1549301742-The_Theory_of_Difference_Schemes__Samarskii

718 Methods for Solving Grid Equations

Having involved vk = (E + w 2 A 2 ) zA. with furter e!in1ination of zk+in from

(18), we derive the equation

( 1 g)

whose transition operator S arranges itself as a product S = 51 52 with the

multipliers

(20)

This is showing the gateway to subsequent considerations: srnce

it is necessary to evaluate the nonn 11 5'1 S2 11 and find mmw 1 ,w 2 11 S1 S2 11

with the aid of the well-established relation.

As further developments occur, we need an auxiliary lemma.

Lemma Let an operator A : H 1-+ H be in line with the conditions

(21) A=A*>O, oE::;A::;t:,,E, b>O.

Then the norm 11 S(w) 11 of the operator S( w) = ( E +wA)-^1 ( E -wA) attains

for w = w 0 = ~ the minimal value

. , l -j17
mm II S(w) II= II S(wo) II= '
w 1 + v0

where

For the most part, the proof is connected with further treatn1enL of

S(w) as the transition operator of the two-layer scheme

(22) (E + w A) Yk+I 2w - Yk +A Yk -_ O ' k = 0, 1, ... , Yo E H,

with the operator B = E + wA involved. This supports the view that

Y1.:+1 = S'(w) Y1.:, II Yk+l II < p II Yk 11,

where p = p(w) =II S(w) 11 needs to be minimized.