Difference schemes as operator equations. General formulations 131of approximation, that is, approximation and col'l'ectness (stability) of a
scheme imply its convergence.
Until now we spoke about convergence of a scheme and approximation
on a fixed element u of the space 5(^1 ). However, if u belongs to the domain
of a linear operator A from 5(l) into 8(^2 ), then A 1t = f, f E 8(^2 ). Hence,
u can be adopted as a solution to the equation(34) Au= f, u E 5(l),
and, therefore, one can speak about the approximation of this equation by
a difference scheme. The only reason we did not introduce equation (34) for
it is that no restriction on the operator A was imposed in the definitions.
Everywhere we have dealt only with an element u E 5(l).
However, if u is a solution to some equation like (34), then one can
speak, as usual, about the approximation of equation (34) by scheme (21) on
a solution of equation (34), about the convergence to a solution of equation
(34), etc.
Once started with the notion of approxi1nation to a.n element f from
5(^2 ) by a set {'Ph} from { 8~^21 }, we can speak about the approximation of
f by the elements 'Ph as well as about the approximation of an operator A
by Ah:
1) 'Ph approximates f with order h if(35)- an operator Ah approximates an operator A with order n if for any
u E 5(l)
(36) II Ah uh -P~2
l(Au)ll(h) = llAh(P~1
lu)-P~2
l(Au)ll( 2 h)= o( I h In).
Obviously, if conditions (35) and (36) are satisfied, then scheme (21)
is of the nth order of approximation on the solution u to equation (34).
With the relation Pf^2 l( f - Au) = 0 in view,
1/J"( u) ='Ph - Ah uh