A fast and well-conditioned spectral method for singular integral equations

TL;DR

This paper introduces a fast, well-conditioned spectral method using Chebyshev and ultraspherical polynomials for solving univariate singular integral equations efficiently, with applications in physics and engineering.

Contribution

The authors develop a novel spectral approach that reformulates singular integral equations into almost-banded systems, enabling efficient solutions with proven stability and broad applicability.

Findings

Achieves ${ m O}(m^2 n)$ computational complexity for solving systems.

Reduces complexity to ${ m O}(m n)$ with pre-caching for multiple right-hand sides.

Provides spectrally accurate solutions for complex physical problems.

Abstract

We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in ${\cal O}(m^2n)$ operations using an adaptive QR factorization, where $m$ is the bandwidth and $n$ is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to ${\cal O}(m n)$ operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The Julia software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface.