The Fine Error Estimation of Collocation Methods on Uniform Meshes for Weakly Singular Volterra Integral Equations

Abstract

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.