A Petrov-Galerkin method for integro-differential equations with a memory term

We investigate the numerical solution of an integro-differential equation with a memory term. For the time discretization we apply the continuous Petrov-Galerkin method considered by Lin et al. [SIAM J. Numer. Anal., 38, 2000]. We combined the Petrov-Galerkin scheme with respect to time with continuous finite elements for the space discretization and obtained a fully discrete scheme. We show optimal error bounds of the numerical solutions for both schemes, and compare our theoretical error bounds with the results of numerical computations.