1549301742-The_Theory_of_Difference_Schemes__Samarskii

116 Basic Concepts of the Theory of Difference Schemes

On account of the Cauchy-Bunyakovskil inequality,

l(f, y). I= 1(77, Y,,)I < 117711 · llvrll·

Substituting this estimate into (54) and eliminating 11 Yx JI fr01n both sides

of the resulting inequality, we arrive at the relations

By Lemma 1,

llYllc < l/Yx]//2

and, consequently, we derive the relation

(56) llYllc < ~ llJll(-1)'

thereby justifying the desired estimate. Further development is connected

with the accurate account of the enor z = y - u, where y is a solution of

problem (51) and 11 is a solution of the original differential problem ( 43).

Upon substituting y = z + u into (51) the problem arises for the error z:

(57) Z 0 =ZN = Q,

Applying estimate (56) to problem (57) yields

llzllc < ~111/>ll(-l) ·

However, we have stated in Section 1.3 that 111/>ll(-l) < Mh^2 , where h =
max 1 <i<N h;. Therefore, scheme (51) on an arbitrary non-equidistant grid
wh co;v~rges in the space C with the rate O(h^2 ).

2.5 DIFFERENCE SCHEMES AS OPERATOR EQUATIONS.

GENERAL FORMULATIONS

Difference sche1nes for the simplest differential equations have been con-
sidered in preceding sections, the basic topics in the theory of difference
schemes have been introduced for them as well as all the tricks and turns
available for investigating stability and convergence of such schemes have
been demonstrated with a great success.
In this section a unified interpretation of difference equations as op-
erator equations in an abstract space is carried out and, after this, the
corresponding definitions of approximation, stability and convergence are
presented. This approach is quite applicable in mathematical physics for
stationary problems.