Abstract. Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels.

In the first part of this paper we study the regularity properties of solutions of linear Volterra integro-differential equations with weakly singular or other nonsmooth kernels. We then use these results in the analysis of two piecewise polynomial collocation methods for solving such equations numerically. The main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of local superconvergence at the collocation points.

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We examine the differential properties of the solution of the linear integral equation of the second kind, whose kernel depends on the difference of arguments and has an integrable singularity at the point zero. The derivatives of the solution of the equation have singularities at the end points of the domain of integration, and we derive precise estimates for these singularities.