  In this paper we derive the fourth-order asymptotic expansions of the
trapezoidal approximation for the fractional integral
  and the $L1$ approximation for the Caputo derivative. We use the expansion of
the $L1$ approximation to obtain the three point compact approximation for the
Caputo derivative \begin{equation*}
\dfrac{1}{\Gamma(2-\alpha)h^\alpha}\sum_{k=0}^{n} \delta_k^{(\alpha)}
y_{n-k}=\dfrac{13}{12}y^{(\alpha)}_n-\dfrac{1}{6}y^{(\alpha)}_{n-1}+\dfrac{1}{12}y^{(\alpha)}_{n-2}+O\left(h^{3-\alpha}\right),
\end{equation*} with weights $\delta_0^{(\alpha)}=1-\zeta(\alpha-1),\;
\delta_n^{(\alpha)}=(n-1)^{1 -\alpha}-n^{1-\alpha},$ $$
\delta_1^{(\alpha)}=2^{1-\alpha}-2+2\zeta(\alpha-1),\;
\delta_2^{(\alpha)}=1-2^{2-\alpha}+3^{1-\alpha}-\zeta(\alpha-1),$$
$$\delta_k^{(\alpha)}=(k-1)^{1-\alpha}-2k^{1-a}+(k+1)^{1-\alpha},\quad
(k=3\cdots,n-1),$$ where $y$ is a differentiable function which satisfies
$y'(0)=0$. The numerical solutions of the fractional relaxation and the
time-fractional subdiffusion equations are discussed.
