Exponential convergence of Gauss-Jacobi quadratures for singular integrals over high dimensional simplices

Galerkin discretizations of integral operators in R d require the evaluation of integrals R S (1) R S (2) f (x, y) dydx where S (1) , S (2) are d-dimensional simplices and f has a singularity at x = y. In [3] we constructed a family of hp-quadrature rules Q N with N function evaluations for a class of integrands f allowing for algebraic singularities at x = y, possibly non-integrable with respect to either dx or dy (hypersingular kernels) and Gevrey-δ smooth for x = y. This is satisfied for kernels from broad classes of pseudodifferential operators. We proved that Q N achieves the exponential convergence rate O(exp(−rN γ)) with the exponent γ = 1/(2dδ + 1). In this paper we consider a special singularity x − y α with real α, appearing frequiently in appplications, and prove that an improved convergence rate with γ = 1/(2dδ) is achieved if a certain one-dimensional Gauss-Jacobi quadrature rule used in the " singular direction ". We also analyze approximation by tensor Gauss-Jacobi quadratures in the " regular directions ". We illustrate the performance of the new Gauss-Jacobi rules on several numerical examples and compare it to the hp-quadratures from [3]. 1. Introduction and notations. A basic problem in the numerical analysis of Galerkin discretizations of singular integral equations involves computation of double integrals of the type