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

Iain Smears

doi:10.5802/jcm.142

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arXiv · 2507.13188

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

Iain Smears

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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.

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Iain Smears. 2025-07-17. On the efficiency of a posteriori error estimators for parabolic partial differential equations in the energy norm. https://doi.org/10.5802/jcm.142

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