1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference approximation of elementary differential operators 71

The same is still valid in the grid norm of the space L2:

( )

N-l 1/2

111/JllL 2 = ~ ni 1/J[ = O(h).

i=l

vVe claim that in an alternative norm known as the negative norm

the local approximation error 1jJ is of order 2 and behaves like

(29)

Indeed, rewriting 1jJ as

h = max h·.

1 <i<N '

and taking into account that v;^11 = v;~ 1 + 0( h;+ 1 ), we establish the decom-

position

o/, 'f"i -- h

(^2) v'. (^11) - h^2 v (^111) o
z+1 1+1 Bn i i +·i= 1/J 1/J. i+, 1/J
'
0
with 1/J; = O(h^2 ) in any suitable norm. The main term 1/J; in the established
0
decomposition 1/J; =1/!; +1/!; is of "divergent form", thereby justifying that
h^2 k+1 v^111 k+1 - h^2 k vk^111 hi+1^2 v'.z+1^11 - h1 1^2 v^111
k=l k=l^6 6
Since ISil < Mh^2 , we find that
0
111/J 11(-1) = (
N-1 ) 1/2
L his; = O(h^2 ),
i=l
which in combination with the relations
0
111/J 11(-1) < 111/J 11(-1) + 111/J* 11(-1),