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

Abstract

We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. 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. We derive novel a posteriori estimates for the norms of L ∞ ⁡ ( 0 , T ; L 2 ⁡ ( Ω ) ) \operatorname {L}_\infty (0,T;\operatorname {L}_2(\Omega )) and the higher order spaces, L ∞ ⁡ ( 0 , T ; H 1 ⁡ ( Ω ) ) \operatorname {L}_\infty (0,T;\operatorname {H}^1(\Omega )) and H 1 ⁡ ( 0 , T ; L 2 ⁡ ( Ω ) ) \operatorname {H}^1(0,T;\operatorname {L}_2(\Omega )) , with optimal orders of convergence.