1549301742-The_Theory_of_Difference_Schemes__Samarskii

70 Basic Concepts of the Theory of Difference Schemes

Remark 3 If the grid wh is non-equidistant, that is, h = (h 1 , ••• , hN ),

where N is the total number of nodes, it is sensible to deal with h =

max 1 <::i<::N h; or the mean square value I h I·

Several examples add interest and help in understanding the outlined

theory.

Example 1. The difference approximation on a non-equidistant

grid. In working in the space H 0 = C(^4 ) [O, 1] comprising all the functions

defined on the segment 0 < x < 1 we refer to the operator L with the values

and take on this segment an arbitrary non-equidistant grid

wh={x;, i=O,l, ... ,N, Xo=O,xN=l}.

On account of Example 6 in the preceding section the difference operator

V; = v(x;),

defined at the node X; on the irregular three-point pattern ( ~:;_ 1, X;, X;+ 1 ),
is associated with the operator Lv. vVith this, we rewrite Lh v as

with more compact notations

V-x,' , = h·

1

In Section 1.2 we have found the local approximation error taking now
the form

i=l,2, ... ,N-l.

From here it is easily seen that the operator Lh is of order 1 with respect
to the grid norm of the space C;

h = l<::i<::N max h,·..