In this report, both one-dimensional and two-dimensional problems will be considered as the numerical experiments to investigate the convergence of the numerical solutions. Moreover, by assuming the existence of the error asymptotic expansions at each node point, the convergence order can be evaluated by using the uniform refinement meshes. In one- dimensional problem, we will consider the following differential operator as the experiment
where f(x) is the external forcing function. In the experiments, a continuous external forcing function will be worked out to verify the theoretical result in the survey paper [1], and then the concentrated loading and jump discontinuous input data will be taken as the examples in the experiment, trying to obtain the similar error asymptotic expansions. Afterward, a Poisson equation will be used for the two-dimensional experiment
where f(x,y) is also the external forcing function
and a square region will be chosen as the domain 
in this report. The same as the one-dimensional problem, the concentrated
loading and jump discontinuous input data will also be taken as the examples
in the experiment, however, we will try to obtain the error asymptotic
expansions at different node points.
In addition, Green's function and Fourier series will also be used to
find the exact solutions those will be compared with the numerical solutions.
And the numerical solutions are obtained by the method of finite difference
from the experiments in this study. Therefore, some definitions and basic
results will be presented in the next section and will be used later.