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As is well known, using piecewise linear polynomial collocation (PLC) andpiecewise quadratic polynomial collocation (PQC), respectively, to approximatethe 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 globalconvergence rate $\mathcal{O}\left(h^2\right)$ and$\mathcal{O}\left(h^{4-\gamma}\right)$ in [Atkinson and Han, TheoreticalNumerical Analysis, Springer, 2009].  Formally, following nonlocal models can be viewed as Fredholm weakly singularintegral 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 significantdifferences for the models in these two fields. In the first part of this paperwe prove that the weakly singular integral by PQC have an optimal localtruncation 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 elementjunction 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 numericalexperiments including two-dimensional case are given to illustrate theeffectiveness of the presented method.

A sharp error estimate of piecewise polynomial collocation for nonlocal problems with weakly singular kernels