I'm trying to solve a nonlinear elliptic equation $$(n(u)u')' = f(u)$$ and have a crucial misunderstanding.
I suppose the procedure of solving some nonlinear equation consists of:
- Choosing a proper approximation method (FEM, FVM, FDM etc.).
- Linearization of the resulting system of nonlinear algebraic equations.
- Choosing a proper method for solving the algebraic system.
Is it possible to invert steps 1 and 2, i.e. can I linearize the PDE first, and only after that perform some discretization? Are all of these steps strictly necessary and do they need to be in this order?