1549301742-The_Theory_of_Difference_Schemes__Samarskii

Two-layer iteration schemes

On the other hand, it follows from the foregoing that

yielding

(14)

<II A-^1 II ·II Bvk 112

<II A-^1 II · II B 112 • 11vk11^2

llBll^2 · llvki12

b

659

By inserting (13') and (14) in (13), it is possible to show that the bilaterial

estimate

II zk+1 II! = 11 S'z" II! < P

2

II zk II!

becomes valid wit.h p^2 = 1 - 2rbb./ll B 112 < 1, assuring estimate (12) and

the inequality 11 ;; 71 llA < p^11 11 z 0 llA and justifying the convergence of the

iterations, since p^11 ~ 0 as n ~ co. The same estin1ate is certainly true

with the correction w 11 = B-^1 (Ay 11 - f).

Remark Condition (11) for fixed B may be viewed as a selection rule for
those values of T for which the iterations converge. For exa1nple, for the
explicit scheme with the identity operator B = E condition (11') is ensured
if all the eigenvalues are subject to the relation

1 1

).k (E - - TA)= 1 - - T >.Jc(A) > 0

2 2

or, what amounts to the same,

1

1-2rllAll > 0.

Thus, the iterations converge for any T < 2/11 A II. Let us stress here that
the estimate obtained for p is too rough for detennination of the total
number n( c;, N) of the necessary iterations and indicates 1nainly the true
order in n as N ~ 1:x:i,