246 Difference Schernes for Elliptic EquationsThe nodes on its boundary (for i 1 = 0, N1 or i 2 = 0, N2) except for
four points (0, 0), (0, / 2 ), (/ 1 , 0), (/ 1 , / 2 ) are called boundary nodes and
are labeled crosses in Fig.10. They constitute the set /h = {(i 1 h 1 , i 2 h 2 )}.
The set of all inner and boundary nodes is known as the grid w h = w h + ih
in a rectangle Go.
Following established practice, at each of the inner nodes x E wh we
compose a five-point regular "cross" pattern, whose nodes x±l.,, o: = 1, 2,
belong to w h, that is, either to w h or to ih. For this reason at all inner
nodes the Laplace operator L':;.u can be replaced by the difference operatorAu = 1lx-1 1 x + 1lx• ' 2·'-' .,. 2.
In this view, it seems reasonable to approximate the right-hand side - f(x)
of equation ( 1') by the grid function -1P( x) so as to achieve the error IP( x )-
f ( x) = O(I h^2 I), f(x) E C(^2 J. Assuming the function J(x) to be continuous,
in what follows we accept IP( x) = f ( x).
As a final result problem (1') is associated with the Dirichlet difference
problem relating to the determination of a grid function y( x) defined on
the grid wh, satisfying at the inner nodes, that is, on wh the equation(19) Ay=-J(x),
and taking the assigned values on the boundary ih(20) y(x) = μ(x), x E lh.
It is worth noting here that the grid w h (Go) becomes rectangular for
h 1 f h 2. In the case where h 1 = h 2 = h it refers to a square grid. More
a detailed expression for Ay on any square grid is of the form
Let IP= 0. The equation Ay = 0 can be solved with respect to y:
The value of y at the center of the pattern is the arithmetic mean of the
values of y at the remaining four nodes of the pattern. This formula gives a
difference analog of the formula for the mean value of a harmonic function.