark:

A is symmetric and positive definite

Proof:

1

111 )()(,

M

iii

TM xVxvVVV

),( vva

1

1

1

1

)(,)(M

jjj

M

iii xVxVa

1

1

1

1

)(),(M

ijiji

M

j

xxaVV

1

1

1

1

M

ijiji

M

j

VaV

AVV Tcoercively 2

1),( vvvaAVV T 2

0'v

0)0(,0'0'02

0 vCvvvAVV T