Fast superconvergent solvers for weakly singular Hammerstein equations

This article investigates discrete versions of the projection-type and modified projection-type methods for solving Hammerstein integral equations with a weakly singular kernel. The approximating operator utilized is either the orthogonal projection or an interpolatory projection onto a space of piecewise polynomials of degree <= r - 1 with respect to a graded partition of [0, 1]. The study reveals that the proposed methods achieve optimal rates of convergence, demonstrating that the numerical quadrature used to estimate the integrals maintains the same rates of convergence as the continuous methods. The theoretical findings are supported by numerical experiments.