Can Nonlinear PDEs Be Solved with Newton's Method in n-Dimensional Domains?

[sunon77]

For a general dynamic system: dXi/dt = Fi(X1, X2,...,Xn), i=1,...,n,
Q.1
do you have some ideas of the existence conditions of following PDE:
a) (grad U, grad U + F) = 0 in n-dimension domain, (,) is inner product;
b) U >=0

Does it need a first type or second type of boundary condition?

Q.2
If solution does exist, how to solve it numerically?
- I tried to write in differential equation in matrix format, then apply Newton method. It is very difficult to have a converge scheme.

A more detailed simple example is attached to explain the solution that I tried.

Many thanks for your ideas or comments

 

[Eynstone]

I despair of finding a general convergence scheme as convergence is very sensitive to the nature of Fi's.