discrete nite element approximations for linear parabolic integro-diierential equations with integrable kernels,Volterra projection onto nite element spaces and applications to integro-diierential and related equations, SIAM J. Nonsmooth data error estimates for approximation of an evolution equation with a positive-type memory term, Math. Error estimates with sharp constants for a fading memory Volterra problem in linear solid viscoelasticity, SIAM J. Error estimates for semi-discrete nite element methods for parabolic integro-diierential equations, Math. 21 where K(t; s) is singular kernel, for example 0 K(t; s) g(t; s)K (t ? s) with g(t; s) smooth and bounded and K 2 L 1 (0; 1). In this case K in (A2) should be replaced by K = max t>0 Z t 0 K(t; s)e (t?s) ds; max 0<< 1 in order to satisfy (A2). It is easy to see from a simple calculation that R(t) = O(e ?(?1)t) and Z t 0 R(t ? s)R(s)ds = O(te ?(?1)t); so that ^ R(t) = O(te ?(?1)t). This implies that (3.9) may not be the best convergence rate estimates. (R2) Since jju(t) ? u h (t)jj jju(t) ? u 1 (t)jj + jju 1 (t) ? u 1 h (t)jj + jju 1 h (t) ? u h (t)jj so that if the assumptions of Theorems 5.2-5.4, Theorem 3.2 and Theorem 4.4 are satissed we have jju(t) ? u h (t)jj = O(h r + te ?t + e ?t) for the semi-discrete approximation and jju(t n) ? u n h jj = O(h r + t + te ?t + e ?t) for the backward Euler scheme. Therefore if te ?t n + e ?t h r + t or t …

Multigrid methods with nested subspaces and inherited forms are analyzed in an abstract framework that permits application to Linear systems of the type that have to be solved at each time level in time-stepping methods for finite element approximations of parabolic problems. Convergence rates that are independent of the space and time steps are obtained in an appropriate time step dependent norm.