Another Point I would like to mention is that I checked constant values for 
neumann_value( for example, 0.7) and the solver converges. So I think 
problem comes from the way I compute neumann_value from old_solution.

I has followed step-7, in which non-homogeneous Neumann boundary values are 
applied using exact_solution for obtaining neumann_value. somewhere in 
documentation it is stated that:

"Finally we define an object denoting the exact solution function. We will 
use it to compute the Neumann values at the boundary from it. Usually, one 
would of course do so using a separate object, in particular since the 
exact solution is generally unknown while the Neumann values are 
prescribed. We will, however, be a little bit lazy and use what we already 
have in information. Real-life programs would to go other ways here, of 
course."

So the question is how to use FEM solution to obtain Neumann_values?

Thanks,

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