1549301742-The_Theory_of_Difference_Schemes__Samarskii

664 Methods for Solving Grid Equations

The well-established expansion of the function

In 1 + x 1 ( 1 1 )^3

1 - x =^2 :r + 3 (1 + x)^3 + (1 - x)3 x ' O<x<l,

where 0 < x < x, is aimed at establishing the relations

l+x

In > 2 x,

1-x

In _..!:._ = In^1 + /[ > 2 .J{,

P1 l - /[

and, hence, inequality (32) holds true for

(33)

This est.in1ate is 1nore convenient in practical i1nplementations than esti-

mate (32).

4. The simple iteration schen1e. By forn1ally setting n = 1 in formula (29)
   the preceding is referred to as the simple iteration method

(34)

with the para111eter T 0 incorporated:

( 34')

2

Ta= ---

l! + /2

Here t 1 =cos~ = 0, T 1 = T 0 and

(35)

The equation for the residual 1·k = A Yk - f reduces to Yk+l = Syk,
S = E-T 0 A. As far as the operator T 1 =Sis concerned, fonnulas (20) and
( 35) together i1nply the estimate for the norm of the transition operator

' 1-~

11 5 11 = Po = l + ~ ·

By making n iterations of the si1nple iteration n1ethod we find that

II rn II <pg II ro 11.