Compatibility of Collocation Error Analyses for Volterra Integral Equations with Smooth and Weakly Singular Kernels

The numerical solution by piecewise polynomial collocation and iterated collocation of Volterra integral equations (VIEs) of the second kind has been extensively studied and apparently sharp convergence results are known for the cases of a smooth kernel K(t, s) and a weakly singular kernel (t-s)-αK(t,s), where α∈(0,1) is a parameter. If one takes the formal limit as α→0, then the weakly singular VIE reduces to the smooth VIE, but the known collocation error bounds for the weakly singular VIE do not become the collocation error bounds for the smooth VIE — the error bounds for the smooth VIE are typically of a higher order. In the current paper this anomaly is explained and new sharper collocation and iterated collocation error bounds are derived for the weakly singular VIE that blend exactly as α→0 with known error bounds for the smooth VIE. This analysis is substantially different from previous VIE collocation analyses, e.g., it constructs a remarkable new decomposition of the solution of the weakly singular VIE, it investigates in detail the dependence on α of the matrices associated with collocation, it establishes a new Gronwall inequality, and the dependence of the error on the parameter α is traced precisely throughout the work. Numerical experiments are presented to illustrate our theoretical results.