# Three steps of pde numerical solution and nonlinear equation

I'm very new here. I'm trying to solve nonlinear elliptic equation $$(n(u)u')' = f(u)$$ and face with crucial misunderstanding. As I suppose, the procedure of solving some nonlinear equation consist of:

1. Choosing of proper approximation method (FEM, FVM, FDM etc.)
2. Linearization of system of nonlinear algebraic equations
3. Choosing of proper method for solving algebraic system

But, what if i wanna linearize pde first, and only after that i perform some discretization? Do this steps connected to each other, or it is absolutely doesn't matter? Each step is important?

What should i do if my nonlinear diffusion coefficient $n(u)$ changes in $10^{15}$ times in very little area? What should i do if it could have discontinuities, but my variable is continues?