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

As is well known, piecewise linear polynomial collocation (PLC) and piecewise quadratic polynomial collocation (PQC) are used to approximate the weakly singular integrals $$\begin{equation*}I(a,b,x) =\int^b_a \frac{u(y)}{|x-y|^\gamma}\textrm{d}y, \quad x \in (a,b),\quad 0< \gamma <1,\end{equation*}$$which have local truncation errors $\mathcal{O} (h^2 )$ and $\mathcal{O} (h^{4-\gamma } )$, respectively. Moreover, for Fredholm weakly singular integral equations of the second kind, i.e., $\lambda u(x)- I(a,b,x) =f(x)$, $\lambda \neq 0$, the global convergence rates are also $\mathcal{O} (h^2 )$ and $\mathcal{O} (h^{4-\gamma } )$ by PLC and PQC in Atkinson (2009, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press). In this work we study the following nonlocal problems, which are similar to the above Fredholm integral equations: $$\begin{equation*}\int^b_a \frac{u(x)-u(y)}{|x-y|^\gamma}\textrm{d}y =f(x), \quad x \in (a,b),\quad 0< \gamma <1. \end{equation*}$$In the first part of this paper we prove that the weakly singular integrals $I(a,b,x)$ have optimal local truncation error $\mathcal{O}(h^4\eta _i^{-\gamma } )$ by PQC, where $\eta _i=\min \left \{x_i-a,b-x_i\right \}$ and $x_i$ coincides with an element junction point. Then the sharp global convergence orders $\mathcal{O}\left (h\right )$ and $\mathcal{O} (h^3)$ by PLC and PQC, respectively, are established for nonlocal problems. Finally, numerical experiments are shown to illustrate the effectiveness of the presented methods.