Pproduct integration-collocation methods for noncompact integral operator equations

We discuss the numerical solution of a class of second-kind integral equations in which the integral operator is not compact. Such equations arise, for example, when boundary integral methods are applied to potential problems in a two-dimensional domain with corners in the boundary. We are able to prove the optimal orders of convergence for the usual collocation and product integration methods on graded meshes, provided some simple modifications are made to the underlying basis functions. These are sufficient to ensure stability, but do not damage the rate of convergence. Numerical experiments show that such modifications are necessary in certain circumstances.