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# Exponential Convergence of Gauss--Jacobi Quadratures for Singular Integrals over Simplices in Arbitrary Dimension

## Chernov, Alexey ; Schwab, Christoph

SIAM Journal on Numerical Analysis, 2012, Vol.50(3), pp.1433-1455 [Peer Reviewed Journal]

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• Title:
Exponential Convergence of Gauss--Jacobi Quadratures for Singular Integrals over Simplices in Arbitrary Dimension
• Author/Creator: Chernov, Alexey ; Schwab, Christoph
• Language: English
• Subjects: Gauss–Jacobi Quadrature ; Numerical Integration ; High Dimensional Integrands ; Hypersingular Integrals ; Integral Equations ; Gevrey Regularity ; Exponential Convergence
• Is Part Of: SIAM Journal on Numerical Analysis, 2012, Vol.50(3), pp.1433-1455
• Description: Galerkin discretizations of integral operators in $\mathbb{R}^{d}$ require the evaluation of integrals $\int_{S^{(1)}}\!\int_{S^{(2)}}\!f(x,y)\,dydx$, where $S^{(1)},S^{(2)}$ are $d$-dimensional simplices and $f$ has a singularity at $x=y$. In [A. Chernov, T. von Petersdorff, and C. Schwab, M$2$AN Math. Model. Numer. Anal., 45 (2011), pp. 387--422] we constructed a family of $hp$-quadrature rules ${Q}_N$ with $N$ function evaluations for a class of integrands $f$ allowing for algebraic singularities at $x=y$, possibly nonintegrable with respect to either $dx$ or $dy$ (hypersingular kernels) and Gevrey-$\delta$ smooth for $x\ne y$. This is satisfied for kernels from broad classes of pseudodifferential operators. We proved that $Q_N$ achieves the exponential convergence rate $\mathcal{O}(\exp(-rN^\gamma))$ with the exponent $\gamma = 1/(2d\delta+1)$. In this paper we consider a special singularity $\|x-y\|^\alpha$ with real $\alpha$ which appears frequently in appplication and prove that...
• Identifier: ISSN: 00361429 ; E-ISSN: 10957170 ; DOI: 10.1137/100812574