# Exponential convergence of hp quadrature for integral operators with Gevrey kernels

• Volume: 45, Issue: 3, page 387-422
• ISSN: 0764-583X

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## Abstract

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

## How to cite

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Chernov, Alexey, von Petersdorff, Tobias, and Schwab, Christoph. "Exponential convergence of hp quadrature for integral operators with Gevrey kernels." ESAIM: Mathematical Modelling and Numerical Analysis 45.3 (2011): 387-422. <http://eudml.org/doc/276355>.

@article{Chernov2011,
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(–rNγ) with constants r, γ > 0. },
author = {Chernov, Alexey, von Petersdorff, Tobias, Schwab, Christoph},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Numerical integration; hypersingular integrals; integral equations; Gevrey regularity; exponential convergence; numerical integration; exponential convergence; numerical examples; Galerkin discretizations; quadrature rules},
language = {eng},
month = {1},
number = {3},
pages = {387-422},
publisher = {EDP Sciences},
title = {Exponential convergence of hp quadrature for integral operators with Gevrey kernels},
url = {http://eudml.org/doc/276355},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Chernov, Alexey
AU - von Petersdorff, Tobias
AU - Schwab, Christoph
TI - Exponential convergence of hp quadrature for integral operators with Gevrey kernels
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/1//
PB - EDP Sciences
VL - 45
IS - 3
SP - 387
EP - 422
AB - 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(–rNγ) with constants r, γ > 0.
LA - eng
KW - Numerical integration; hypersingular integrals; integral equations; Gevrey regularity; exponential convergence; numerical integration; exponential convergence; numerical examples; Galerkin discretizations; quadrature rules
UR - http://eudml.org/doc/276355
ER -

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