250 Difference Schemes for Elliptic Equationswh I ()' be the set of all near-boundary nodes with respect to Xa and wh l ()' be
the set of those near-boundary nodes, which are irregular in the direction
xa. Obviously, w~~ a C w~, a· The notation w~ stands for the set of all
near-boundary nodes (that is, near-boundary at least in one direction)
and w~ stands for the set of all irregular nodes (that is, irregular at least
in one of the directions x 1 or x 2 ). Let (:;h be the complement of w~ to wh:
wh = w~ + (:;,,. All the nodes belonging to (:;h are called strictly inner
0
nodes. The symbol wh a is in c01nmon usage for all strictly inner nodes
'
with respect to xa (that is, the nodes adjacent to the node x E (:;ha in the
'
direction Ox a are inner ones).
0
In Fig. 11 the aforementioned nodes wh are depicted by the signs o,
the nodes being irregular only with respect to x a (CY = 1, 2) - by the signs
~CY' the nodes being irregular both with respect to x 1 and with respect to
x 2 - by the signs ~ 1 2 and, finally, near-boundary nodes regular both with
'
respect to x 1 and with respect to x 2 - by the signs •.
The grid wh in view is supposed to be connected, it being understood
that any two inner nodes can be joined by a polygonal line, the parts of
which are parallel to the coordinate axes and vertices coincide with inner
nodes of the grid. Then at least one of the four nodes x(±a), CY= 1, 2, of
the five-point pattern (x(±li), x, x(±^12 l) (regular 01· irregular) falls within
the collection of inner nodes. The assumption on connectedness of the grid
entails some limitations both on the choice of spacings h 1 and h 2 and on
the shape of the domain and its position with respect to the grid w h for
fixed h 1 and h 2 •
Examples of disconnected and connected grid domains are shown in
Fig. 12.a and 12.b, respectively. The assumption on connectedness for a
domain with a nanow bridge will be satisfied if we make the step ha small
enough or refine the grid in this part of the domain. Fig. 12.b illustrates
the case where the connectedness of the grid is stipulated by the proper
choice of its step h 1 rather than by successive grid refinements.
The procedure of constructing a grid in the plane domain we have
described above can easily be generalized to the case of an arbitrary p-
dimensional domain. A grid so constructed is a result of the intersection of
hyperplanes (planes for p = 3 or straight lines for p = 2)
Xa°' ' ='la. h a) iCY = 0,±1,. .. , CY= 1,2,. .. ,p,
where ha > 0. The preceding classification of nodes remains unchanged
here.