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

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.