A Fourier extension based numerical integration scheme for fast and high-order approximation of convolutions with weakly singular kernels

TL;DR

This paper introduces a Fourier extension-based numerical integration scheme that achieves high-order, fast, and efficient approximation of convolutions with weakly singular kernels, overcoming issues with Gibbs oscillations.

Contribution

It presents a novel $O(n ext{log} n)$ Fourier extension method that improves convergence and simplicity in approximating weakly singular convolution integrals.

Findings

Achieves high-order convergence in numerical integration.

Effectively removes Gibbs oscillations in Fourier approximations.

Demonstrates efficiency and accuracy through numerical experiments.

Abstract

Computationally efficient numerical methods for high-order approximations of convolution integrals involving weakly singular kernels find many practical applications including those in the development of fast quadrature methods for numerical solution of integral equations. Most fast techniques in this direction utilize uniform grid discretizations of the integral that facilitate the use of FFT for $O(n\log n)$ computations on a grid of size $n$. In general, however, the resulting error converges slowly with increasing $n$ when the integrand does not have a smooth periodic extension. Such extensions, in fact, are often discontinuous and, therefore, their approximations by truncated Fourier series suffer from Gibb's oscillations. In this paper, we present and analyze an $O(n\log n)$ scheme, based on a Fourier extension approach for removing such unwanted oscillations, that not only converges with high-order but is also relatively simple to implement. We include a theoretical error analysis as well as a wide variety of numerical experiments to demonstrate its efficacy.