$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
The nonlinear algebraic equations
Find \( u\in V \) such that
$$
\int_0^L \dfc(u)u^{\prime}v^{\prime}\dx + \int_0^L auv\dx =
\int_0^L f(u)v\dx - Cv(0),\quad \forall v\in V
$$
\( \forall v\in V\ \Rightarrow\ \forall i\in\If \), \( v=\baspsi_i \).
Inserting \( u=D+\sum_jc_j\baspsi_j \)
and sorting terms:
$$
\sum_{j}\left(
\int\limits_0^L \dfc(D+\sum_{k}c_k\baspsi_k)
\baspsi_j^{\prime}\baspsi_i^{\prime}\dx\right)c_j =
\int\limits_0^L f(D+\sum_{k}c_k\baspsi_k)\baspsi_i\dx -
C\baspsi_i(0)
$$
This is a nonlinear algebraic system
