Methods for designing difference schemes 2152) the variational difference methods (the Ritz rnethod and Bubnov-
Galerkin rnethods) and the finite element method;
3) the method of the summator identity;
4) the method of approximating a variational functional.- The integro-interpolational method (IIM). In Section 2 we have already
studied the IIM, but its possibilities and potential have not been illustrated
in full measure. Here we consider other ways of its applications by appeal
to the problem
(1)(2)(ku^1 )^1 - q(x)u = -f(x),
x -- (^0) '
vVe introduce an equidistant grid on the segment 0 < x < 1
wh ={xi= ih, i = 0, 1, ... , N, hJV = 1}.
x = 1,
Integration of equation ( 1) is accomplished over the segrnent xi < x < xi+ll
leading to
(3)
x;+1
wi+i - wi = ;· (qu - J) dx =
Xi
w = ku'.
Here, in contrast to Section 2, the flow -w is taken at the same node X;
as the unknown function u, which is sought. Therefore, the intention is to
use instead of wi+i/ 2 the approximation (wi+i + w;)/2 by accepting
(4)
where ai+i is sorne functional of k( x) on the segment X; < x < X;+i satis-
fying the relation ai = k(x;_ 112 ) + O(h^2 ). In the course of the elimination
of w; from ( 3) and ( 4) we find that
Subtracting from here the relevant expression