I am trying to solve convection diffusion type equations by a modification: $$-\epsilon \Delta u+ b \nabla u+cu=f_1,$$ with the following boundary conditions: $$u=g~ on ~\Gamma_D,~~~~~~~~~~~~~~ \epsilon \frac{\partial u}{\partial n}=g_N ~~on ~~\Gamma_N.$$
The idea is to solve another equation: $$-\nu\Delta u+ \nabla u+\frac{c}{b}u=f_2,$$ $$u=g~ on ~\Gamma_D,~~~~~~~~~~~~~~ \epsilon \frac{\partial u}{\partial n}=g_N ~~on ~~\Gamma_N.$$
where $\nu=\frac{\epsilon}{b}$ and then use $\nu+\delta$ instead of $\nu.$ so we should solve $$$$ $$-(\nu+ \delta) \Delta u+ \nabla u+\frac{c}{b}u=f_3,$$ $$u=g~ on ~\Gamma_D,~~~~~~~~~~~~~~ \epsilon \frac{\partial u}{\partial n}=g_N ~~on ~~\Gamma_N.$$ Here $(\nu+\delta)=\frac{h}{2}*coth(\frac{h}{2*\nu})$, where $h$ is the mesh size.
Now the question is that what the relations are between $f_1,f_2,f_3.$ I used $f_2=f_3$ and $f_2=\frac{f_1}{b}$. Is it true?
Also if $b$ be a vector I used $||b||_\infty$, but the results are acceptable just for some examples. Am I wrong in this way?