314 Difference Schemes with Constant CoefficientsThis type of situation is covered by the following assertion.If scheme (II) is stable with respect to the right-hand side and
approximates problem (I), then it converges and the order of
accuracy coincides with the order of approximation.Upon substituting the esti1nates for the approximation error obtained
in Section 3 into (34) we find that(} = ~ , u E Cj ,
(35) II Yj - uj II= O(h^4 + r^2 ),
{O(h^2 +r^2 ),
(} = (}* ' 'U E Cf '
O(h^2 +r), (}-/:-~, (}-1-(J.,So far we have investigated stability and convergence in mean, that is, in
the grid nonn of the space L 2 (wh). l\IIeanwhile, tnany situations exist in
which a uniform estimate, that is, an estimate for the error of a solution
will be of practical significance with regard to the norm- Stability and convergence in the space C. In this section we expound
certain devices for obtaining uniform esti1nates for the problem (16) solu-
tion:- the maximum principle;
- the energy method which makes it possible to establish stability
in the space C with respect to the right-hand side on account of
embedding theore1ns; - the representation of a solution in integral form in terms of the grid
function of the impulse point source (Green's function).
The tnaximum principle and, in particular, Theorem 3 in Chapter 1,
Section 2 will be quite applicable once we rearrange problem (II) supplied
by homogeneous boundary conditions ( sche1ne (16)) with obvious modifi-
cations and minor changes. The traditional tool for carrying out this work
is connected with
(}TA f) - f) = -y - (1 - (}) T Ay - T zp = -F'
(36)
Yo = o, YN = 0'