Vector Curl and Gradient, Lecture Notes - Mathematics - 4, Study notes of Mathematics

Download Vector Curl and Gradient, Lecture Notes - Mathematics - 4 and more Study notes Mathematics in PDF only on Docsity!

For boundary value problem:Find

u

such that

2 u

f in Ω u

g D on

D and ∂u ∂n

g N on

N

The Galerkin finite element solution

u h

n ∑ j = u j φ j

n

n ∂ ∑ j = n

u j φ j

∈ H

1 E

is found from the solution of:

A

u = f

where

A

a i,j

, a i,j

Ω

φ j

φ i u = (u 1

u 2

u n

T

and

f = (f 1

f 2

f n

T

with

f i

Ω φ i f

∂ Ω N φ i g N

n

n ∂ ∑ j = n

u j

Ω

φ j

φ i

Minimisation Form As before we assume

D

Define the functional

F

H

1 E

R

by

F

u

Ω

u

u

Ω uf

∂ Ω N ug N

Then

u

∈ H

1 E

minimises

F

, ie.

F

u

F

v

for all v

∈ H

1 E

M

if and only if

Ω

u

v

Ω vf

∂ Ω N vg N

for all v

∈ H

1 E 0

which is the weak form we will here denote by

W

With this minimisation form, the Galerkin finite elementmethod is simply seen as :Choose an approximation subspace

S

0

⊂ H

1 E 0

and find

w

S

0

so that for a particular function

r

which satisfies

r

g D

on

D

u h

r

w

minimises

F

Note that if

g D

(homogeneous Dirichlet boundary

conditions) then

r

can be taken as

r

, so simply need to

minimise

F

u h

over

u h

S

0

⊂ H

1 E 0

If

g D

(homogeneous Dirichlet boundary conditions)

thenminimise

F

u h

over

u h

S

0

⊂ H

1 E 0

In this context one might expect that the better is theapproximation of

u

by

u h

the smaller will be

F

u h

F

u

For larger

n

one might be able to make this approximation

better.(certainly

F

u h

F

u

since

u h

S

0

⊂ H

1 E 0 , u

∈ H

1 E 0

We consider the estimation of errors later, but firstlyconsider how the Galerkin finite element method might beimplemented on a computer.

FEM – p.8/