Discrete reliability for high-order Crouzeix--Raviart finite elements

In this paper, the adaptive numerical solution of a 2D Poisson model problem by Crouzeix-Raviart elements ($\operatorname*{CR}_{k}$ $\operatorname*{FEM}$) of arbitrary odd degree $k\geq1$ is investigated. The analysis is based on an established, abstract theoretical framework: the \textit{axioms of adaptivity} imply optimal convergence rates for the adaptive algorithm induced by a residual-type a posteriori error estimator. Here, we introduce the error estimator for the $\operatorname*{CR}_{k}$ $\operatorname*{FEM}$ discretization and our main theoretical result is the proof ot Axiom 3: \textit{discrete reliability}. This generalizes results for adaptive lowest order $\operatorname*{CR}_{1}$ $\operatorname*{FEM}$ in the literature. For this analysis, we introduce and analyze new local quasi-interpolation operators for $\operatorname*{CR}_{k}$ $\operatorname*{FEM}$ which are key for our proof of discrete reliability. We present the results of numerical experiments for the adaptive version of $\operatorname*{CR}_{k}$ $\operatorname*{FEM}$ for some low and higher (odd) degrees $k\geq1$ which illustrate the optimal convergence rates for all considered values of $k$.