Lower bounds, elliptic reconstruction and a posteriori error control of parabolic problems

A popular approach for proving a posteriori error bounds in various norms for evolution problems with partial differential equations uses reconstruction operators to recover conforming objects from the approximate solutions. So far, lower bounds in reconstruction-based a posteriori error estimators have been proven only for time-discrete schemes for parabolic problems; the proof of lower bounds for fully discrete schemes in reconstruction-based a posteriori error estimators has eluded. In this work, we provide a complete framework addressing this issue for energy-type norms. We consider Backward Euler discretizations and time-discontinuous Galerkin schemes, combined with dynamically changing conforming finite element methods in space, approximating linear parabolic problems. The results presented include sharp upper and lower a posteriori error bounds. Localized versions of the lower bounds are also considered.