Superconvergent flux recovery of the Rannacher-Turek nonconforming element

TL;DR

This paper establishes superconvergence estimates for the Rannacher--Turek nonconforming element, demonstrating that a specially designed flux recovery operator achieves superconvergence for second order elliptic equations on cubical meshes.

Contribution

It introduces a superconvergent flux recovery technique for the Rannacher--Turek element, extending superconvergence results to cubical meshes in 2D and 3D.

Findings

Supercloseness of corrected flux to Raviart--Thomas interpolant

Design of a superconvergent recovery operator based on local averaging

Superconvergence of the recovered flux to the exact flux

Abstract

This work presents superconvergence estimates of the nonconforming Rannacher--Turek element for second order elliptic equations on any cubical meshes in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$. In particular, a corrected numerical flux is shown to be superclose to the Raviart--Thomas interpolant of the exact flux. We then design a superconvergent recovery operator based on local weighted averaging. Combining the supercloseness and the recovery operator, we prove that the recovered flux superconverges to the exact flux. As a by-product, we obtain a superconvergent recovery estimate of the Crouzeix--Raviart element method for general elliptic equations.