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Galerkin discretizations of integral equations in ${ℝ}^d$ require the evaluation of integrals $I={\int }_{S^{\left(1\right)}}{\int }_{S^{\left(2\right)}}g(x,y)\mathrm{d}y\mathrm{d}x$ where , are -simplices and has a singularity at = . We assume that is Gevrey smooth for $\ne$ and satisfies bounds for the derivatives which allow algebraic singularities at = . This holds for kernel functions commonly occurring in integral equations. We construct a family of quadrature rules ${𝒬}_N$ using function evaluations of which achieves exponential...

Galerkin discretizations of integral equations in ${ℝ}^d$ require the evaluation of integrals $I={\int }_{S^{\left(1\right)}}{\int }_{S^{\left(2\right)}}g(x,y)\mathrm{d}y\mathrm{d}x$ where , are -simplices and has a singularity at = . We assume that is Gevrey smooth for $\ne$ and satisfies bounds for the derivatives which allow algebraic singularities at = . This holds for kernel functions commonly occurring in integral equations. We construct a family of quadrature rules ${𝒬}_N$ using function evaluations of which achieves exponential...