$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
The group finite element method
Since \( u \) is represented by \( \sum_j\basphi_j u(\xno{j}) \), we may use the
same approximation for \( f(u) \):
$$
f(u)\approx \sum_{j} f(\xno{j})\basphi_j
$$
\( f(\xno{j}) \): value of \( f \) at node \( j \). With
\( u_j \) as \( u(\xno{j}) \), we can write
$$
f(u)\approx \sum_{j} f(u_{j})\basphi_j
$$
This approximation is known as the group finite element method
or the product approximation technique. The index \( j \) runs over
all node numbers in the mesh.
