$$
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\newcommand{\If}{\mathcal{I}_s} % for FEM
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\newcommand{\setl}[1]{#1^-}
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$$
Finite element approximation vs finite differences
With Trapezoidal integration of \( (f,\basphi_i) \), the finite element
metod essentially solve
$$
\begin{equation}
u + \frac{h^2}{6} u'' = f,\quad u'(0)=u'(L)=0,
\end{equation}
$$
by the finite difference method
$$
\begin{equation}
[u + \frac{h^2}{6} D_x D_x u = f]_i
\end{equation}
$$
With Simpson integration of \( (f,\basphi_i) \) we essentially solve
$$
\begin{equation}
[u + \frac{h^2}{6} D_x D_x u = \bar f]_i,
\end{equation}
$$
where
$$ \bar f_i = \frac{1}{3}(f_{i-1/2} + f_i + f_{i+1/2}) $$
Note: as \( h\rightarrow 0 \), \( hu''\rightarrow 0 \) and \( \bar f_i\rightarrow f_i \).