A numerical method for two-dimensional Hammerstein integral equations

Abstract

In this paper we investigate a collocation method  for the approximate solution of Hammerstein integral equations in two dimensions. We start with a special type of piecewise linear interpolation over triangles for a reformulation of the equation. This leads to a numerical integration scheme that can then be extended to a domain in $\r^2$, which is used in collocation. We analyze and prove the convergence of the method and give error estimates. Because the quadrature formula has a higher degree of precision than expected with linear interpolation, the resulting collocation method is superconvergent at the nodes. We show the applicability of the proposed scheme on a numerical example and discuss future research ideas in this area.