  In this paper, several two-grid finite element algorithms for solving
parabolic integro-differential equations (PIDEs) with nonlinear memory are
presented. Analysis of these algorithms is given assuming a fully implicit time
discretization. It is shown that these algorithms are as stable as the standard
fully discrete finite element algorithm, and can achieve the same accuracy as
the standard algorithm if the coarse grid size $H$ and the fine grid size $h$
satisfy $H=O(h^{\frac{r-1}{r}})$. Especially for PIDEs with nonlinear memory
defined by a lower order nonlinear operator, our two-grid algorithm can save
significant storage and computing time. Numerical experiments are given to
confirm the theoretical results.
