$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Alternative: evaluation of finite element terms at nodes gives great simplifications
Idea: integrate \( \int f(u)v\dx \) numerically with a rule that samples \( f(u)v \)
at the nodes only. This involves great simplifications, since
$$ \sum_k u_k\basphi_k(\xno{\ell}) = u_\ell$$
and
$$ f\basphi_i(\xno{\ell}) =
f(\sum_k u_k\underbrace{\basphi_k(\xno{\ell})}_{\delta_{k\ell}})
\underbrace{\basphi_i(\xno{\ell})}_{\delta_{i\ell}}
= f(u_\ell)\delta_{i\ell}\quad \neq 0\mbox{ only for } f(u_i)$$
(\( \delta_{ij}=0 \) if \( i\neq j \) and \( \delta_{ij}=1 \) if \( i=j \))
