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Exponential convergence of hp quadraturefor integral operators with Gevrey kernels

Published online by Cambridge University Press:  11 October 2010

Alexey Chernov ,
Tobias von Petersdorff  and
Christoph Schwab
Alexey Chernov
Affiliation:
Hausdorff Center for Mathematics and Institute for Numerical Simulation, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany.
Tobias von Petersdorff
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
Christoph Schwab
Affiliation:
Seminar für Angewandte Mathematik, ETH Zürich, 8092 Zürich, Switzerland. schwab@sam.math.ethz.ch
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Abstract

Galerkin discretizations of integral equations in $\mathbb{R}^{d}$ requirethe 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 singularityat x = y. We assume that g is Gevrey smooth for x $\ne$ y andsatisfies bounds for the derivatives which allow algebraic singularitiesat x = y. This holds for kernel functions commonly occurring in integralequations. We construct a family of quadrature rules $\mathcal{Q}_{N}$ usingN function evaluations of g which achieves exponential convergence|I – $\mathcal{Q}_{N}$ | ≤C exp(–r N γ ) with constants r, γ > 0.

Keywords

Numerical integrationhypersingular integralsintegral equationsGevrey regularityexponentialconvergence
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 3 , May 2011 , pp. 387 - 422
Copyright
© EDP Sciences, SMAI, 2010

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