Accuracy of the Discretization Matrix of a Radiative Transfer Problem

The numerical solution, either of a weakly singular Fred Holm integral equation of the second kind or of the spectral problem associated, using projection methods such as classical Galerkin, Kantorovich or Sloan (iterated Galerkin) requires the evaluation of a discretization matrix which represents the integral operator restricted to a finite dimensional space. The accuracy of the approximate solution depends not only on the projection method used but also on the dimension of the discretization subspace, on the basis chosen for this subspace, and on the precision of the evaluation of this discretization matrix. In this work we study the accuracy of the discretization matrix of a particular weakly singular integral operator whose kernel is defined by a first exponential integral function. The discretization of this problem yields formulae for the matrix elements in terms of the third exponential integral function. We discuss different strategies of evaluating this discretization matrix and show its accuracy.