Difference approximation of elementary differential operators 53Example 4. A grid in a two-din1ensional dom.ain. Our final example
is connected with a con1plex domain G in the plane x = ( x 1 , x 2 ) of rather
complicated configuration with the boundary r. One way of proceeding is to
subdivide into sets of rectangles by equally spaced grid lines parallel to Ox 2
defined by x~ii) = i 1 h 1 , i 1 = 0, ±1, ±2, ... , h 1 > 0, and equally spaced grid
lines parallel to Ox 1 defined by x~i^2 ) = i 2 h 2 , i 2 = 0,±1,±2, ... ,h 2 > 0,
as shown in Fig. 3. As a final result we obtain the grid with the nodes
(i 1 h 1 ,i 2 h 2 ), iu i 2 = 0,±1,±2, ... in the plane (x 1 ,:i; 2 ). It see1ns clear
that the lattice so constructed is equidistant in each of the directions Ox 1
and Ox 2 • We are only interested in the representative grid nodes from the
domain G = G + r, including the boundary r. The nodes (i 1 h 1 , i 2 h 2 )
inside Gare called inner nodes and the notation wh is used for the whole
set of such nodal points. The points of the intersection of the straight lines
x(ii). h cl (i^2 ). h..^0 ±1 ±2^0. h h b cl
1 = z^1 1 an· x 2 = z^2 2 , z^1 , z^2 = , , ,... , wit t e oun · ary
rare known as boundary nodes. The set of all boundary nodes is denoted
by 'Yh. In Fig. 3 the boundary and inner nodes are quoted with the marks
- and o, respectively.
As can readily be observed, there are boundary nodes with the dis-
tance to the nearest inner nodes smaller than h 1 or h 2. In spite of the
obvious fact that the grid in the plane is equidistant in J; 1 and x 2 both, the
grid w h = w h + 'Yh in the domain c; is non-equidistant near the boundary.
This case will be the subject of special investigations in Chapter 4.
The approach we have described above depends for its success on
replacing the domain G of the argument x by the grid wh, that is, by a
finite-dimensional set of points x; belonging to the domain G. The grid
functions y( x;) will be quite applicable in place of the functions u( x) of the
continuous argument x E G with xi being a node of the grid wh = {xd.
Also, we may attempt the grid function y(xi) in vector form. This is clue
to the fact that by enumerating the nodes in some order x 1 , x 2 , ••• x N
the values of the grid function at those nodes arrange themselves as the
components of the vector
y = (Y1, · · · , Y;, · · · YN) ·
If the domain G of fanning the grid is finite, then so is the dimension
N of the vector Y. In the case of an infinite domain G the grid consists of
an infinite nmnber of nodes and thereby Y becomes an infinite-di1nensional
vector. Within the framework of this book the sets of grids wh depending
on the step h as on the paran1eter will be given special investigation, so
that the relevant grid functions Y1i ( x) will depend on the parameter h or on