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

Abstract

Galerkin discretizations of integral equations in $ \mathbb{R}^d$ require the evaluation of integrals $ I = \int_{S^{{(1)}}}$ $\int_{S^{{(2)}}}$ $g(x,y)dydx$ 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 $Q_N$ usin $N$ function evaluations of $g$ which achieves exponential convergence | $ I - Q_N$ | $\leq C$ exp$(-rN^{\gamma})$ with constants $r, \gamma$ > 0.