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

Abstract

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 an improved convergence rate with $\gamma = 1/(2d\delta)$ is achieved if a certain one-dimensional Gauss--Jacobi quadrature rule is used in the (univariate) „singular coordinate.” We also analyze approximation by tensor Gauss--Jacobi quadratures in the „regular coordinates.” We illustrate the performance of the new Gauss--Jacobi rules on several numerical examples and compare it to the $hp$-quadratures from [A. Chernov, T. von Petersdorff, and C. Schwab, M$2$AN Math. Model. Numer. Anal., 45 (2011), pp. 387--422].