1549301742-The_Theory_of_Difference_Schemes__Samarskii

686 Methods for Solving Grid Equations

when performing current manipulations with the operators B and R:

(By,y) = ((E-wR1)(E-wR2)y,y) +2w(Ry,y)

= ( ( E - w R 2 ) y, ( E - w R 2 ) y) + 2 w (Ry, y)

= ll(E-wR2)Yll^2 +2w(Ry,y)

1

2".2w(Ry,y)=- 0 (Ry,y).

I 2

0

From such reasoning it seems clear that (Ry, y) < f 2 (By, y).

The next step is to find a maximal value of the ratio

0 0

fi 2w8

~(w)= .'.f

2

= l+w8+(w^2 8~)/4

by observing that
0
d~ =28 l-w28~/4 2·
dw ( 1 + w8 + (w^28 ~ )/4)
0
From calculus it is known that the maximal value of ~(w) is attained for
2 0
w = w 0 = ./8/S.' since ~"(w) < 0 for w = w 0 • Upon substituting w 0 =
8~
2 /i)/8 into (28) we derive formulas (30).

Theorem 2 Let under the conditions of Lemma 2 the inequalities

(32)

hold simultaneously for opera.tors A= A > 0 and R = R > 0. Then for
ATM given by formula (23) with optinml set of Chebyshev's parameters

(33)

where

2

To= fl+ f2 '

1 - ~

Po= l + ~,

0 0 0 8

fl=C1f11 f2=C2f21 fi=2(l+y'f/)'

c - fl

c, - f2 '