Orthogonality relations of Crouzeix-Raviart and Raviart-Thomas finite element spaces

Identities that relate projections of Raviart-Thomas finite element vector fields to discrete gradients of Crouzeix-Raviart finite element functions are derived under general conditions. Various implications such as discrete convex duality results and a characterization of the image of the projection of the Crouzeix-Ravaiart space onto elementwise constant functions are deduced.