  Optimal convergence rates of adaptive finite element methods are well
understood in terms of the axioms of adaptivity. One key ingredient is the
discrete reliability of a residual-based a posteriori error estimator, which
controls the error of two discrete finite element solutions based on two nested
triangulations. In the error analysis of nonconforming finite element methods,
like the Crouzeix-Raviart or Morley finite element schemes, the difference of
the piecewise derivatives of discontinuous approximations to the distributional
gradients of global Sobolev functions plays a dominant role and is the object
of this paper. The nonconforming interpolation operator, which comes natural
with the definition of the aforementioned nonconforming finite element in the
sense of Ciarlet, allows for stability and approximation properties that enable
direct proofs of the reliability for the residual that monitors the equilibrium
condition. The novel approach of this paper is the suggestion of a
right-inverse of this interpolation operator in conforming piecewise
polynomials to design a nonconforming approximation of a given coarse-grid
approximation on a refined triangulation. The results of this paper allow for
simple proofs of the discrete reliability in any space dimension and multiply
connected domains on general shape-regular triangulations beyond newest-vertex
bisection of simplices. Particular attention is on optimal constants in some
standard discrete estimates listed in the appendices.
