1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference schemes as operator equations. General formulations 137

on wh and vanishing for i = 0 and i = N under the inner product structure

( y, v) = L;:,~^1 Yi v; h in H.

As a first step towards the ~ solution of ~ this problem, the operator A is

defined by the relation Ay = -Ay, where Ay = Ay for y ED, so that

(49) (Ay),:=-(ayx)x,i• i=l,2, ... ,N-l, (Ay) 0 =(Ay)N=0

and the vector <pis taken to be <p = (0, <p 1 , <p 2 , ... , <pN_,, 0). The next

step is to recast the problem in view as an operator equation

Ay=<p.

It is easy to see from Green's formulae

and

0
where y, v E D, that the operator A is self-adjoint and positive definite,
that is, A > b E, b = 8 c 1. Recall that we have established in Section 3
frorn Chapter 2 that

This implies that the inverse A-^1 exists and ( A-^1 )* = A-^1 > 0. More
exactly,

(50)

1 1

E < A-^1 < E

II A II fj.

Let us show that the negative norm 11 'PI IA_, of operator ( 49) 1s repre-
sentable by

(51)

(52)

i-1

Si= L h 'Pk'

k=l

(

N h ) 2 ( N h )-l

~ ai Si ~ ai '

i = 2, 3, ... , N,