Considerations on Double Exponential-Based Cubatures for the Computation of Weakly Singular Galerkin Inner Products

Abstract

Abstract-Highly accurate and efficient cubatures based on the double exponential quadrature rules are presented for the computation of weakly singular integrals arising in Galerkin mixed potential integral equation formulations. Due to their unique ability to handle non-smooth kernels, the proposed integration schemes can safely replace (in a "plug-n-play" sense) the traditional Gauss-Legendre rules in the existing singularity cancellation and singularity subtraction methods. Numerical examples using RWG basis functions confirm the excellent performance of the proposed method.Index Terms-Double exponential quadrature, method of moments (MoM), mixed potential integral equation (MPIE), RWG basis functions, weakly singular integrals.

Weighted Averages and Double Exponential Algorithms

“…This function is so fast decreasing that it will "kill" the original singularities at infinity, no matter which type they were in the original integral (25). Despite these facts, transforming a finite integration interval [−1, 1] into an infinite one [−∞, ∞] may not appear as the best possible strategy to simplify numerical calculations.…”

“…The improvent obtained by the optimization of the DE parameter is particularly evident for low numbers of sampling points. After [25].…”

“…Although, as mentioned, the historical choice has been η = π/2 and this choice has given consistently excellent results in onedimensional integrals, nothing prevents to try to optimize this parameter for multidimensional integrals. Indeed an exhaustive study for 4D-integrals of type (31) shows that a lower value η = 0.3π (closer to unity) is a better choice in this case [25], [35]. As a typical example, Fig.…”

“…However, the idea was quickly extended to multidimensional singular integrals and successfully applied to the double surface integrals arising in the Integral-Equation treatment of metallic scatterers when discretized with a Galerkin's method [22]- [25]. A final development of paramount relevance for this paper is the transformation, obtained by T. Ooura, of the DE quadrature into a form that can be applied to Fourier transforms of slowly decaying functions [26], [27].…”

Analytical computation of impedance integrals with power-law Green's functions

“…A disadvantage of this technique is that F (r) can still have non-smooth features, such as the presence of singular derivatives for ν = −1 when V b and V t share a part of their support. A solution to this problem has been proposed in [5], by using a double-exponential-based quadrature rule for the outer integration. This scheme results in a wellcontrolled error, as opposed to when a direct product Gauss-Legendre quadrature rule would be used.…”