Exponential convergence of hp quadrature for integral operators with Gevrey kernels

Abstract

Galerkin discretizations of integral equations in ℝd require the evaluation of integrals I = ∫S(1) ∫S(2) g(x, y)dydx where S(1), S(2) are d-simplices and g has a singularity at x = y. We assume that g is Gevrey smooth for x ≠ y and satisfies bounds for the derivatives which allow algebraic singularities at x = y. This holds for kernel functions commonly occurring in integral equations. We construct a family of quadrature rules QN using N function evaluations of g which achieves exponential convergence |I -QN| ≤ C exp(-rNγ) with constants r, γ > 0. © EDP Sciences, SMAI, 2010.