The sun1marized approxiination n1ethod 605- problem (15) has a unique sufficiently s1nall solution;
- the intersection of the domain G with any straight lines Ca parallel
to the axis 0 x 0 may consist only of a finite number of intervals; - it is possible to compose in the clo111ain Ga connected grid wh with
steps ha, o· = l, 2, ... ,p (for more detail sc>e Chapter <3, Section 1).
To 111a.ke our exposition 111ore transparent, it is supposed that the
intersection of the domain G with the line Ca is a unique interval.
Thus, the set w h of all inner nodes contains the points x = ( x 1 , x 2 , ... ,
xp) E G of the intersection of the hyperplanes xa = iaha, ia = 0, ±1, ... ,
Cl' = 1, 2, ... , p, while the set ih of all boundary nodes consists of the points
of the intersection of the lines Ca, o· = 1, 2, ... , p, passing through all inner
nodes x E w h, with the boundary f.
Here we retain the san1e notations as was clone in Chapter 6, Section
1: ih a is the set of all boundary nodes in the direction xa; ih is the set of
all boundary ' nodes x E f; wh is the set of all near-boundary nodes in the
,a
direction x ~; u. w 1 i is the set of all near-boundary nodes; w 1 i,a is the set of all
irregular nodes in the direction xa; w~* is the set of all irregular nodes and
w h is the set of all regular nodes.
The difference approxi1nation of the operator Lo: at a node x is con-
structed on the pattern consisting of the three nodal points x<-^10 l, x,
x< +^10 ). In view of this, the difference operator Au ~ La is taken to be
( 16)
A y = ·y- = _^1 (y(+la) _ 2y + y(-ln))
O' :Va:Vo: h2
a
at the regular nodes and( 17) Aa 1; <. = ·1; ~-'-'O·"l1 .. ,.
1 ( yl + l Q ) - y - y - yl - 1 ") ) xl-la) E...,
f ia h a h* CY ' I h,a l- 1 (y(+la) y y y(-1^0 )) x(+la)E...,
h CY h* CY h CY l I h,cr l
at the irregular nodes, where h:, is the distance from an irregular node x
to one of the boundary nodes: .rl+^10 l or .r,(-l.,J. But it 111ay happen that
both nodes x(+l.,) and x<-^10 l, which are neighboring to x E w~ CY' belong
'
to the boundary, that is, x<±^10 ) E /h ,,. In that case
'
(18)
l ( y< + j G ) _ y y _ y< -) Q ) )
A CY y--- h h* -----h* l
o o+ a-