$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Variable coefficient; linear system (full derivation)
\( v=\baspsi_i \) and \( u=B + \sum_jc_j\baspsi_j \):
$$
(\dfc B' + \dfc \sum_{j\in\If} c_j \baspsi_j', \baspsi_i') =
(f,\baspsi_i), \quad i\in\If
$$
Reorder to form linear system:
$$ \sum_{j\in\If} (\dfc\baspsi_j', \baspsi_i')c_j =
(f,\baspsi_i) + (aL^{-1}(D-C), \baspsi_i'), \quad i\in\If
$$
This is \( \sum_j A_{i,j}c_j=b_i \) with
$$
\begin{align*}
A_{i,j} &= (a\baspsi_j', \baspsi_i') = \int_{\Omega} \dfc(x)\baspsi_j'(x)
\baspsi_i'(x)\dx\\
b_i &= (f,\baspsi_i) + (aL^{-1}(D-C),\baspsi_i')=
\int_{\Omega} \left(f\baspsi_i + \dfc\frac{D-C}{L}\baspsi_i'\right) \dx
\end{align*}
$$
