1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternative-triangular 1nethod 687

estimate (18) is true and it suffices to perform n iterations for the valirlit.r

of the inequalit.r

where

(34) n 0 (t:) = fji_ In (2/t:).

V C1 2 /2 \111

Proof In complete agreement with the preceding theorem with known co-

efficients / 1 and / 2 , arising from the operator inequalities / 1 B < A < / 2 B,

we deduce frorn (27) and (32) that

0

meaning /1 = c1 I 1 ,

0

mean mg f? -= c., - 1 '). -

The conditionality parameter for w = w 0 is equal to

Theorem 2 and the results of Section 2 apply equally well to such a setting.

As stated in Section 2, the condition qn < E is ensured by n > In ~ / ( 2 Jf,).

Substituting here ~ = 11 /1 2 and taking into account that ~ < .:i 2y'ri,

C2

we derive the sufficient condition (34), which can easily be verified in one

particular case of interest:

The total number of iterations required in this connection is no less than

( )

_ In (2/ E)

n 0 E - ;r; .;;:::.

2v2~TJ

5. ATM for the difference Dirichlet problem. The trace of those ideas can
   clearly be seen in tackling the difference Dirichlet problem associated with
   Poisson's equation in the rectangle (J = {O < xa < l°', o: = 1, 2}: