I have the following eliptic 1-D problem $$-\mu u'' + \beta u' = 1$$ $$u(0) = u'(1) = 1$$ where $\mu = 10e^{-5}$ and $\beta = 1$. For this specific problem I am using the following space steps $h=[0.1, 0.01]$
What surprises me is that this problem doesn't present oscillations. I think it is related to the Péclet number, $\mathrm{Pe}$. When the problem had a source term (right-hand side function) equals to zero I know that the Péclet number should be less than 1 to avoid this oscillations.
Is this lack of oscillations related to the local Péclet number of my problem?
I tried to use finite differences to see if I can find why this does not present oscillations but I don't know how to proceed from here
$$(\textrm{Pe}-1)u_{i+1} + 2u_i - (\textrm{Pe}+1)u_{i-1} = \frac{h^2}{\mu}$$
Any ideas on how to proceed or if I'm actually in the right direction?