256 Difference Schemes for Elliptic Equations
In the case corresponding to Fig. 13.c we deduce that
where nl = ~(h~- + h~+) and n2 = ~(h2 + h;).
Let w h ( G) be a grid in a p-dimensional domain and
near-boundary irregular node. Then
(33)
* _ 1 ( y( + l & ) _ y Y _ y( -la) )
Ay - -----
ex - h a h* a+ h* a-
x E wh* ' ex be a
Substituting this expression into the equation A *y = -f and regarding
formally the node x to be irregular in all directions x ex, we finally get
(34)
This is acceptable for h; = h;+ = nf3 = hf3 when x happens to be regular
along some direction xf3. But if x is regular in all directions J:o:, then
h:+ = h: =no:.= hex for all CY= 1,2, ... ,p, leading to fonnula (32).
Comparison of (32) and (34) shows that these equations can be represented
in the canonical form
(35) A(x) y(x) = B(x, ~) y(0 + F(x),
~EPatt'(x)
where Patt' ( x) is the set consisting of 2p nodes of the (2p+ 1 )-point "cross"
pattern with center at the point x except for the node x itself, that is,
~-::/- x. We call the set Patt(x)'(x) the neighborhood of the node x. The