A parametric study of the Double Exponential algorithm utilized in weakly singular integrals

The Double Exponential (DE) quadrature rule is modified in order to efficiently integrate the observation domain of the 4-D weakly singular integrals arising in Mixed Potential Integral Equation (MPIE) formulations. Although, the original DE rule already guarantees numerically exact results, it results in poor convergence when compared to widely used interpolatory quadratures like Gauss Legendre (GL), in the cases in which only a few sampling points are considered. The proposed modification, based on the parametrization of the DE transformation, overcomes this weakness: it achieves higher accuracy for a small number of sampling points without additional computational effort while for a large number of evaluation points the behavior of the original DE is recovered. Furthermore, the universality of the proposed technique is outlined, demonstrating that it is satisfactorily applicable to a vast variety of source and observation domains with different geometrical characteristics.