A Posteriori Error Estimates for Parabolic Problems via Elliptic Reconstruction and Duality

We use the elliptic reconstruction technique in combination with a duality approach to prove a posteriori error estimates for fully discrete backward Euler scheme for linear parabolic equations. As an application, we combine our result with the residual based estimators from the a posteriori estimation for elliptic problems to derive space-error estimators and thus a fully practical version of the estimators bounding the error in the L∞(0, T ; L2(Ω)) norm. These estimators, which are of optimal order, extend those introduced by Eriksson and Johnson [EJ91] by taking into account the error induced by the mesh changes and allowing for a more flexible use of the elliptic estimators. For comparison with previous results, an application of our abstract results using residual estimators is provided.