Finite Element Techniques for Fluid Flow

WEIGHTED RESIDUAL

AND

VARIATIONAL METHODS

17

Hence

5c

u

=

a

i

+ b

i(

x2

-

ax

)(y

2

-

b

y) (

f

)

Evaluating (f )

at

the

centre point

x =

a/2,

y =

b/2

gives

5 fl2fe2

"<

=

32<VTP

(g)

Let

us now

solve

the

same example

but

using

a

trigonometric

series

to

approximate

u.

We assume, taking symmetry into considera-

tion, that

.

knx . Iny

u

=

LL

(X

ki

sin sin

— (h)

IT

a b

This expression satisfies

the

boundary conditions

and the

sinusoidal

functions

are

orthogonal,

i.e.

f

Jo

.

mux . ηπχ

sin

sin dx = 0 for

m

Φ n

a

a

(i)

=

- for

m

= n

2

After substituting (h) into (c)

and

taking orthogonality into account,

we have

a

rb

Ι«τ2κ2

^2/2

«·<ΓΠ-

Jo

Jo

\

a

n

2

k

2

n

2

l

2

\

. Λπ