Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems

We derive aposteriori error estimates for fully discrete approximations to solutions of linear parabolic equations on the space-time domain. The space discretization uses finite element spaces, that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto (2003). We derive novel optimal order aposteriori error estimates for the maximum-in-time and mean-square-in-space norm and the mean-square in space-time of the time-derivative norm.