132 Basic Concepts of the Theory of Difference Schemesprovided conditions (35)-(36) hold. From such reasoning it seems clear that
the converse assertion, in general, fails to be true, that is, relations (32) do
not imply conditions (35) and (36).
Let us stress once again that in order to estimate the order of accuracy
of a scheme, it is necessary to estimate its order of accuracy only on a
solution of the original problem.
So far we have always preassumed that the operator Ah is linear
(scheme (21) is linear). If Ah is nonlinear (scheme (21) is nonlinear), the
preceding arguments need minor changes only in the concept of stability.
In dealing with a nonlinear scheme(21*) in r h E 8(h^2 ). 'where Yh is a solution and Yh is a solution with the right-hand side 'Ph E
B~^2 ), scheme (21 *) is said to be stable if there are positive constants h 0 > 0
and M > 0 independent of the parameter h and disregarding to the choice
of 'Ph and 'Ph such that for I h I < h 0 the inequality(22*)is satisfied for any 'Ph, 'Ph E B~^2 ). We note in passing that for a linear
scheme with 'Ph = 0 and Yh = 0 this implies (22). All of the above defi-
nitions of approximation and convergence remain valid. Theorem 1 is also
true. However, its proof follows another reasoning: instead of (30) it is
more cinvenient to write downwhere 1/Jh is the approximation error (residual) on the element u E B(^1 ).
Denote by Yh a solution of equation (21 ), Yh = uh and make use of the
stability condition (22). We obtain in this direction estimate (33)
thereby completing the proof of Theorem 1 for a nonlinear scheme as well.- Some a priori estimates. We now consider several simplest a priori
estimates for a solution to equation (21), the form of which depends on the
subsidiary information on the operator of a scheme. These estimates are
typical for difference elliptic problems.
For simplicity of writing we will omit the subscript h if this does not
cause an ambiguity. The equation we must solve is of the form
(37) Ay=r.p,