$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Incorporation of the Neumann condition in the variational formulation
Note: \( v\neq 0 \) only on \( \partial\Omega_N \) (since \( v=0 \) on \( \partial\Omega_D \)):
$$ \int_{\partial\Omega} a\frac{\partial u}{\partial n} v\ds
= \int_{\partial\Omega_N} \underbrace{a\frac{\partial u}{\partial n}}_{-g} v\ds
= -\int_{\partial\Omega_N} gv\ds
$$
The final variational form:
$$
\int_{\Omega} (\v\cdot\nabla u + \alpha u)v\dx =
-\int_{\Omega} a\nabla u\cdot\nabla v \dx
- \int_{\partial\Omega_N} g v\ds
+ \int_{\Omega} fv \dx
$$
Or with inner product notation:
$$
(\v\cdot\nabla u, v) + (\alpha u,v) =
- (a\nabla u,\nabla v) - (g,v)_{N} + (f,v)
$$
\( (g,v)_{N} \): line or surface integral over \( \partial\Omega_N \).
