1549301742-The_Theory_of_Difference_Schemes__Samarskii

136 Basic Concepts of the Theory of Difference Schemes

5. Negative norms. In a priori estimates ( 41) and ( 42) we were dealing with
   the negative norm 11 'PI IA -1. Just for this reason the question of computing
   negative norms for the sin1plest operators will be of great importance. First
   of all we quote a result of simple observations:

( 46) if

0 0

where A = A* > 0 and A = A* > 0. This follows from the equivalence of

the operator inequalities

( 47) and

Our first goal is to prove the equivalence between the inequalities Q > 0

and L*Q L > 0, where Q, L: Hf--+ Hand L-^1 exists. Indeed,

(L*QLy, y) = (QLy, Ly)= (Qv, v),

0 0

where v = Ly and y = L-^1 v. Accepting A> /A or Q =A - I A> 0 and

0

setting L = L* = A-^1!^2 , we obtain C-1 E > 0, where

Since the operator c-^1 /^2 = ( c-^1 /^2 )* > O does exist, the inequality C -

/ E > 0 is equivalent to

0 0

c-1/2(C-1E)c-1/2 = E-,c-1 = E-1Alf2A-l Al/2 > 0.

0 0

By merely setting L = L* = A-^1 /^2 we get A-^1 -1A-^1 > 0, what means

that inequalities ( 47) are equivalent as required.

Example 4 Consider the first boundary-value problem

Ay=(ayx)x=-<p(xi), xi=ih, i=l,2, ... ,N-1,

( 48)

h N = 1, Yo = YN = 0, ai > c 1 > 0, i = 1, 2, ... , N,

on the uniform grid wh = {xi= i h, i = 0, 1, 2, ... 'N' h = 1 IN}. One
0
assumes, as usual, that Hh = Dh is the space of all grid functions defined