C ¤ B ‚ — ¤ ‚ Q¤ Y¥

t

¨

3

¢& &

(

£

Integrating the above equality over and using the divergence theorem, we obtain

£¢ £¢ £¢

h9 ¤ ¤ ‚ ¤ B c

—

9 9

3

¦ ¦ ¦

C

£¢ ¥¢

¤ ‚ 9 ¤ ˜ ˜¤ Y¥

t

—

9 ¨

3 ¢& &

2

¦

¤

The above equality can be viewed as a generalization of the standard integration by part ¤

formula in calculus. Green™s formula results from (2.28) by simply taking a vector which

Q¤

is itself a gradient of a scalar function , namely, , V V

h 9 £ ¢ ¦¢

£ ¥¢

9 ¤˜ —

‚ 3

9V

¦ ¦

V V ¤

¡ ‚ ¤˜ V

Observe that . Also the function is called the normal derivative and is

V V

denoted by

¤

¤˜ ˜ V

V

With this, we obtain Green™s formula

£¢ £¢

9 ¤˜ ¥ ¢ —

¡ Y¥

t

h 9

V

9V ¨

3 ¢& &

¦ ¦

V ¤

We now return to the initial problem (2.24-2.25). To solve this problem approximately, it

is necessary to (1) take approximations to the unknown function , and (2) translate the V

equations into a system which can be solved numerically. The options for approximating

are numerous. However, the primary requirement is that these approximations should be

V

in a (small) ¬nite dimensional space. There are also some additional desirable numerical

properties. For example, it is dif¬cult to approximate high degree polynomials numerically.

To extract systems of equations which yield the solution, it is common to use the weak

formulation of the problem. Let us de¬ne

0 6

¤¢

£ £¢

C B ‚ ‚ 9

x 9

V V

—

¦ ¦

V V 6

0

¦ ¦ ¦ ¦

`

Y

£¢

C y¡ B g9 ¡

¦

Y

‚

An immediate property of the functional is that it is bilinear. That means that it is linear

B‚

with respect to and , namely, V

6 0 C 6 B ‚6 — C 0 B ‚0 C 6 6 — 0 0

V V V V

¦ ¦ ¦ ¦ ¦ ¦

6 0 C 6 B ‚ 6 — C 0 B ‚ 0 C 6 6 — 0 0 ©‚

B

V @ @ @ V @ V @ @

d u„

u n £ § ©

¢ ¡£

§ ¥ © U© © § ¥ £ © ¡ §

¥ © ¡

¥

C B

6¥ £

Notice that denotes the -inner product of and in , i.e.,

V V

£¢

C x 9C B C

B BV

¦ "

¦ ¦

V

then, for functions satisfying the Dirichlet boundary conditions, which are at least twice

differentiable, Green™s formula (2.29) shows that

C ¡B C B ‚ 3

V V

The weak formulation of the initial problem (2.24-2.25) consists of selecting a subspace of

6

¨

reference of and then de¬ning the following problem:

¥

Y

t

C y¡ B C B©‚

¨ £ ¢ &¥

¨ ¨

Find such that

V V

¨

In order to understand the usual choices for the space , note that the de¬nition of the

weak problem only requires the dot products of the gradients of and and the functions V

¡ 6¥ ¨

and to be “integrable. The most general under these conditions is the space of

B0