1549301742-The_Theory_of_Difference_Schemes__Samarskii

252 Difference Schemes for Elliptic Equations

where h: is the distance between the nodes x and x(-la), or

A * -y -1 ( y( + l") -y -----y - y( -1"))

(23b) ex - h ex h* a h a for

where h* u is the distance between the nodes x and x(+^10 ).

In this context, it is worth noting that the reader can encounter the

case when x(-lc.) E /h °' and x(+la) E /h "" If this happens, it is recom-

mended to refer to the ' difference operator '

(23c) for

where h :± f ha is the distance between x and x(±^10 ).
Typical situations corresponding to the forms (23a)-(23c) for p = 2 are
shown in Fig. 13. Using one of the formulae (11) and (23) for approximating
Lexu = i:r by the difference operator, we get instead of (1) the difference
equation Ay + IP( x) = 0 for all x E wh, where A = L~=l Aex. Here the
exact value YI 'Yh = μ( x) is taken on the grid boundary ih.
Finally, we arrive at the Dirichlet difference problem of determin-
ing a grid function y( :i:) defined for i: E ::i h = w h + /1i, satisfying at the
inner nodes the equation

(24)

(25)

Ay + IP(x) = 0

A*y + IP(x) = 0

at the regular nodes,

at the irregular nodes,

and taking the assigned values at the boundary nodes x E ih:

(26) y = μ(x), x E ih.

By analogy with Section 3 we formulate conditions for the accuracy
of a scheme under the agreement that y(x) is a solution of the difference
problem (24)-(26) and u = u(x) is a solution of the original problem (1).
Substitution of y = z + u into (24)-(26) yields

Az = -4' at the regular nodes,

(27) A'z = -t/!* at the irregular nodes,

z = (^0) on lh'