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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?

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    $\begingroup$ 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. $\endgroup$ – whpowell96 May 18 '20 at 0:41
  • $\begingroup$ 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. $\endgroup$ – Maxim Umansky May 18 '20 at 15:56

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