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I present an accurate and efficient technique for numerical evaluation of singular 6-dimensional integrals over tetrahedon-product domains, with applications to calculation of Galerkin matrix elements for discretized volume-integral-equation (VIE) solvers using Schaubert-Wilton-Glisson (SWG) and other tetrahedral basis functions. My method extends the generalized Taylor-Duffy strategy---used to handle the singular textit{triangle}-product integrals arising in discretized surface-integral-equation (SIE) formulations---to the tetrahedron-product case; it effects an exact transformation of a singular 6-dimensional integral to an nonsingular lower-dimensional integral that may be evaluated by simple numerical cubature The method is highly general and may---with the aid of automatic code generation facilitated by computer-algebra systems---be applied to a wide variety of singular integrals arising in various VIE formulations with various types of tetrahedral basis function, of which I present several examples. To demonstrate the accuracy and efficiency of my method, I apply it to the calculation of matrix elements for the volume electric-field integral equation (VEFIE) discretized with SWG basis functions, where the method yields 12-digit or higher accuracy with low computational cost---an improvement of many orders of magnitude compared to existing techniques.
We present a generic technique, automated by computer-algebra systems and available as open-source software cite{scuff-em}, for efficient numerical evaluation of a large family of singular and nonsingular 4-dimensional integrals over triangle-product
Evaluation of basic integrals over Gaussian functions, traditionally utilized for electronic structure computations on molecules and solids, is discussed in a pedagogical form.
By using Poissons summation formula, we calculate periodic integrals over Gaussian basis functions by partitioning the lattice summations between the real and reciprocal space, where both sums converge exponentially fast with a large exponent. We dem
We consider singular integral operators and maximal singular integral operators with rough kernels on homogeneous groups. We prove certain estimates for the operators that imply $L^p$ boundedness of them by an extrapolation argument under a sharp con
We prove certain $L^p$ estimates ($1<p<infty$) for non-isotropic singular integrals along surfaces of revolution. As an application we obtain $L^p$ boundedness of the singular integrals under a sharp size condition on their kernels.