$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Numerical integration of nonlinear terms
Trapezoidal rule with the nodes only gives the finite difference form of \( [f(u)]_i \):
$$
\int_0^L f(\sum_k u_k\basphi_k)(x)\basphi_i(x)\dx
\approx h\sum_{\ell=0}^{N_n-1} f(u_\ell)\delta_{i\ell} - \mathcal{C}
= h{\color{red}f(u_i)}
$$
(\( \mathcal{C} \): boundary adjustment of rule, \( i=0,N_n-1 \))
