Piecewise polynomial collocation for fredholm integro-differential equations with weakly singular kernels

In the first part of this paper we study the regularity properties of solutions of initial-or boundary-value problems of linear Fredholm integro-differential equations with weakly singular or other nonsmooth kernels. We then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically. The main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of convergence of numerical solutions for all values of the nonuniformity parameter of the underlying grid.