A popular class of methods for solving weakly singular integral equations is the class of piecewise polynomial collocation methods. In order to implement those methods one has to compute exactly certain integrals that determine the linear system to be solved. Unfortunately those integrals usually cannot be computed exactly and even when analytic formulas exist, their straightforward application may cause unacceptable roundoff errors resulting in apparent instability of those methods in the case of highly nonuniform grids. In this paper fully discrete analogs of the collocation methods, where integrals are replaced by quadrature formulas, are considered, corresponding error estimates are derived.
Kangro, R., & Kangro, I. (2009). On fully discrete collocation methods for solving weakly singular integral equations. Mathematical Modelling and Analysis, 14(1), 69-78. https://doi.org/10.3846/1392-6292.2009.14.69-78
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