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 $t\rightarrow\infty$.