1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference schemes as operator equations. General formulations 121

and (y, w) = Lt,:~^1 Yi wi h. Making use of the Green formula from Section

3 and substituting the relevant expressions for A -v and A+ v, we find that

-(y, Av)= (-Ay, v) - (Yx,O V 0 - Yx,N vN) +(Yo vx,O - YN vx,N),

-0.5 h (Yo A-v + YN A+ v) = -yo ( v.T, o - CT1 Vo)+ YN ( v,, N + CT2 vN).

'

Since Yx,o = cr 1 y 0 + 0.5hA-y and Y.r:,N = -cr 2 yN - 0.5hA+ y, we arrive

at the chain of the relations

[ y, Av ] = -(A y, v) - 0 .5 h ( v 0 A - Y + v N A+ y) = [A y, v]

as required.

We are going to show that if cr 1 > c 1 > 0 and cr 2 > c 1 > 0, then the

operator A is positive definite:

( 11) [A y, y l > 2c1 l[Y]l2.

1 + c 1

With this aim, we follow the sa1ne procedures as we did in the proof of

Lemma 1 in Section 3. Namely, those ideas are connected with attempting

the function y^2 ( x) in the fonn

and applying then the E-inequality. The outcome of this is

( l)(x )

2

y

2

(x) < (1 + E) y; + 1 + ~ x'f.h Y;c(x') h

By the Cauchy-Bunyakovskil inequality,

implying that