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(-rNγ) with constants r, γ > 0.
Mots clés : numerical integration, hypersingular integrals, integral equations, Gevrey regularity, exponential convergence
@article{M2AN_2011__45_3_387_0,
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://www.numdam.org/articles/10.1051/m2an/2010061/}
}
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 PY - 2011 SP - 387 EP - 422 VL - 45 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2010061/ DO - 10.1051/m2an/2010061 LA - en ID - M2AN_2011__45_3_387_0 ER -
%0 Journal Article %A Chernov, Alexey %A von Petersdorff, Tobias %A Schwab, Christoph %T Exponential convergence of $hp$ quadrature for integral operators with Gevrey kernels %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2011 %P 387-422 %V 45 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2010061/ %R 10.1051/m2an/2010061 %G en %F M2AN_2011__45_3_387_0
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://www.numdam.org/articles/10.1051/m2an/2010061/
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