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

Chernov, Alexey; von Petersdorff, Tobias; Schwab, Christoph

doi:10.1051/m2an/2010061

Exponential convergence of

h p

quadrature for integral operators with Gevrey kernels

Chernov, Alexey ; von Petersdorff, Tobias ; Schwab, Christoph

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 3, pp. 387-422.

Résumé

Galerkin discretizations of integral equations in $ℝ^{d}$ require the evaluation of integrals $I = \int_{S^{(1)}} \int_{S^{(2)}} g (x, y) d y d x$ 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 $\neq$ 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(-rN^γ) with constants r, γ > 0.

Zbl

DOI : 10.1051/m2an/2010061

Classification : 65N30
Mots-clés : numerical integration, hypersingular integrals, integral equations, Gevrey regularity, exponential convergence

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     author = {Chernov, Alexey and von Petersdorff, Tobias and Schwab, Christoph},
     title = {Exponential convergence of $hp$ quadrature for integral operators with {Gevrey} kernels},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {387--422},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {3},
     year = {2011},
     doi = {10.1051/m2an/2010061},
     zbl = {1269.65143},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2010061/}
}

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Chernov, Alexey; von Petersdorff, Tobias; Schwab, Christoph. Exponential convergence of $hp$ quadrature for integral operators with Gevrey kernels. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 3, pp. 387-422. doi : 10.1051/m2an/2010061. http://archive.numdam.org/articles/10.1051/m2an/2010061/

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