The Enriched Crouzeix–Raviart Elements are Equivalent to the Raviart–Thomas Elements

Abstract

For both the Poisson model problem and the Stokes problem in any dimension, this paper proves that the enriched Crouzeix–Raviart elements are actually identical to the first order Raviart–Thomas elements in the sense that they produce the same discrete stresses. This result improves the previous result in literature which, for two dimensions, states that the piecewise constant projection of the stress by the first order Raviart–Thomas element is equal to that by the Crouzeix–Raviart element. For the eigenvalue problem of the Laplace operator, this paper proves that the error of the enriched Crouzeix–Raviart element is equivalent to that of the first order Raviart–Thomas element up to higher order terms.

Appendix: Basis Functions and Convergence Analysis of the ECR Element

For any $K\in \mathcal {T}$, we give the basis functions of the shape function space $\mathrm{ECR}(K)$. Suppose the coordinate of the centroid $\mathrm{mid}(K)$ is $(M_1,M_2\ldots ,M_n)$. The vertices of $K$ are denoted by $a_i,1\le i\le n+1$ and the barycentric coordinates by $\lambda _1,\lambda _2,\ldots ,\lambda _{n+1}$. Let $H=\sum _{i<j}|a_i-a_j|^2$, then the basis functions are as follows

$$\begin{aligned}&\phi _K = \frac{n+2}{2}-\frac{n(n+1)^2(n+2)}{2H}\sum ^{n}_{i=1}(x_i-M_i)^2,\\&\phi _j=1-n\lambda _j-\frac{1}{n+1}\phi _K,\quad 1\le j\le n+1. \end{aligned}$$

For any $v\in V_\mathrm{ECR}$, by the definition of $V_\mathrm{ECR}$ in Subsect. 2.4, $\int _{E}[v]dE=0\text { for all }E\in \mathcal {E}(\Omega )$ and $\int _EvdE=0\text { for all }E\in \mathcal {E}(\partial \Omega )$. From the theory of [23], there holds that

$$\begin{aligned} \Vert \nabla _\mathrm{NC}(u-u_\mathrm{ECR})\Vert \lesssim \Vert \nabla u-{\Pi _0}\nabla u\Vert +osc(f), \end{aligned}$$

where

$$\begin{aligned} osc(f)=\left( \sum _{K\in \mathcal {T}}h_K^2\left[ \inf _{\bar{f}\in P_r(K)}\Vert f-\bar{f}\Vert ^2_{L^2(K)}\right] \right) ^{1/2}, \end{aligned}$$

$r\ge 0$ is arbitrary. The convergence of the ECR element follows immediately.