I am trying to solve a nonlinear poisson equation of the form:
$u_{xx} + f(u_y)u_{yy} = 0$.
I would like to use Newton's method to handle the nonlinearity, however I am not sure when/how to discretize the differential terms.
If I identify $F[u]$ with the LHS of the above I could immediately write down (where n represents iterations):
$u^{n+1} = u^n - F'(u^n)^{-1}F[u]$
However, I am not sure how to proceed from here. I can write down a discretized version of $F$ but I do not know how to write down $F'^{-1}$.