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

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.