On the Direct Evaluation of Weakly Singular Integrals in Galerkin Mixed Potential Integral Equation Formulations

Abstract

Weakly singular integrals over coincident triangles, arising in the Galerkin discretization of mixed potential integral equation formulations, are calculated using a direct evaluation method. The proposed method utilizes a series of coordinate transformations, together with a re-ordering of the integrations, in order to reduce the dimensionality of the original four-dimensional (4D) weakly singular integrals into 1D numerical integrations of smooth functions. The final formulas can be easily evaluated with a standard Gaussian quadrature rule, resulting in a scheme with great accuracy and efficiency properties. Numerical results for the comparison of the proposed method with both singularity subtraction and singularity cancellation methods, often used for the evaluation of multidimensional singular integrals, are presented, indicating the superior overall performance of the direct evaluation scheme.