ABSTRACT
Second kind Fredholm integral equations with weakly singular kernels typically have solutions that are nonsmooth near the boundary of integration. On the basis of certain regularity properties of the exact solution, this chapter discusses the piecewise polynomial collocation and Galerkin methods on graded grids to solve nonlinear multidimensional weakly singular integral equations. The superconvergence effect at collocation points for the collocation method is studied and global convergence estimates for Galerkin approximations are derived. The chapter presents a theorem which shows how to choose scaling parameter and collocation points so that for this method the superconvergence phenomenon at collocation points would take place.
