278 Difference Schemes for Elliptic EquationsWe list below the properties of the difference Laplace operator acting from
0 0
rlh into s:th C rlh. By definition,- The operator A is self-adjoint:
0
(16) (Ay,v) = (y,Av) for all y,v E rlh.To make sure of it, it is straightforward to verify the chain of the relationsN2-I N1-l
(A1 y, v) = L h 2 L h 1 (v A1 Y)i 1 i 2
i2=1 i1=lwith a simple observation that the operator A 1 is self-adjoint on the gridThis is due to the fact that the order of summation over i 1 and i 2 may be
interchanged. In a similar way we obtain
(A2y,v)=(y,A2v),
which leads to (16).- The operator -A is positive definite:
(17) (-Ay,y) > bllYll^2 ,
where
4 2 7!' h I 4 , 2 7!' h 2 8 8
b = 2 sin 2I + - 1 2 srn 2I > (i + (i = b 0 •
hi I^12 2 I 2
This property follows immediately from the relationAminllYll^2 < (-Ay,y),
which is certainly true with b =Amin (see Section 1).