In the previous experiment, the input data is a constant function
on each subinterval. In order to verify the existence of the asymptotic
error expansion for any kind of input data, we randomly selected two forcing
functions involving polynomials. We consider the following jumped discontinuous
function as the example
According to (3.2) and (3.3), we decompose our problem into the following problems,
and
Again we use the same Green's function (1.1.12)
to find the exact solution for the problem (1.4.1).
The same as the last experiment, we have
if 
, then
On the other hand, if 
, then
Therefore, the exact solution for the problem (1.4.1) is
Again, using the same notations as (1.1.13)
and (1.1.14), we can obtain the same approximation
for the differential operator as (1.1.15).
And we extend the concept of the previous experiment for taking the function
value at the nodal point x = 1/2 to be the average of the function
values of the neighbouring nodal points. So that we can form a matrix equation
to find the nodes 
, that is
where A is an order (n-1) square matrix, u and
b
are 
column matrices, such that
 |
 |
 |
 |
|
| h=1/4 |
|
|
|
|
| h=1/8 |
|
|
|
|
| h=1/16 |
|
|
|
|
| h=1/32 |
|
|
|
|
| h=1/64 |
|
|
|
|
| h=1/128 |
|
|
||
| h=1/256 |
|
TABLE 4.2
|
r(1)
|
|
|
|
|
|
r(2)
|
|
|
|
|
|
r(3)
|
|
|
|
|
|
r(4)
|
|
|
||
|
r(5)
|
|
|
||
|
r(6)
|
|
Table 4.1 and Table 4.2
show the numerical results at the nodal point x = 1/2. It is obvious
that we can have the identical asymptotic error expansion as (1.1.19)
for such kind of input data. The extrapolating technique at this nodal
point is effective, we get a numerical solution with error 
through extrapolating from the three solutions corresponding to h
= 1/4, 1/8 and 1/16. This error is less than the numerical solution error 
corresponding to h = 1/256. We also investigate the numerical solution
error at other nodal points, and the identical asymptotic error expansion
has been obtained. [See Appendix D.4a
-D.4d].
