1549301742-The_Theory_of_Difference_Schemes__Samarskii

Other iterative 1nethods 731

2p"

Comparison of (8) with the resulting expression for qn =^12 , valid for

1 + P1 n

Chebyshev's scheme, shows that both schemes are of the same asymptotic

order a.s ~ _,_ 0:

n = n(c) = o(~ In~).

Here n is, as usual, the nun1ber of iterations. But in practica.l in1ple1ne11-

tations Chebyshev's schenw with known values fl and f 2 is preferable,

because the extra iterations and storage are necessary for later use of the

three-layer scheme concerned. What is more, the second scheme depends

1nore significantly on the errors in specifying fl and f 2 than the first one.

Remark The passage from explicit three-layer sche1nes to implicit ones

can be acc0111plished by the replacen1ents of A by B-^1 A and f by B-^1 f,

so that

or

(10) Byk+l = (1 + o:) (B - T 0 A) Yk - o:Byk-l + (1 + o:) T 0 f,

By 1 = B Yo - T 0 Ay 0 + T 0 f, k = 1, 2, ... , given Yo E ]{.

Observe that equation (10) e1nerged fron1 the identity

Bu= (1 +er) (Bu - T 0 A u) - o:Bu + (1 + o:) T 0 f

with further indication of iteration nmnber in the appropriate positions, if

any. forn1ulas ( 4) for T 0 and er are still valid with known spectral bounds

f1 and 12

( 11) fl B < A < f 2 B , fl > 0, B = B* > 0,

for the operator A acting in a certain space H B (not in the entire space
H), making it possible to establish for a solution of problem (10) instead
of (7) the estimate

( 12)

with qn still subject to (8). This is acceptable if operator (76) is involved
in the fra1nework of ATM in Section 3.