$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Computations in a reference cell \( [-1,1] \)
$$
\begin{align*}
\tilde F_r^{(e)} &=
\int_{-1}^1\left(
\dfc(\tilde u^{-})\tilde u^{-\prime}\refphi_r^{\prime} +
(a-f(\tilde u^{-}))\refphi_r\right)\det J\dX -
C\refphi_r(0)
\\
\tilde J_{r,s}^{(e)} &=
\int_{-1}^1
(\dfc^{\prime}(\tilde u^{-})\tilde u^{-\prime}\refphi_r^{\prime}\refphi_s +
\dfc(\tilde u^{-})\refphi_r^{\prime}\refphi_s^{\prime}
+ (a - f(\tilde u^{-}))\refphi_r\refphi_s)\det J\dX
\end{align*}
$$
\( r,s\in\Ifd \) (local degrees of freedom)
