1549301742-The_Theory_of_Difference_Schemes__Samarskii

426 Stability Theory of Difference Schemes

that is, there exists an inverse operator B-^1. By inserting y = B-^1 x it is

plain to establish the relations llB-^1 xll2 = (B-^1 x,x) < llB-^1 xll · llxll,

giving, II B-^1 x II < II x II and II B-^1 II < 1. The next step is to rewrite the

scheme in the form

Bf)=By-rAy,

and calculate the inner product

(Bf),!))= (By, y) - T (Ay, y) - T (By, B-^1 Ay) + r^2 (Ay, B-^1 Ay).

For later use, it will be sensible to represent the operator B in the form

B = E + cr r A= (E - cr r A)+ 2cr r A= B* + 2cr r A

and make the obvious transformations

(By, B-^1 Ay) = (B*y, B-^1 Ay) + 2cr r (Ay, B-^1 Ay)

= (y,Ay) +2crr(Ay,B-^1 Ay)

= 2crr(Ay,B-^1 Ay).

As a final result we obtain

yielding

II fJ II < II Y II if(} > 0.5)

or

ll:iJll < PllYll, C 0 = ( l - 2 (}) c 2 ,

if cr < ~ and T 11 A 112 < c 2.
We have obtained through such an analysis the following estimates for
the norm of the transition operator:

llSll < 1 for cr > 0.5,

llSll<p for cr<0.5 and rllAll2<c 2.