1549301742-The_Theory_of_Difference_Schemes__Samarskii

Ho1nogeneous difference schemes on non-equidistant grids 173

which are always valid, we follow the same procedures in the accuracy

analysis as we did on equidistant grids in Section 3. This amounts to

regarding q, f E cC^2 ) to be continuous and establishing the relations:

17; = 0( hi) for i # n + 1 , 1Jn+i = 0(1).

Only for the best scheme (3) we might achieve 1Jn+i = O(hn+i) and

h^2 i (qu - f)' i-1 /2

8

of,* 'Yi = 1lx, - i + of,* 'Yi ' 1Ji = 1/J7 = O(liT)

for all i = 1, 2, ... , N - 1.

Using estimate (10) behind we deduce that the best scheme (3)-( 4)

retains the second-order accuracy in the class of discontinuous coefficients

on an arbitrary sequence of non-equidistant grids, while any scheme ( 4)

appears to be of order 1.

In the class of continuous coefficients k, q, f E cC^2 l [O, 1] any

scheme ( 4) retains the second-order accuracy on an arbitrary

sequence of non-equidistant grids.

Before we undertake the proof of the last assertion, it is worth men-

tioning the estimate obtained in Chapter 2, Section 4 for the operator

equation A z = 1/J:

0 0 0
where A 2: c 1 A, A= A> 0 and A= A> 0. In the case of interest
0 0
Ay = -Yxx, y ED,
0
where D = H is the space of grid functions defined on wh and vanishing
for i = O,N. The negative norm ll0lla A- 1 was estimated in Chapter 2,

Section 4 as follows:

Here Ti^2 denotes, as usual, the mean square of hI and so the desired assertion
follows immediately from the preceding.