$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Boundary function
$$ u^n(\x) = u_0(\x,t_n) + \sum_{j\in\If}c_j^n\baspsi_j(\x)$$
$$
\begin{align*}
\sum_{j\in\If} & \left(\int_\Omega \baspsi_i\baspsi_j\dx\right)
c^{n+1}_j = \sum_{j\in\If}
\left(\int_\Omega\left( \baspsi_i\baspsi_j -
\Delta t\dfc\nabla \baspsi_i\cdot\nabla\baspsi_j\right)\dx\right) c_j^n - \\
&\quad \int_\Omega\left( u_0(\x,t_{n+1}) - u_0(\x,t_n)
+ \Delta t\dfc\nabla u_0(\x,t_n)\cdot\nabla
\baspsi_i\right)\dx \\
& \quad + \Delta t\int_\Omega f\baspsi_i\dx -
\Delta t\int_{\partial\Omega_N} g\baspsi_i\ds,
\quad i\in\If
\end{align*}
$$
