Equilibrated residual error estimates are p-robust

Equilibrated residual error estimators applied to high order finite elements are analyzed. The estimators provide always a true upper bound for the energy error. We prove that also the efficiency estimate is robust with respect to the polynomial degrees. A complete theory is provided for tensor product elements. In the case of simplicial elements, the theorem was first based on a conjecture, but the conjecture was quite recently proved by Costabel and McIntosh [M. Costabel, A. McIntosh, On Bogovskiıˇ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains. IRMAR-Preprint 08-40, Rennes, August 2008]. Numerical evidence for the p-robustness is provided.