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A new quadratic and cubic polynomial enrichment of the Crouzeix-Raviart finite element

Francesco Dell'Accio, Allal Guessab, Federico Nudo

TL;DR

The paper addresses improving the nonconforming Crouzeix-Raviart finite element by introducing quadratic and cubic polynomial enrichments via weighted line integral degrees of freedom. It develops two rigorous finite element constructions, a quadratic one with $\mathcal{C}=(T,\mathbb{P}_2(T),\Sigma_{2,T}^{\mathrm{enr}})$ and a cubic one with $\mathcal{S}=(T,\mathbb{P}_3(T),\Sigma_{3,T}^{\mathrm{enr}})$, and provides explicit bases and interpolation operators $\Pi_2^{\mathrm{enr}}$ and $\Pi_3^{\mathrm{enr}}$ that reproduce $\mathbb{P}_2(T)$ and $\mathbb{P}_3(T)$, respectively. These enrichments rely on two-parameter families of admissible DOFs defined by $\mathcal{F}_j^{\mathrm{enr}}$ and $\mathcal{L}_j^{\mathrm{enr}}$, with orthogonality properties ensuring well posedness for $\alpha,\beta>-\tfrac{1}{2}$. Numerical experiments demonstrate substantial accuracy gains over the standard CR element, with the cubic enrichment achieving the best performance on finer meshes, indicating a practical enrichment strategy for nonconforming FEM on triangles.

Abstract

In this paper, we introduce quadratic and cubic polynomial enrichments of the classical Crouzeix--Raviart finite element, with the aim of constructing accurate approximations in such enriched elements. To achieve this goal, we respectively add three and seven weighted line integrals as enriched degrees of freedom. For each case, we present a necessary and sufficient condition under which these augmented elements are well-defined. For illustration purposes, we then use a general approach to define two-parameter families of admissible degrees of freedom. Additionally, we provide explicit expressions for the associated basis functions and subsequently introduce new quadratic and cubic approximation operators based on the proposed admissible elements. The efficiency of the enriched methods is compared to the triangular Crouzeix--Raviart element. As expected, the numerical results exhibit a significant improvement, confirming the effectiveness of the developed enrichment strategy.

A new quadratic and cubic polynomial enrichment of the Crouzeix-Raviart finite element

TL;DR

The paper addresses improving the nonconforming Crouzeix-Raviart finite element by introducing quadratic and cubic polynomial enrichments via weighted line integral degrees of freedom. It develops two rigorous finite element constructions, a quadratic one with and a cubic one with , and provides explicit bases and interpolation operators and that reproduce and , respectively. These enrichments rely on two-parameter families of admissible DOFs defined by and , with orthogonality properties ensuring well posedness for . Numerical experiments demonstrate substantial accuracy gains over the standard CR element, with the cubic enrichment achieving the best performance on finer meshes, indicating a practical enrichment strategy for nonconforming FEM on triangles.

Abstract

In this paper, we introduce quadratic and cubic polynomial enrichments of the classical Crouzeix--Raviart finite element, with the aim of constructing accurate approximations in such enriched elements. To achieve this goal, we respectively add three and seven weighted line integrals as enriched degrees of freedom. For each case, we present a necessary and sufficient condition under which these augmented elements are well-defined. For illustration purposes, we then use a general approach to define two-parameter families of admissible degrees of freedom. Additionally, we provide explicit expressions for the associated basis functions and subsequently introduce new quadratic and cubic approximation operators based on the proposed admissible elements. The efficiency of the enriched methods is compared to the triangular Crouzeix--Raviart element. As expected, the numerical results exhibit a significant improvement, confirming the effectiveness of the developed enrichment strategy.
Paper Structure (4 sections, 11 theorems, 117 equations, 4 figures)

This paper contains 4 sections, 11 theorems, 117 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A quadratic polynomial enrichment of the Crouzeix--Raviart finite element
  3. A cubic polynomial enrichment of the Crouzeix--Raviart finite element
  4. Numerical experiments

Key Result

Lemma 2.1

Let $\gamma_0,\ldots, \gamma_d$ be nonnegative real numbers. Then the following identity holds where $\left\lvert T_d\right\rvert$ is the volume of $T_d$ and $\Gamma(z)$ is the gamma function Abramowitz:1948:HOM.

Figures (4)

  • Figure 1: Delaunay triangulation of $N=33$, $N=306$, $N=2650$ and $N=23576$ tringles.
  • Figure 2: Loglog plot of errors in $L^1$-norm for approximating the functions $f_1$ (left) and $f_2$ (right). The blue line represents the trend of approximation errors obtained using the standard Crouzeix--Raviart finite element. The magenta line represents the trend of errors obtained with the quadratic enriched finite element $\mathcal{C}$, while the black line represents the trend of errors obtained with the cubic enriched finite element $\mathcal{S}$ with $\alpha=\beta=1$. The comparisons are conducted using Delaunay triangulations from Figure \ref{['Fig:regulatri']}.
  • Figure 3: Loglog plot of errors in $L^1$-norm for approximating the functions $f_3$ (left) and $f_4$ (right). The blue line represents the trend of approximation errors obtained using the standard Crouzeix--Raviart finite element. The magenta line represents the trend of errors obtained with the quadratic enriched finite element $\mathcal{C}$, while the black line represents the trend of errors obtained with the cubic enriched finite element $\mathcal{S}$ with $\alpha=\beta=1$. The comparisons are conducted using Delaunay triangulations from Figure \ref{['Fig:regulatri']}.
  • Figure 4: Loglog plot of errors in $L^1$-norm for approximating the functions $f_5$ (left) and $f_6$ (right). The blue line represents the trend of approximation errors obtained using the standard Crouzeix--Raviart finite element. The magenta line represents the trend of errors obtained with the quadratic enriched finite element $\mathcal{C}$, while the black line represents the trend of errors obtained with the cubic enriched finite element $\mathcal{S}$ with $\alpha=\beta=1$. The comparisons are conducted using Delaunay triangulations from Figure \ref{['Fig:regulatri']}.

Theorems & Definitions (17)

  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • Theorem 2.10
  • ...and 7 more