$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
First-order derivative in the equation and boundary condition; linear system
Insert \( u=C+\sum_jc_j\baspsi_j \) and \( v=\baspsi_i \) in
$$ (u'v') + (bu',v) = (f,v) + Ev(L), \quad\forall v\in V$$
and manipulate to get
$$
\sum_{j\in\If}
\underbrace{((\baspsi_j',\baspsi_i') + (b\baspsi_j',\baspsi_i))}_{A_{i,j}}
c_j =
\underbrace{(f,\baspsi_i) + E \baspsi_i(L)}_{b_i},\quad i\in\If
$$
Observation: \( A_{i,j} \) is not symmetric because of the term
$$
(b\baspsi_j',\baspsi_i)=\int_{\Omega} b\baspsi_j'\baspsi_i dx
\neq \int_{\Omega} b \baspsi_i' \baspsi_jdx = (\baspsi_i',b\baspsi_j)
$$
