1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference sche1nes as operator equations. General for1nulations 123

Exa1nple 3. Non-self-adjoint operators. Let wh = { X; = i h' i =

0, 1, ... , N , h = 1 / N} be a. grid on the segn1ent 0 < x < 1 and let the

difference operators

(14) A- y = Y-' x A+ y = -y_, x

0

!napping the set Qh of grid functions defined on wh and vanishing for i = 0

and i = N into Qh, be such that

Z• --^1 '

i = 2, 3, ... , N - 1,

i = 1, 2, ... , N - 2,

i=N-l.

These forn1ulae show that the operators A - and A+ can be treated under

such an approach as operators from Qh into Qh·

Let (y, v) = L;:~^1 Yi vi h be the inner product in Qh and Qh· It seems

clear that the operators A- and A+ are mutually adjoint:

(15) for any

Indeed, by the summation by parts formula we have -(y, vx) = (Y:t' v) if

y = v = 0 for i = 0 and i = N. This just implies the fact that A- and A+

are operators adjoint to each other. We note in passing that

A-+ A+= -( Yx - Y-) x = -h Ay,

where A y = y_ xx , that is, A- +A+ = -h A. Therefore, the operators A_

and A+ are positive definite:

. h h
(A-y, y) =(A+ y, y) = 2 (-Ay, y) = 2 llvvll^2 > 4hll Yll2.

The last inequality holds true on account of Lemma 3 in Section 3.4.

(16)

Later we will deal with operators fr0111 Qh into Qh of the structure

1

R 1 y = -h Y-, x

1

Ri = h A-,

1

R2 y = h ( -yx ) ,

1

R2 = h A+