A comparison of duality and energy aposteriori estimates for L?(0,T;L2({\Omega})) in parabolic problems

TL;DR

This paper compares duality and energy-based a posteriori error estimates for parabolic problems, introducing improved estimators that account for mesh changes and are applicable to fully discrete schemes.

Contribution

It develops and compares duality and energy a posteriori error estimators for parabolic equations, extending previous methods with mesh change considerations and practical error bounds.

Findings

Duality estimators are of optimal order and flexible.

Energy estimators simplify previous bounds.

Comparison shows advantages of duality over energy estimates in certain scenarios.

Abstract

We use the elliptic reconstruction technique in combination with a duality approach to prove aposteriori error estimates for fully discrete back- ward Euler scheme for linear parabolic equations. As an application, we com- bine our result with the residual based estimators from the aposteriori esti- mation for elliptic problems to derive space-error indicators and thus a fully practical version of the estimators bounding the error in the L \infty (0, T ; L2({\Omega})) norm. These estimators, which are of optimal order, extend those introduced by Eriksson and Johnson (1991) by taking into account the error induced by the mesh changes and allowing for a more flexible use of the elliptic estima- tors. For comparison with previous results we derive also an energy-based aposteriori estimate for the L \infty (0, T ; L2({\Omega}))-error which simplifies a previous one given in Lakkis and Makridakis (2006). We then compare both estimators (duality vs. energy) in practical situations and draw conclusions.