applying neumann boundary condition

[Alborz2020]

Hello Mr. Rao,
thank you for your nice work, I found it quite inspiring and close to the topic that I am working on. however, the way that you applied Neumann boundary condition is a bit vague to me. Generally speaking, we know that for Dirichlet BC the predefined solution just satisfies the values on the boundaries by having D=0 and T takes the values of Tbc (T = Tbc + D * Tnn). For the Neumann boundary condition (gradT . n = Nbc) the derivatives of T are specified while the value of T is not given but it gets some values that might not be correct. Could you please explain a bit more how the mixed-variables output solved this issue?

best regards,

Alborz