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We consider the a posteriori error analysis of fully discrete approximationsof parabolic problems based on conforming $hp$-finite element methods in spaceand an arbitrary order discontinuous Galerkin method in time. Using anequilibrated flux reconstruction, we present a posteriori error estimatesyielding guaranteed upper bounds on the $L^2(H^1)$-norm of the error, withoutunknown constants and without restrictions on the spatial and temporal meshes.It is known from the literature that the analysis of the efficiency of theestimators represents a significant challenge for $L^2(H^1)$-norm estimates.Here we show that the estimator is bounded by the $L^2(H^1)$-norm of the errorplus the temporal jumps under the one-sided parabolic condition $h^2 \lesssim\tau$. This result improves on earlier works that required stronger two-sidedhypotheses such as $h \simeq \tau$ or $h^2\simeq \tau$; instead our result nowencompasses the practically relevant case for computations and allows forlocally refined spatial meshes. The constants in our bounds are robust withrespect to the mesh and time-step sizes, the spatial polynomial degrees, andalso with respect to refinement and coarsening between time-steps, therebyremoving any transition condition.

Equilibrated flux a posteriori error estimates in $L^2(H^1)$-norms for high-order discretizations of parabolic problems