Exponential convergence of Gauss--Jacobi quadratures for singular integrals over simplices in arbitrary dimension (Q2903044)

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 \(x = y\) and is Gervey-\(\delta\) smooth for \(x\neq y\). The authors consider a special singularity \(\|x-y\|^\alpha\) with real \(\alpha\) which appears frequently in application. They construct a family of \(hp\)-quadrature rules \(Q_N\) with \(N\) function evaluations for a class of integrands \(f\) and prove that the convergence rate \(O(exp(-rN^\gamma)\) with the exponent \(\gamma = 1/(2d\delta)\) is achieved if a certain one-dimensional Gauss-Jacobi quadrature rule is used in the (univariate) ``singular coordinate''. They also analyze an approximation by tensor Gauss-Jacobi quadratures in the ``regular coordinates'' and illustrate the performance of the new Gauss-Jacobi rules on several numerical examples.