Iterated galerkin versus iterated collocation for integral equations of the second kind

We consider the numerical solution of one-dimensional Fredholm integral equations of the second kind by the Galerkin and collocation methods and their iterated variants, using spline bases. In particular, we state and prove new superconvergence results for the iterated solutions, under general smoothness requirements on the kernel and solution. We find that the smoothness requirements for the iterated collocation method are more stringent than those for the iterated Galerkin method, and show by example that these more stringent smoothness conditions are in a certain sense necessary. In the light of these results the Galerkin and collocation schernes are compared.