A modified iterated projection method adapted to a nonlinear integral equation

The classical way to tackle a nonlinear Fredholm integral equation of the second kind is to adapt the discretization scheme from the linear case. The Iterated projection method is a popular method since it shows, in most cases, superconvergence and it is easy to implement. The problem is that the accuracy of the approximation is limited by the mesh size discretization. Better approximations can only be achieved for fine discretizations and the size of the linear system to be solved then becomes very large: its dimension grows up with an order proportional to the square of the mesh size. In order to overcome this difficulty, we propose a novel approach to first linearize the nonlinear equation by a Newton-type method and only then to apply the Iterated projection method to each of the linear equations issued from the Newton method. We prove that, for any value (large enough) of the discretization parameter, the approximation tends to the exact solution when the number of Newton iterations tends to infinity, so that we can attain any desired accuracy. Numerical experiments confirm this theoretical result.