  As is well known, using piecewise linear polynomial collocation (PLC) and
piecewise quadratic polynomial collocation (PQC), respectively, to approximate
the weakly singular integral $$I(a,b,x) =\int^b_a \frac{u(y)}{|x-y|^\gamma}dy,
\quad x \in (a,b) ,\quad 0< \gamma <1,$$ have the local truncation error
$\mathcal{O}\left(h^2\right)$ and $\mathcal{O}\left(h^{4-\gamma}\right)$.
Moreover, for Fredholm weakly singular integral equations of the second kind,
i.e., $\lambda u(x)- I(a,b,x) =f(x)$ with $ \lambda \neq 0$, also have global
convergence rate $\mathcal{O}\left(h^2\right)$ and
$\mathcal{O}\left(h^{4-\gamma}\right)$ in [Atkinson and Han, Theoretical
Numerical Analysis, Springer, 2009].
  Formally, following nonlocal models can be viewed as Fredholm weakly singular
integral equations $$\int^b_a \frac{u(x)-u(y)}{|x-y|^\gamma}dy =f(x), \quad x
\in (a,b) ,\quad 0< \gamma <1.$$ However, there are still some significant
differences for the models in these two fields. In the first part of this paper
we prove that the weakly singular integral by PQC have an optimal local
truncation error $\mathcal{O}\left(h^4\eta_i^{-\gamma}\right)$, where
$\eta_i=\min\left\{x_i-a,b-x_i\right\}$ and $x_i$ coincides with an element
junction point. Then a sharp global convergence estimate with
$\mathcal{O}\left(h\right)$ and $\mathcal{O}\left(h^3\right)$ by PLC and PQC,
respectively, are established for nonlocal problems. Finally, the numerical
experiments including two-dimensional case are given to illustrate the
effectiveness of the presented method.
