ABSTRACT
The quantitative and qualitative theory of the difference equations associated with collocation methods for the numerical solution of initial-value problems for ordinary differential equations in piecewise polynomial spaces is now reasonably complete. While this is, to a somewhat lesser extent, also true for Volterra integral equations with smooth kernels, the analogous analysis of (systems of) Volterra difference equations arising in collocation methods for Volterra integral equations whose kernels contain an integrable (= weak) singularity of algebraic or logarithmic type appears to be significantly more difficult and is as yet not well understood.
In this paper I shall describe some of the recent and current work, as well as a number of important open problems, in this field. The discussion will focus on questions concerning the connection between the location of the collocation points and the (quantitative and qualitative) properties of the corresponding collocation difference equations, especially for weakly singular Volterra integral equations of the first kind.
AMS subject classification: 65R20, 45L05, 39A10, 39A11, 39A12
