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We consider energy norm a posteriori error analysis of conforming finiteelement approximations of singularly perturbed reaction-diffusion problems onsimplicial meshes in arbitrary space dimension. Using an equilibrated fluxreconstruction, the proposed estimator gives a guaranteed global upper bound onthe error without unknown constants, and local efficiency robust with respectto the mesh size and singular perturbation parameters. Whereas previous workson equilibrated flux estimators only considered lowest-order finite elementapproximations and achieved robustness through the use of boundary-layeradapted submeshes or via combination with residual-based estimators, thepresent methodology applies in a simple way to arbitrary-order approximationsand does not request any submesh or estimators combination. The equilibratedflux is obtained via local reaction-diffusion problems with suitable weights(cut-off factors), and the guaranteed upper bound features the same weights. Weprove that the inclusion of these weights is not only sufficient but alsonecessary for robustness of any flux equilibration estimate that does notemploy submeshes or estimators combination, which shows that some of the fluxequilibrations proposed in the past cannot be robust. To achieve the fullycomputable upper bound, we derive explicit bounds for some inverse inequalityconstants on a simplex, which may be of independent interest.

Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction-diffusion problems