Long Time Stability of Finite Element Approximations for Parabolic Equations with Memory

In this paper we derive the sharp long time stability and error estimates of nite element approximations for parabolic integro di erential equations First the exponential decay of the solution as t is studied and then the semi discrete and fully discrete approximations are considered using the Ritz Volterra projection Other related problems are studied as well The main feature of our analysis is that the results are valid for both smooth and non smooth weakly singular kernels Introduction In this paper we continue the work by Thom ee and Wahlbin and study the long time stability and error estimates of nite element approximations for parabolic integro di erential equations For simplicity we consider the following parabolic integro di erential equation nd u u x t such that ut Au Z t K t s Bu s ds f t in Q u on u x u x x where Q R d d is a bounded domain with smooth boundary K t is a non negative memory kernel other kernels can be handled in the same way by taking the absolute value jK t j in the analysis and f is a known function A is a symmetric positive de nite second order elliptic operator A d X i j xi aij x xj a x I a x aij x aji x i j d a d X