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

Iain Smears

doi:10.5802/jcm.142

Iain Smears ¹

¹Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom

The SMAI Journal of computational mathematics, Volume 12 (2026), pp. 27-42

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.

Published online: 2026-03-02

DOI: 10.5802/jcm.142

Classification: 65M15, 65M60
Keywords: A posteriori error analysis, parabolic partial differential equations, energy norm

Author's affiliations:

Iain Smears ¹

¹ Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom

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Iain Smears. On the efficiency of a posteriori error estimators for parabolic partial differential equations in the energy norm. The SMAI Journal of computational mathematics, Volume 12 (2026), pp. 27-42. doi: 10.5802/jcm.142

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