2015/07/02 by Richard Mikaël Slevinsky, Sheehan Olver, Slevinsky, Richard Mikael +1 · 1 citation
Physics and Astronomy · Engineering · #Electromagnetic Scattering and Analysis #Numerical methods in engineering #Electromagnetic Simulation and Numerical Methods
paper · pdf · doi:10.48550/arxiv.1507.00596
We develop a spectral method for solving univariate singular integral\nequations over unions of intervals by utilizing Chebyshev and ultraspherical\npolynomials to reformulate the equations as almost-banded infinite-dimensional\nsystems. This is accomplished by utilizing low rank approximations for sparse\nrepresentations of the bivariate kernels. The resulting system can be solved in\n cal O(m2n) operations using an adaptive QR factorization, where m is\nthe bandwidth and n is the optimal number of unknowns needed to resolve the\ntrue solution. The complexity is reduced to cal O(m n) operations by\npre-caching the QR factorization when the same operator is used for multiple\nright-hand sides. Stability is proved by showing that the resulting linear\noperator can be diagonally preconditioned to be a compact perturbation of the\nidentity. Applications considered include the Faraday cage, and acoustic\nscattering for the Helmholtz and gravity Helmholtz equations, including\nspectrally accurate numerical evaluation of the far- and near-field solution.\nThe Julia software package SingularIntegralEquations.jl implements our method\nwith a convenient, user-friendly interface.\n