, but the forcing function requires a high smoothness.
In the Experiment (1.2), the forcing function is a concentrated loading, we used the concept of the delta function to describe this loading which carried a high singularity. Nevertheless, we still obtained the same asymptotic error expansion as the first example for all nodal points, even at the singular point. Moreover, no matter where the concentrated loading acted, the numerical solutions are oscillating convergent and the rates of convergence are satisfactory at all nodal points.
In the Experiment (1.3), the forcing
function is jumped discontinuous at x = 1/2, totally, we have tried
three different possibilities of determination of the function value at
the jumped discontinuous point. In the first two possibilities, that is,
the selection of the function value at this point is 1 or -1. The results
show that the numerical solutions are unstably converged and the rate of
convergence is so slow at all nodal points except for x = 1/2. However,
the numerical solutions converge regularly and the rate of convergence
is quite satisfactory although the order is O(h). For the
last possibility, we selected the function value at the jumped discontinuous
point to be 0. At this point, the error of the numerical solutions is very
small, even better than the extrapolated solutions whereas the result cannot
show the pattern for the asymptotic error expansion. On the other hand,
the numerical solutions at all other nodal points are convergent and we
can obtain the same expansion as the first and the second experiments,
that is
. Therefore, we can conclude that selecting zero to be the function value
at the jumped discontinuous point is reasonable and the best.
In the next two experiments in one-dimensional problems, we have tried
to make the input data more general in order to verify the existence of
the asymptotic error expansion for any kinds of input data. Therefore,
we randomly selected polynomials and trigonometric functions as the external
forcing functions. In the Experiment (1.4),
we choose two different polynomials, and in the Experiment (1.5),
we assigned a linear polynomial in one subinterval and a trigonometric
function in the other subinterval. Moreover, we extended the idea from
the previous experiment, we took the function value at the jumped discontinuous
point to be the average of the function values of the neighbouring nodal
points. However, the numerical solutions are convergent and the rate of
the convergence is
, that is the same as the Experiment (1.3).
In the last section, we have tried the concentrated loading and jumped
discontinuous forcing function in two-dimensional problems, although the
numerical results are not so clear as the one-dimensional problems, we
can still observe the patterns. In the Experiment (2.1),
the external forcing function is a concentrated loading, we used a two
dimensional delta function to represent the loading. Since the domain is
a unit square, we used Fourier sine series to find the exact solution and
finite difference method for computing the approximated solutions. The
numerical solutions are convergent and the rate of convergence is quite
good, that is
, the result is the same as the one-dimensional problems.
In the Experiment (2.2), we divided
the domain to four small equal squares and applying a non-zero constant
function on two small squares diagonally. We also applied Fourier series
and finite difference method to find the exact solution and the numerical
solutions respectively. If we assigned zero to be the function value at
the junctions of the small squares when we were finding the numerical solutions,
we discovered that the solutions are still convergent but the order is
O(h).
If we extended the idea of Experiment (1.3),
taking the function value at the junctions to be the average of the function
values of the neighbouring nodal points, the numerical results show that
the solutions are convergent and the order is
which is better than the previous results. Once again, taking the average
at the junctions will be the best choice when finding the numerical solutions
by the finite difference method.
For the last experiment (2.3) in this section, we also make the input data more general in order to verify the existence of the asymptotic error expansion for any kinds of input data . We also selected two different two-variable polynomials as the external forcing function. It is obvious that the results completely followed the results in the Experiment (2.2).
As a summary, although the solutions of the problems with concentrated loading have a strong singularity, we verified that the solutions are convergent and the existence of the asymptotic error expansion which makes use of the Richardson extrapolation in accelerating the convergence.