On the efficiency of a posteriori error estimators for parabolic partial differential equations in the energy norm

TL;DR

This paper proves the efficiency of a posteriori error estimators for the heat equation discretized by implicit Euler and finite elements, highlighting the influence of the solution reconstruction on estimator performance.

Contribution

It demonstrates the efficiency of error estimators in the energy norm for parabolic PDEs, considering a specific reconstruction of the numerical solution.

Findings

Efficiency depends on the choice of norm and solution notion.

The proposed reconstruction improves the estimator's effectiveness.

Results are validated for the heat equation with implicit Euler and finite elements.

Abstract

For the model problem of the heat equation discretized by an implicit Euler method in time and a conforming finite element method in space, we prove the efficiency of a posteriori error estimators with respect to the energy norm of the error, when considering the numerical solution as the average between the usual continuous piecewise affine-in-time and piecewise constant-in-time reconstructions. This illustrates how the efficiency of the estimators is not only possibly dependent on the choice of norm, but also on the choice of notion of numerical solution.