Some properties of difference elliptic operators 275
in the sense of the inner product
N 1 -1N 2 -1
(7) (y,v)= LL y(i 1 h 1 ,i 2 h 2 )v(i 1 h 1 ,i 2 h 2 )h 1 h 2
ii =l i2=lL y(x) v(x) h 1 h 2.
xEwh
Therefore, any function f(x 1 , x 2 ) defined on the grid wh and vanishing on its
boundary can always be expanded on the systen1 of functions {vkik 2 (x 1 , x 2 )}:
N
f(x) = L ckvk(x),
k=l
In the case of the second boundary-value problem with 8v/8nlr = 0,
the boundary condition of second-order approximation is irnposed on ih as a
first preliminary step. It is not difficult to verify directly that the difference
eigenvalue problem of second-order approximation with the second kind
boundary conditions is completely posed by
Av+ AV= 0)
(8)CY= 1,2.
Indeed, assuming that u(x) is a solution to the equation 6. u +Au = 0
subject to the condition 8u/8nlr = 0, the error of approximation for the
boundary conditiozi is represented by
v 1 - = ( ux, + ~ h 1 A 2 u + ~ h 1 Au) I
X1=0= :: I + ~
1
(6.u +Au) I + O(h;) = O(h;).
1 X1=0 X1=0