ABSTRACT
For an arbitrary rectangle (Figure 5.1), choose the x-y axes as shown and number the corners as shown. We now wish to approximate a function f(x,y) in this region. If the function varies linearly along an edge, it is uniquely determined along that edge by its values at the corners and continuity between elements that share those corners will be achieved. This will be the case if
f x y c c x c y c xy x y xy ck( , ) { },= + + + = 1 2 3 4 1 (5.1)
since the function is linear on an edge where either x or y is constant. The four coefficients can be expressed in terms of the four nodal values of the function, fk = f(Xk,Yk), where Xk = ±a/2 and Yk = ±b/2 are the coordinates of the kth node:
{fk} = L{ck} (5.2)
where
L =
− − +
+ − −
+ + +
− + −
1
2 2 4
1 2 2 4
1 2 2 4
1 2 2 4
a b ab
a b ab
a b ab
a b ab
.
