$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Treating the right-hand side; Trapezoidal rule
$$ (f,\basphi_i) = \int_{\xno{i-1}}^{\xno{i}} f(x)\frac{1}{h} (x - \xno{i-1}) dx
+ \int_{\xno{i}}^{\xno{i+1}} f(x)\frac{1}{h}(1 - (x - x_{i})) dx
$$
Cannot do much unless we specialize \( f \) or use numerical integration.
Trapezoidal rule using the nodes:
$$ (f,\basphi_i) = \int_\Omega f\basphi_i dx\approx h\half(
f(\xno{0})\basphi_i(\xno{0}) + f(\xno{N})\basphi_i(\xno{N}))
+ h\sum_{j=1}^{N-1} f(\xno{j})\basphi_i(\xno{j})
$$
\( \basphi_i(\xno{j})=\delta_{ij} \), so this formula collapses to one term:
$$
(f,\basphi_i) \approx hf(\xno{i}),\quad i=1,\ldots,N-1\thinspace.
$$
Same result as in collocation (interpolation)
and the finite difference method!
