Superconvergence in collocation methods for Volterra integral equations with vanishing delays

In this paper, we investigate the optimal (global and local) convergence orders of the (iterated) collocation solutions for second-kind Volterra integral equations with vanishing delays on quasi-geometric meshes. It turns out that the classical global convergence results still hold under certain regularity conditions of the given functions. In particular, it is shown that the optimal local superconvergence order p=2mp=2m can be attained if collocation is at the mm Gauss(–Legendre) points, which contrasts with collocations both on uniform meshes and on geometric meshes. Numerical experiments are performed to confirm our theoretical results.