Superconvergent methods for solving two-dimensional Hammerstein integral equations

Abstract

In this paper, we introduce the superconvergent degenerate kernel method and the superconvergent Nyström method for the numerical solution of two-dimensional Hammerstein integral equations of the second kind. By employing piecewise polynomial interpolation of degree $r$, we prove that, under symmetry conditions on both the triangulation and the interpolation nodes, convergence orders of $2r+3$ and $2r+4$ are achieved for the approximate solutions and their iterated versions, respectively. Furthermore, we discuss computational aspects related to the construction of the corresponding nonlinear systems, and we present numerical examples to illustrate the theoretical results obtained.