Fast Spectral Galerkin Method for Logarithmic Singular Equations on a Segment

Abstract

We present a fast Galerkin spectral method to solve logarithmic singular equations on segments. The proposed method uses weighted first-kind Chebyshev polynomials. Convergence rates of several orders are obtained for fractional Sobolev spaces $\tilde{H}^{-1 ⁄ 2}$ (or $H^{-1 ⁄ 2}_{00}$). Main tools are the approximation properties of the discretization basis, the construction of a suitable Hilbert scale for weighted $L^2$-spaces and local regularity estimates. Numerical experiments are provided to validate our claims.