This work develops polynomial-degree-robust (p-robust) equilibrated a posteriori error estimates of finite element methods, based on $H^1$ auxiliary space decomposition. The proposed framework employs the fictitious space lemma for preconditioning and $H^1$ regular decomposition to decompose the finite element residual into $H^{-1}$ residuals that are further controlled by p-robust equilibrated a posteriori error analysis. As a result, we obtain novel p-robust a posteriori error estimates of conforming methods for the $H(\rm curl)$ and $H(\rm div)$ problems and the mixed method for the biharmonic equation. We also prove guaranteed a posteriori upper error bounds under convex domains or certain boundary conditions. Numerical experiments demonstrate the effectiveness and p-robustness of the proposed error estimators for the $H(\rm curl)$ conforming and the Hellan--Herrmann--Johnson methods.

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