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 occuring in integral equations. We construct a family of quadrature rules $Q_N$ using $N$ functions evaluations of $g$ which achieves exponential convergence | $I - Q_N$ | $\leq C$ exp(-$rN^{\gamma}$) with constants $r,\gamma$ > 0.