$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Treating the right-hand side; Simpson's rule
$$ \int_\Omega g(x)dx \approx \frac{h}{6}\left( g(\xno{0}) +
2\sum_{j=1}^{N-1} g(\xno{j})
+ 4\sum_{j=0}^{N-1} g(\xno{j+\half}) + f(\xno{2N})\right),
$$
Our case: \( g=f\basphi_i \). The sums collapse because \( \basphi_i=0 \) at most of
the points.
$$
(f,\basphi_i) \approx \frac{h}{3}(f_{i-\half} + f_i + f_{i+\half})
$$
Conclusions:
- While the finite difference method just samples \( f \) at \( x_i \),
the finite element method applies an average (smoothing) of \( f \) around \( x_i \)
- On the left-hand side we have a term \( \sim hu'' \), and \( u'' \)
also contribute to smoothing
- There is some inherent smoothing in the finite element
method
