1549301742-The_Theory_of_Difference_Schemes__Samarskii

Mathematical apparatus in the theory of difference schemes 115

scheme (51). We have shown in Section 1.3 that the local approximation

error 1jJ = u_. xx + f of scheme (51) is a quantity of order O(h;) and

(52) 111/J lie < M hmax ·

Estimate (52) indicates an effective reduction of the speed of conver-

gence of scheme (51) on the non-equidistant grid wh in comparison with

scheme ( 44) on the equidistant grid. However, as we have stated above, if

the error of approximation is evaluated not in the grid norm of the spaces

C or L 2 , but in a specially constructed negative norm II· 11(-l)' then the er-

ror of approximation on any non-equidistant grid will be of the same order

O(h^2 ). Namely, the negative norm

is good enough for our purposes. What has been said above implies that in
the further derivation of a priori estimates for problem (51) the right-hand
side should be evaluated in the negative norm 11 · 11(-l)·
Let us obtain this a priori estimate by multiplying equation (51) by
Y;h; and summing over all grid nodes of wh. In terms of the inner products
the resulting expression can be written as

(53) (yxi:' Y). + (f, Y). = 0.

Via transform of the first summand in (53) by the Green difference formula

(8) it is not difficult to establish the relation

(54)

As we will see later, it will be sensible to deal with the function TJ( x) specified

by the relations

(55) T/ x •,z. = Ji ' i=l,2, ... ,N-1,

While solving problem (55) it reduces to -TJ(x;) = L,~=-/ fkhk.
The inner product on the right-hand side of (54) is modified by the
summation by parts formula (7) into

(f, Y). = (i7,,,, Y). = -( TJ, Y;;].