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.