$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\dfc}{\alpha} % diffusion coefficient
$$
We can solve 1D Poisson/Laplace equation by going to infinity in time-dependent diffusion equations
Looking at the numerical schemes, \( F\rightarrow\infty \) leads to the
Laplace or Poisson equations (without \( f \) or with \( f \), resp.).
Good news: choose \( F \) large in the BE or CN schemes and one
time step is enough to produce the stationary solution for
\( t\rightarrow\infty \).
