aV
7 &&
on ¥
p C 0 Q p B
6 Q #B
#
7 7
£
where is now the rectangle and its boundary. Both intervals can be ¥
—6˜ C 0˜
—
0 ¦
discretized uniformly by taking points in the direction and points in the
6
¦
directions:
— 6 ˜ 111)z }r 6 ™( 6 — 0 ˜ i1)1p #P 0 s@ 0
" 78 "… a
7
¦ ¦
where
6— Q 6 ˜ X6 0— Q 0 ˜ 0
Since the values at the boundaries are known, we number only the interior points, i.e.,
0˜
6˜
©¦© " 0 "B
6 "… 7 7
¦
the points with and . The points are labeled from
C
the bottom up, one horizontal line at a time. This labeling is called natural ordering and is
6˜ 0˜
shown in Figure 2.5 for the very simple case when and . The pattern of the ¤ ¢
matrix corresponding to the above equations appears in Figure 2.6.
d u„
u n £ § ©
¢ § ¥ © U© © § ¥ £ © ¡ §
¥ © ¡ £
£
¥
¥ ¢
D T
¡
Pattern of matrix associated with the ¬nite
¤ ¢
difference mesh of Figure 2.5.
To be more accurate, the matrix has the following block structure:
¢ ¢ 34 @8
9
3
˜ 35 6 8
3
A ¢ 5 ˜ 3 3
3 A3
˜ ¢
with 3 3
3
˜ 3
( "&" £( £ ¥ ¤(
© ! © ( &%
%
'&c

The ¬nite element method is best illustrated with the solution of a simple elliptic Partial
Differential Equation in a twodimensional space. Consider again Poisson™s equation (2.24) 6
£
with the Dirichlet boundary condition (2.25), where is a bounded open domain in
and its boundary. The Laplacean operator
¥
6 6
¡ —
6 66
¦ ¦
0
appears in many models of physical and mechanical phenomena. Equations involving the
more general elliptic operators (2.5) and (2.6) can be treated in the same way as Pois
son™s equation (2.24) and (2.25), at least from the viewpoint of the numerical solutions
techniques.
An essential ingredient for understanding the ¬nite element method is Green™s for
£
mula. The setting for this formula is an open set whose boundary consists of a closed ¤ ¡¢ ¤
and smooth curve as illustrated in Figure 2.1. A vectorvalued function , which
¥
£
is continuously differentiable in , is given. The divergence theorem in twodimensional 3
spaces states that
¤¢
£ ¦¢
¥
¤ ¤˜ ¤ Y¥
t
h 9
9 ¨
&&
¦
div ¤
” ” ”n
uu u ™
n
¢
£ ¤§ ¥ §¢
¡
¡© £ £ ¥ £ ¤ § ¨¡ ¢
§¥ © ©
6
The dot in the righthand side represents a dot product of two vectors in . In this case it is
¤ ¤˜
between the vector and the unit vector which is normal to at the point of consideration ¥
¤ ¡¢ Q¤
and oriented outward. To derive Green™s formula, consider a scalar function and a vector
function . By standard differentiation,
}¤ ‚ — ¤ C % C ¤ B ‚
3 B
¤