1549301742-The_Theory_of_Difference_Schemes__Samarskii

Iterative alternating direction methods 719

Along these lines, it is worth noting that we are still in the framework

of the general stability theory outlined in Chapter 6, Section 2 for difference

schemes like (22) asserting that a necessary and sufficient condition for the

estimate I IYk+i I ID :S p 11 Yk I ID to be valid for any 0 < p < 1 and any operator

D = D* > 0 commuting with A is

l~p(E+wA)<A<

1

2

:p(E+wA).

This is acceptable if we agree to consider D = E or D = A. In this regard,

it should be noted that the preceding is equivalent to the bilateral operator

estimate

-E<A<-E ~^1

w - - ~w ' where

l-p

~ = 1 + p and

Putting these together with (21) we conclude that

1-~

p=

l+~

1 1

(23) §._ < - b ' ~w-- > ~ or c, cw -~' < - so that c, c^2 <_ 17,

w

it being understood that the minimum of p will be attainable in the case

of the 111axin1al value of~ in (23). This is certainly so with

f-o - '

w

1

-=~,

~w

whose use permits us to establish the chain of the relations

w=~=ViJ= l -w

b b Vt!S.-^0

and arrive at

min p(w) = p(w 0 ) = l - /ii.

w l+Vfi

However, the same result can be obtained by a simple observation that

1

p =II S'(w) II= inax 1-w.\I ,\ ,

b:S.\S:b. 1 +w

where ,\ = .\(A) is an eigenvalue of the operator A. Plain calculations show

that

mm. p () w = mm. max 11-w.\I ,

w w b:S.\:Sb. 1 + w /\

(

mm max 1-wb w~-1)

-'- 6 - <w < - 6 l 1 + W D ' W ~ + 1

1-Vfi

=p(w 0 )= l+Vfi for w=w 0 ,