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. References S. Larsson, V. Thomee and L. Wahlbin, Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method, Math. Comp., 67, 45--71 (1998). T. Lin, Y. Lin, M. Rao and S. Zhang, Petrov-Galerkin methods for linear Volterra integro-differential equations, SIAM J. Numer. Anal., 38, 937--963 (2000). doi:10.1137/S0036142999336145 W. McLean, I. H. Sloan and V. Thomee, Time discretization via Laplace transformation of an integro-differential equation of parabolic type, Numer. Math., 102, 497--522 (2006). N. Y. Zhang, On fully discrete Galerkin approximations for partial inregro-differential equations of parabolic type, Math. Comp., 60, 133--166 (1993). http://www.jstor.org/pss/215315