1549301742-The_Theory_of_Difference_Schemes__Samarskii

Some properties of difference elliptic operators^279

3. The operator A admits the bilateral estimate

(18)

where f; is the sarne as before and

4 2 7!' h I 4 2 7!' h 2 4 4

6 = >-max = 2 cos --+ 2 cos -21 < -12 + z2 = 60.

h I 2 11 h2 2 I 2

From the relations just established and the lower estimate for f; we deduce

that

(19)

These properties provide support for the view that the operator A = -A
is more convenient for many things than the operator A. The operator A
0
can be treated as an operator either from Hh = Dh into D1i C Hh or from
H h = s:th into H1i by merely setting

Ay= -Ay,

~ 0

where Ay = Ay for y E rlh. In accordance with what has been said above,

we have in the space nh

bE<A<6E

with E denoting the identity operator.
0
Any function f E rlh, defined on the grid w h and vanishing on the
boundary lh or defined only on wh, can be expanded into a series of the
eigenfunctions of the operator A:

N

·f(x)= L ckvk,

k=I

so that L k = lN c~ =II f 112 , where vk are the eigenvectors of the operator
A: Avk = >-k vb N = (N1 - l)(N2 -1).
In the case of the second eigenvalue boundary-value problem (8) the
space H1i = [2h comprises all the functions defined on the grid w h; the inner
product (,) on H h is to be understood in the sense ( 14) and the operator
A is defined as a sum