I'm asking here cause I have a doubt about approximate solution of this problem:
\begin{equation} \begin{cases}-\epsilon u''+bu'=0 \\ u(0) = 0, u(1)=1 \end{cases} \end{equation}
which is a diffusion transport problem. I found the approximate solution using FEM in Matlab, and I tested for $\epsilon = 10^{-1}$ and $\epsilon = 10^{-3}$. In the second case I have the convection term $b$ dominating over diffusion term $\epsilon$. So I'm using an Upwind stabilization scheme by substituiting $\epsilon$ with $\epsilon_h = \epsilon(1+\psi(Pe))$ where $\psi(t) = t$ and $Pe$ is Pèclet number ($\frac{bh}{2\epsilon}$).
My question is, should I substitute $\epsilon$ with $\epsilon_h$ just in my approximation I find using FEM, so basically when I assemble the final matrix ($A= \epsilon_h*K + b*H$) or even in the original problem? (I mean when I find the exact solution at the beginning that I will compare to approximate solution, do I have to consider $\epsilon$ or $\epsilon_h$?).
Thank you in advance for your help!
You only have to substitute in the matrix assembly because if you do the substitution in the original problem then it doesn't remain the diffusion transport problem.