Optimal superconvergence analysis for the Crouzeix-Raviart and the Morley elements

In this paper, an improved superconvergence analysis is presented for both the Crouzeix-Raviart element and the Morley element. The main idea of the analysis is to employ a discrete Helmholtz decomposition of the difference between the canonical interpolation and the finite element solution for the first-order mixed Raviart–Thomas element and the mixed Hellan–Herrmann–Johnson element, respectively. This in particular allows for proving a full one-order superconvergence result for these two mixed finite elements. Finally, a full one-order superconvergence result of both the Crouzeix-Raviart element and the Morley element follows from their special relations with the first-order mixed Raviart–Thomas element and the mixed Hellan–Herrmann–Johnson element respectively. Those superconvergence results are also extended to mildly structured meshes.