It is well known that for the second-kind Volterra integral equations (VIEs) with weakly singular kernel, if we use piecewise polynomial collocation methods of degree m to solve it numerically, due to the weak singularity of the solution at the initial time \(t = 0\), only \(1 - \alpha \) global convergence order can be obtained on uniform meshes, comparing with m global convergence order for VIEs with smooth kernel. However, in this paper, we will see that at mesh points, the convergence order can be improved, and it is better and better as n increasing. In particular, 1 order can be recovered for \(m = 1\) at the endpoint. Some superconvergence results are obtained for iterated collocation methods, and a representative numerical example is presented to illustrate the obtained theoretical results.

众所周知，对于第二种Volterra积分方程（VIES）与弱奇异核，如果我们使用的程度分段多项式搭配方法米来解决它数值，由于溶液的在初始时间的弱奇异\（ t = 0 \），在均匀网格上只能获得\（1-\ alpha \）全局收敛阶，与具有光滑核的VIE的m全局收敛阶相比。但是，在本文中，我们将看到在网格点处，收敛阶数可以得到改善，并且随着n的增加而越来越好。特别是，可以针对\（m = 1 \）恢复1个订单在端点。对于迭代配置方法，获得了一些超收敛结果，并给出了一个有代表性的数值例子来说明所获得的理论结果。