$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Relevance of this stationary 1D problem
1. As stationary limit of a diffusion PDE
$$ u_t = (\alpha(u)u_x)_x + au + f(u) $$
(\( u_t\rightarrow 0 \))
2. The time-discrete problem at each time level arising from a Backward Euler scheme for a diffusion PDE
$$ u_t = (\alpha(u)u_x)_x + f(u) $$
(\( au \) comes from \( u_t \), \( a\sim 1/\Delta t \), \( f(u) := f(u) - u^{n-1}/\Delta t \))
