1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference sche1nes as operator equations. General formulations 119

The operator A is positive definite, that is, (Ay, y) > 8IIy112· This is

a consequence of Lemma 3 in Section 3.4.

The norm of the operator A is equal to

(5)

4 7rh 4

11 A 11 = h 2 cos 2 2 < h 2

Indeed, the norm of a self-adjoint positive operator in a. finite-dimensional

space Qh is equal to its greatest eigenvalue: II A II= >.N_ 1. In that case, in

complete agreen1ent with the results of Section 4.2, we might have

and, therefore, formula (5) holds true. In addition,

(Ay, Y) < llAll · llYll^2 ·

The next quantity we will introduce is an operator with the values

(6) Ay=-(ayx)x+dy,

Due to the second Green formula it is self-adjoint. In turn, the first Green

formula assures us of the validity of the relation

(7) (Ay, y) =(a, y~] + (dy, y),

where y~ x = (y_ x )^2. This implies that

(8)

meaning that A is positive definite. It seems clear from fonnulae (7) and

(8) that its norm satisfies the estimate II A II< 4 c 2 /h^2 + c 3.

Remark Here the operator A = Ah depends only on the grid step h playing
for the moment the role of parameter. vVhen wh = { X; E [ 0, 1], i
0, 1, ... , N, x 0 = 0, xN = 1} is a non-equidistant grid, its step h;
x; - x;_ 1 itself becomes a grid function or an N-dimensional vector h
(h 1 , h 2 , .•• , hN). The operator A admits the form A= -A, where