# When to discretize nonlinear Poisson Equation

I am trying to solve a nonlinear poisson equation of the form:

$$u_{xx} + f(u_y)u_{yy} = 0$$.

In trying to get a handle on this problem, it seems like there are two approaches. I could either discretize this directly, and end up with a nonlinear system of equations for $$u_{i,j}$$ or I could apply newtons method first and then discretize. Which is preferable and why?

• For many of the "simple" discretizations i.e. Galerkin FEM, finite differences, the order does not matter and you should get the same system of equations in the end unless your $f$ is very nasty. – whpowell96 May 18 '20 at 0:41
• What is the domain, general or a simple form like rectangle? What form is $f(u_y)$? What are the boundary conditions? Is the right-hand-side zero? With some knowledge of this one could perhaps find some shortcuts here. – Maxim Umansky May 18 '20 at 15:56