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?


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