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

• Published in: Journal of computational physics Vol. 332; pp. 290 - 315
• Main Authors: Slevinsky, Richard Mikael, Olver, Sheehan
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01-03-2017; Elsevier Science Ltd
• Subjects: Acoustic scattering; Adaptive systems; Approximation; Bivariate analysis; Caching; Chebyshev approximation; Computational physics; Conditioning; Factorization; Faraday cage; Helmholtz equations; Integral equations; Linear operators; Mathematical analysis; Operators (mathematics); Polynomials; Singular integral equations; Sparsity; Spectra; Spectral method; Spectral methods; Ultraspherical polynomials; Singular integral equations; Ultraspherical polynomials; Spectral method
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2016.12.009

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 O(m2n) 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 O(mn) 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.