1549301742-The_Theory_of_Difference_Schemes__Samarskii

558 Economical Difference Schemes for Multidimensional Problems

In dealing with non-negative and, generally speaking, non-self-adjoint op-

erators A 1 > 0 and A 2 > 0, it will be sensible to omit for a while any

subscripts and superscripts

Z -- Zn '

allowing a si1npler writing of the ensuing fonnulas:

(E + 0.5 T .iL) z = (E - 0.5 T A2) z + 0.5 r1/i 1 ,

( E + 0.5 T A2) z = (E - 0.5 T Ai) z + 0.5T1f'2.

The triangle inequality gives

(40) ll(E+0.5rAi)zll<ll(E-U.5rA2)zll+0.5rll1/i 1 ll,

(41) ll(E+0.5rA2)zll < ll(E-0.5rAi)zll+0.5rll1fi2ll·

An auxiliary lemma may be useful in the sequel.

Le1n1na 1 If A > 0 is a linear operator in a. Hilbert space H .• then

(42) ll(E-(1-())rA)yll<IJ(E+()rA)yll for ()>0.5

This assertion is an immediate implication of the chain of the relations

II (E +()TA) y 112 - II (E - (l - ())TA) y 112

= 2 T ( Ay, y) + 2 ( () - 0. 5) T^2 11 Ay 112 > 0 '

which are valid for () > 0.5 and the operator A > 0.

Collecting inequalities ( 40) and ( 41) and taking into account estimate

( 42) with the value () = 0.5 incorporated, we proceed to the eli1nination of

z. Following established practice, we arrive at

II (E + 0.5 T A2) z II < II (E + 0.5 T A2) z II+~ (II 1/'1 II+ II 1/'2 II)'

yielding with the aid of inequality ( 42) either