A numerical method for two-dimensional Hammerstein integral equations

In: Studia Universitatis Babeş-Bolyai. Mathematica, Volume 66, Issue 2, p. 267-277

Abstract

"In this paper we investigate a collocation method for the approximate
solution of Hammerstein integral equations in two dimensions. As in [8], col-
location is applied to a reformulation of the equation in a new unknown, thus
reducing the computational cost and simplifying the implementation. We start
with a special type of piecewise linear interpolation over triangles for a refor-
mulation of the equation. This leads to a numerical integration scheme that can
then be extended to any bounded domain in R2, which is used in collocation. We
analyze and prove the convergence of the method and give error estimates. As
the quadrature formula has a higher degree of precision than expected with linear
interpolation, the resulting collocation method is superconvergent, thus requiring
fewer iterations for a desired accuracy. We show the applicability of the proposed
scheme on numerical examples and discuss future research ideas in this area."