A fast and well-conditioned spectral method for singular integral equations. 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(m 2 n) 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.
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References in zbMATH (referenced in 8 articles , 1 standard article )
Showing results 1 to 8 of 8.
- Aurentz, Jared Lee; Slevinsky, Richard Mikaël: On symmetrizing the ultraspherical spectral method for self-adjoint problems (2020)
- Gutleb, Timon S.; Olver, Sheehan: A sparse spectral method for Volterra integral equations using orthogonal polynomials on the triangle (2020)
- Olver, Sheehan; Townsend, Alex; Vasil, Geoffrey: A sparse spectral method on triangles (2019)
- Olver, Sheehan; Xu, Yuan: Orthogonal structure on a wedge and on the boundary of a square (2019)
- Slevinsky, Richard Mikaël: Fast and backward stable transforms between spherical harmonic expansions and bivariate Fourier series (2019)
- Hale, Nicholas; Olver, Sheehan: A fast and spectrally convergent algorithm for rational-order fractional integral and differential equations (2018)
- Slevinsky, Richard Mikaël; Montanelli, Hadrien; Du, Qiang: A spectral method for nonlocal diffusion operators on the sphere (2018)
- Slevinsky, Richard Mikael; Olver, Sheehan: A fast and well-conditioned spectral method for singular integral equations (2017)