Singular quadrature rules and fast convolutions for Fourier spectral methods

TL;DR

This paper introduces a spectral-accurate quadrature scheme for integral operators with weak singularities, enabling high-precision Fourier spectral methods in arbitrary dimensions.

Contribution

It develops a generic, corrected trapezoidal rule with spectral accuracy for weakly singular kernels, including explicit formulas and convergence analysis.

Findings

Achieves spectral accuracy for weakly singular integral operators.

Provides explicit formulas for singularities like log(r) and r^{- u}.

Demonstrates super-algebraic convergence for smooth data.

Abstract

We present a generic scheme to construct corrected trapezoidal rules with spectral accuracy for integral operators with weakly singular kernels in arbitrary dimensions. We assume that the kernel factorization of the form, $K=\alpha\phi+\widetilde{K}$ with smooth $\alpha$ and $\widetilde{K}$, is available so that the operations on the smooth factors can be performed accurately on the basis of standard Fourier spectral methods. To achieve high precision results, our approach utilizes the exact evaluation of the Fourier coefficients of the radial singularity $\phi$, which can be obtained in arbitrary dimensions by the singularity isolation/truncation described in this article. We provide a complete set of formulas for singularities of the type: $\log(r)$ and $r^{-\nu}$. Convergence analysis shows that the constructed quadrature rules exhibit almost identical rate of convergence to the trapezoidal rule applied for non-singular integrands. Especially, for smooth data, the corrected trapezoidal rules converge super-algebraically.