Three steps of pde numerical solution and nonlinear equation

Question

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:

1. Choosing a proper approximation method (FEM, FVM, FDM etc.).
2. Linearization of the resulting system of nonlinear algebraic equations.
3. 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?

Answer 1

Linearization and discretization can be switched. The linearization then happens in function space, where a Newton method can be employed based on the Fréchet derivative of the differential operator.

For each Newton iteration, a linear problem needs to be solved. In your case, this problem may be elliptic (depending on $n$). For actual computation, this linear problem needs to be discretized using the method of your choice (probably FEM if the problem is elliptic).

Since the Newton iteration now happens in function space, each linear problem could be solved on its own individual mesh, provided mesh-transfer operators (e.g., interpolation) from the previous mesh are available.

For each Newton step, after spatial discretization, a linear equation system needs to be solved. This step appears last and cannot be switched around.

See Step 15 of the deal.ii tutorial for a worked out example of this approach.

Answer 2

You can't invert steps 1 and 2. Because different numerical methods have different linearization forms.