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.

在本文中，对 Crouzeix-Raviart 单元和 Morley 单元进行了改进的超收敛分析。分析的主要思想是对一阶混合 Raviart-Thomas 元和混合 Hellan-Herrmann-Johnson 元分别采用正则插值和有限元解之间差异的离散亥姆霍兹分解。这特别允许证明这两个混合有限元的完全一阶超收敛结果。最后，Crouzeix-Raviart 元素和 Morley 元素分别与一阶混合 Raviart-Thomas 元素和混合 Hellan-Herrmann-Johnson 元素的特殊关系，得到了完全一阶超收敛结果。这些超收敛结果也扩展到温和结构的网格。