1549301742-The_Theory_of_Difference_Schemes__Samarskii

Two-layer iteration sche111es 655

The usual practice involves the numerical solution of problem (5) with

a prescribed accuracy c > 0 (a relative accuracy II Yk - u II/II Yo - u II), it

being understood that the calculations should be terminated if

(6) II Yk - U II < c II Yo - il II.

In connection with inconvenience caused by the unknown vector 11, it seems

reasonable to replace this condition by the inequality for the residual

(7)

In the general case the accepted view is the termination condition of

the type

(8)

where D :::: D* > 0 is some operator. By merely setting D = A^2 we deduce

from ( 8) inequality (7).

We are now interested in the governing equation related to the residual

zk = Yk - tt. Since Att = f, we might have

(9) k=0,1,2, ... ,

where z 0 E H is known. As far as Bk = B is independent of the subscript

k, the correction wk = B-^1 rk satisfies the h01nogeneous equation

Indeed, (4) implies that

Applying the operator A twice to both sides of the preceding equality and
taking into account that

we establish the h01nogeneous equation for the correction wk.