Petrov-Galerkin methods for linear Volterra integro-differential equations

In this paper we study a class of Petrov-Galerkin solutions that have global optimal convergence rates for linear Volterra integro-differential equations. These solutions also possess certain local and global superconvergence. Asymptotic expansions of the errors in these solutions are established which can be used to form higher order approximations by Richardson extrapolation and defect corrections. Several postprocessing techniques are introduced to enhance these solutions. As by-products, these higher order numerical approximations can be used to generate a posteriori error estimators. Representative numerical results are also provided.