$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Application of the group finite element method
$$ \int_0^L f(u)\basphi_i\dx \approx
\int_0^L (\sum_j \basphi_jf(u_j))\basphi_i\dx
= \sum_j (\underbrace{\int_0^L \basphi_i\basphi_j\dx}_{\mbox{mass matrix }M_{i,j}}) f(u_j)$$
Corresponding part of difference equation for P1 elements:
$$ \frac{h}{6}(f(u_{i-1}) + 4f(u_i) + f(u_{i+1}))$$
Rewrite as "finite difference form plus something":
$$ \frac{h}{6}(f(u_{i-1}) + 4f(u_i) + f(u_{i+1}))
= h[{\color{red}f(u)} - \frac{h^2}{6}D_xD_x f(u)]_i$$
This is like the finite difference discretization of
\( -u'' = f(u) - \frac{h^2}{6}f''(u) \)
