ESAIM: Mathematical Modelling and Numerical Analysis

Table of Contents - May 2011 - Volume 45, Issue 03  

Research Article

Exponential convergence of hp quadrature for integral operators with Gevrey kernels

Chernov, Alexeya1, von Petersdorff, Tobiasa2 and Schwab, Christopha3

a1 Hausdorff Center for Mathematics and Institute for Numerical Simulation, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany.

a2 Department of Mathematics, University of Maryland, College Park, MD 20742, USA.

a3 Seminar für Angewandte Mathematik, ETH Zürich, 8092 Zürich, Switzerland. schwab@sam.math.ethz.ch

Abstract

Galerkin discretizations of integral equations in $\mathbb{R}^{d}$ require the evaluation of integrals $I = \int_{S^{(1)}}\int_{S^{(2)}}g(x,y){\rm d}y{\rm d}x$ 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 $\ne$ 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 $\mathcal{Q}_{N}$ using N function evaluations of g which achieves exponential convergence |I – $\mathcal{Q}_{N}$ | ≤ C exp(–r N γ ) with constants r, γ > 0.

(Received January 25 2009)

(Online publication October 11 2010)

Key Words:

  • Numerical integration;
  • hypersingular integrals;
  • integral equations;
  • Gevrey regularity;
  • exponential convergence

Mathematics Subject Classification:

  • 65N30
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