Solving a Boundary Value Problem: Non-Uniform vs. Uniform Partitioning

[evinda]

Hello! (Wave)Consider the boundary value problem
$\left\{\begin{matrix}
- \epsilon u''+u'=1 &, x \in [0,1] \\
u(0)=u(1)=0 &
\end{matrix}\right.$
where $\epsilon$ is a positive given constant.
I have to express a finite difference method for its numerical solution.
How can we know whether it is better to use non-uniform partition or uniform partition?

 

[Ackbach]

Well, you can always do the experiment and see. This BVP has an exact answer that's not too difficult to obtain. If you do the substitution $v=u'$, you get a first-order linear DE, $v'-v/\varepsilon=-1/\varepsilon$, with solution
$$v=1+C_1 e^{x/\varepsilon}.$$
Integrating once yields
$$u=x+\varepsilon C_1 e^{x/\varepsilon}+C_2.$$
Applying the boundary conditions yields the system
\begin{align*}
\varepsilon C_1+C_2&=0 \\
\varepsilon C_1 e^{1/\varepsilon}+C_2&=-1,
\end{align*}
with solution
\begin{align*}
C_1&=\frac{1}{\varepsilon(1-e^{1/\varepsilon})} \\
C_2&=\frac{1}{e^{1/\varepsilon}-1}.
\end{align*}
Hence, the exact solution is
$$u(x)=x+\frac{e^{x/\varepsilon}-1}{1-e^{1/\varepsilon}}.$$
You can use this to compare how good the two numerical solutions are.