Analysis and estimation of quadrature errors in weakly singular source integrals of the method of moments

Journal article, 2018

The method of moments (MoM) is used for the numerical solution of electromagnetic field integral equations. Weakly singular integrals over surfaces in 3 dimensions (3D) are routinely evaluated for the impedance matrix setup and for post-processing. Available numerical integration schemes range from direct application of Gauss-Legendre product-rule quadrature, to singularity and near-singularity cancellation, coordinate transformation schemes. This paper presents a general, explicit, pole-based, a priori procedure to estimate quadrature errors in the numerical evaluation of weakly singular and near-singular, 3D surface integrals in the MoM. It is based on an error theorem for linear Gaussian quadrature, which involves the analytic extension of the integrand into the complex plane. Errors are linked to poles in the complex plane. New closed-form estimates are presented for direct Gaussian product-rule integration, polar-coordinate integration, and the Radial-Angular-R 1 -Sqrt singularity cancellation scheme, for triangle integration domains. This work can serve as a foundation/template for further, 3D MoM-related work to identify appropriate quadrature schemes according to their error characteristics; for automatic selection of optimal schemes and quadrature orders in a computer implementation of the MoM; and for local and global estimation of MoM quadrature errors. This work can be specialized to the MoM for surfaces in 2D.