Projection methods based on grids for weakly singular integral equations

Abstract

For the solution of a weakly singular Fredholm integral equation of the 2nd kind defined on a Banach space, for instance L^1([a,b]), the classical projection methods with the discretization of the approximating operator on a finite dimensional subspace usually use a basis of this subspace built with grids on [a,b]. This may require a large dimension of the subspace. One way to overcome this problem is to include more information in the approximating operator or to compose one classical method with one step o iterative refinement. This is the case of Kulkarni method or iterated Kantorovich method. Here we compare these methods in terms of accuracy and arithmetic workload. A theorem stating comparable error bounds for these methods, under very weak assumptions on the kernel, the solution and the space where the problem is set, is given.