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 He −1/2 (or H00−1/2). Main tools are the approximation properties of the discretization basis, the construction of a suitable Hilbert scale for weighted L2-spaces and local regularity estimates. Numerical experiments are provided to validate our claims.