Superconvergence of both the Crouzeix-Raviart and Morley elements

TL;DR

This paper introduces a novel approach to prove superconvergence of Crouzeix-Raviart and Morley elements by leveraging their equivalences with certain mixed finite elements, demonstrating a half-order superconvergence through postprocessing.

Contribution

It presents a new method that uses equivalences with mixed elements to establish superconvergence results for nonconforming finite elements.

Findings

Superconvergence of Crouzeix-Raviart and Morley elements proven.

Half order superconvergence achieved via postprocessing.

Method exploits special conformity of discrete stresses.

Abstract

In this paper, a new method is proposed to prove the superconvergence of both the Crouzeix-Raviart and Morley elements. The main idea is to fully employ equivalences with the first order Raviart-Thomas element and the first order Hellan-Herrmann-Johnson element, respectively. In this way, some special conformity of discrete stresses is explored and superconvergence of mixed elements can be used to analyze superconvergence of nonconforming elements. Finally, a half order superconvergence by postprocessing is proved for both nonconforming elements.