Cancellation Integration Scheme for the Evaluation of Strongly Singular Terms on Curved Triangles in Boundary Integral Equations

In this paper, we present a numerical integration scheme for the highly accurate and efficient computation of strongly singular integrals occurring in Galerkin discretization of boundary integral equations using curved triangles. In particular, we focus on the strongly singular integral involving the magnetic boundary integral operator for a pair of identical quadratic triangles. The present scheme allows for the exact cancellation of the singularity by combining basic geometrical arguments and a series of changes of variables. The resulting integrands prove to be continuous functions, which can be evaluated numerically using a simple Gauss 1-D quadrature rule. The paper is complemented by a validation method, providing an effective and systematic way of assessing the precision of an integration scheme. Numerical results are provided to demonstrate the accuracy and the efficiency of the new integration scheme.