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This paper establishes the equivalence of conforming Courant finite element method and nonconforming Crouzeix-Raviart finite element method in the sense that the respective energy error norms are equivalent up to generic constants and higher-order data oscillations in a Poisson model problem. The Raviart-Thomas mixed finite element method is better than the previous two whereas the conjecture of the converse relation is proved to be false.
This paper completes the analysis of comparison initiated by Braess in Calcolo (2010). Two numerical benchmarks illustrate the comparison theorems and the possible strict superiority of the Raviart-Thomas mixed finite element method. Applications include least-squares finite element methods and equality of approximation classes for concepts of optimality for adaptive finite element methods.