The Dirichlet difference problem for Poisson's equation 255Remark Quite often, the Dirichlet problem is approximated by the method
based on the difference approximation at the near-boundary nodes of the
Laplace operator on an irregular pattern, with the use of formulae (14)
instead of ( 16) at the nodes x E w~. However, in smne cases the difference
operator so constructed does not possess several important properties in-
trinsic to the initial differential equation, namely, the self-adjointness and
the property of having fixed sign. For this reason iterative methods are
of little use in studying grid equations and will be excluded from further
consideration.
- The canonical form of a difference equation. VVe now consider the
(2p + 1 )-point scheme Ay = -f at a regular node
which admits an alternative form of writing
(32)
To avoid cumbersome calculations, we concentrate primarily on the
two-dimensional case. Fig. 6 demonstrates that at a regular node
Let x E w~ 1 be an irregular node. In the case corresponding to Fig. 13.a
we obtain '
1
A2 Yo = h 2 (Y2 - 2 Yo + Y4) ·
2From the equation A* y = Ar y + A 2 y = -f we find that
(2 ri^1 2 )
h 1 2 h* 1 + h^2 2