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 Pclet number, $\mathrm{Pe}$. When the problem had a source term (right-hand side function) equals to zero I know that the Pclet number should be less than 1 to avoid this oscillations.Is this lack of oscillations related to the local Pclet 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?
FEM oscillations for polynomials of degree 1
finite element;finite difference