Exponential convergence of hp quadrature for integral operators with Gevrey kernels

¹ Hausdorff Center for Mathematics and Institute for Numerical Simulation, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany.
² Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
³ Seminar für Angewandte Mathematik, ETH Zürich, 8092 Zürich, Switzerland. schwab@sam.math.ethz.ch

Received: 25 January 2009

Abstract

Galerkin discretizations of integral equations in require the evaluation of integrals 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 using N function evaluations of g which achieves exponential convergence |I – | ≤ C exp(–rN^γ) with constants r, γ > 0.