Equilibrated flux a posteriori error estimates in L²(H¹)-norms for high-order discretizations of parabolic problems

We consider the a posteriori error analysis of fully discrete approximations of parabolic problems based on conforming hp-finite element methods in space and an arbitrary order discontinuous Galerkin method in time. Using an equilibrated flux reconstruction we present a posteriori error estimates yielding guaranteed upper bounds on the L²(H¹)-norm of the error, without unknown constants and without restrictions on the spatial and temporal meshes. It is known from the literature that the analysis of the efficiency of the estimators represents a significant challenge for L²(H¹)-norm estimates. Here we show that the estimator is bounded by the L²(H¹)-norm of the error plus the temporal jumps under the one-sided parabolic condition h² ≲ τ. This result improves on earlier works that required stronger two-sided hypotheses such as h ~ τ or h² ~ τ; instead, our result now encompasses practically relevant cases for computations and allows for locally refined spatial meshes. The constants in our bounds are robust with respect to the mesh and time-step sizes, the spatial polynomial degrees and the refinement and coarsening between time steps, thereby removing any transition condition.