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.