Optimal superconvergence results for Volterra functional integral equations with proportional vanishing delays

In this paper, we develop a new technique to study the optimal convergence orders of collocation methods for Volterra functional integral equations with vanishing delays on quasi-geometric meshes. Basing on a perturbation analysis, we show that for m collocation points, the global convergence order of the collocation solution is only m. However, the collocation solution may exhibit superconvergence with order m+1 at the collocation points. In particular, the local convergence order may attain 2m−1 at the nodes, provided that the collocation is based on the m Radau II points. Finally, some numerical examples are performed to verify our theoretical results.