(^296) Difference Schemes for Elliptic Equations
with x 1 = h;/12 and x 2 = h;/12 incorporated. The operators A 1 and A 2
are self-adjoint:
(Acx y, v) = (y, Acx v), a=l,2, y,vE}fh,
positive definite:
,(CY) = i sin2 7rhcx >!
1 h2 O' 2 l O' - /2 CY '
and commuting: Ai A2 = A2 A 1. In view of this,
By virtue of the relations
we arrive at
yielding
I
4 2 7r hO' 4
I I ACY I = J;2 COS '2/ < J;2 ,
O' CY O'
< X1 I I A 1 I I A 2 + X2 I I A 2 I I A 1
<~(A1+A2),
~A< A'= Ai+ A2 - (x 1 + x 2 ) Ai A2 <A,
A= A 1 + A2,
~II Az II< II A' z II < II Az II·
a=l,2,
The operators A and A' are commuting and self-adjoint and, therefore,
for the equation A' z = 1/; the estimate 11 A' z 11 < ~ 111/J 11 is certainly true.
By the embedding theorem from Section 4,
12 312
II z lie < 2 Jr;z; Az < 4 jz;--r; 111/J II,
1 2 1 2