$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Non-homogeneous Neumann conditions
A natural physical flux condition:
$$
-\dfc(u)\frac{\partial u}{\partial n} = g,\quad\x\in\partial\Omega_N
$$
Integration by parts gives the boundary term
$$
\int_{\partial\Omega_N}\dfc(u)\frac{\partial u}{\partial u}v\ds
$$
Inserting the nonlinear Neumann condition:
$$ -\int_{\partial\Omega_N}gv\ds$$
(no nonlinearity)
