Legendre spectral projection methods for linear second kind Volterra integral equations with weakly singular kernels

In this paper, Galerkin and iterated Galerkin methods are applied to approximate the linear second kind Volterra integral equations with weakly singular algebraic kernels using Legendre polynomial basis functions. We discuss the convergence results in both L2 and infinity norms in two cases: when the exact solution is sufficiently smooth and non-smooth. We also apply Legendre multi-Galerkin and iterated Legendre multi-Galerkin methods and derive the superconvergence rates. Numerical results are given to verify the theoretical results.