Our talk introduces a global superconvergence in maximum norm for the derivatives of the time-continuous Galerkin finite element solutions of parabolic integro-differential equation of the form

ut + A(t)u + Z t

0

B(t, s)u(s)ds = f(t) in Ω × J, u = 0 on ∂Ω × J, u(0) = u0(x) x ∈ Ω, where Ω ⊂ R

2 is an open bounded domain with smooth boundary ∂Ω, J = (0, T) with T > 0, A(t) is a self-adjoint positive definite linear elliptic partial differential operator of second order, and B(t, s) an arbitrary second-order linear partial differential operator, both with coefficients depending smoothly on x, t, and, for the latter, s in the closure of

their respective domains.

First, we recall a regularized Green’s functions with memory terms and present some estimates for them. Secondly, we consider the global superconvergence in W 1,∞-norm by virtue of the supercloses between the finite element solution and the interpolation function of the exact solution of the problem, and between the Ritz-Volterra projection of the exact solution and its interpolant. In the end, we devote to the global superconvergence analysis in W1,∞-norm for hyperbolic integro-differential equations, Sobolev and viscoelasticity type equations, on the basis of which some a posteriori error estimators are presented.