1549301742-The_Theory_of_Difference_Schemes__Samarskii

720 Methods for Solving Grid Equations

thereby completing the task of motivating the desired minimum.

Via transforms (9)-(10) available for the operators A 1 and A2 and the

parameters w 1 and w 2 we accept

making it possible to reduce the operators specified by (20) to

Their members satisfy the relations

TJE <A~< E, C\'=l,2, TJ>O.

In the preceding reasoning the member T) and the parameters p, q, r are

given by formulas (14)-(LS). With the inequality llSll < ll,5'1 II· ll,5'2 II in

view, the lemma asserts that for w = w 0 = /if/ o

min 11,S\ II =min II S'2 II =

1

• ,fij
  w w l+j1]

and a solution of problen1 ( 18) obeys the a priori estimate

with w 1 and w 2 still subject to the relations

(25)

In particular, T) = b / ~ for b1 = b2 = b and ~1 = ~2 = ~.
From (24) it follows that the condition pn < c of the termination of
iterations is ensured if

nC1l(c) = ln(l/c) ,

2~

where

2 ,fij

~=.

1 + T)