When considering a weakly singular Fredholm integral equation of the 2nd kind  using projection methods,  the evaluation of a discretization matrix A_n, which represents the integral operator $T_n$ restricted to a finite dimensional space $X_n$, is required.

The precision of the approximate solution depends, not only on the projection method used, but also on dimension of the discretization subspace,
on the basis of this subspace, and on the precision of the evaluation of this discretization matrix.

The choice of the basis must take in account the properties of the space where the problem is set, and the discontinuities of the kernel and of the source term $f$.

We study cases where the basis of  $X_n $ should be as simple as possible, but based on a grid that can include the discontinuities of the kernel,  or the regions where boundary layers can occur, near the boundaries of the domain when f has a discontinuity.

So nonuniform grids are interesting since for a given n we may obtain a better approximation by distributing those points in a nonuniform way.