Discrete Reliability for Crouzeix--Raviart FEMs

The proof of optimal convergence rates of adaptive finite element methods relies on Stevenson's concept of discrete reliability. This paper proves the general discrete reliability for the nonconforming Crouzeix--Raviart finite element method on multiply connected domains in any space dimension. A novel discrete quasi-interpolation operator of first-order approximation involves an intermediate triangulation and acts as the identity on unrefined simplices, to circumvent any Helmholtz decomposition. Besides the generalization of the known application to any dimension and multiply connected domains, this paper outlines the optimality proof for uniformly convex minimization problems. This discrete reliability implies reliability for the explicit residual-based a posteriori error estimator in any space dimension and for multiply connected domains.