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Two one-parameter families of nonconforming enrichments of the Crouzeix-Raviart finite element

Federico Nudo

TL;DR

The study addresses improving the accuracy of the nonconforming Crouzeix–Raviart finite element by introducing two one-parameter quadratic enrichment families. Each family augments the standard CR space with weighted line-integral functionals and quadratic enrichment functions, yielding enriched DOFs and a reproducing $\mathbb{P}_2(T)$ space with explicit basis functions and stable interpolation operators. The authors prove that both enrichment families form finite elements for admissible parameter ranges ($\alpha>-1$, $\alpha\neq -\frac{6}{7}$ and $\beta>-1$) and provide explicit basis constructions $\psi_i,\zeta_i$ and $\tau_i,\rho_i$, along with operators that preserve all CR functionals and enrichments. Numerical experiments demonstrate that the enriched elements achieve lower $L^1$-norm errors than the standard CR element, with superior performance on meshes with more elements, illustrating practical gains in accuracy for elliptic PDE approximations. This enrichment framework offers a robust and systematic way to enhance nonconforming FEMs without increasing the computational complexity beyond polynomial-based enrichment.

Abstract

In this paper, we introduce two one-parameter families of quadratic polynomial enrichments designed to enhance the accuracy of the classical Crouzeix--Raviart finite element. These enrichments are realized by using weighted line integrals as enriched linear functionals and quadratic polynomial functions as enrichment functions. To validate the effectiveness of our approach, we conduct numerical experiments that confirm the improvement achieved by the proposed method.

Two one-parameter families of nonconforming enrichments of the Crouzeix-Raviart finite element

TL;DR

The study addresses improving the accuracy of the nonconforming Crouzeix–Raviart finite element by introducing two one-parameter quadratic enrichment families. Each family augments the standard CR space with weighted line-integral functionals and quadratic enrichment functions, yielding enriched DOFs and a reproducing space with explicit basis functions and stable interpolation operators. The authors prove that both enrichment families form finite elements for admissible parameter ranges (, and ) and provide explicit basis constructions and , along with operators that preserve all CR functionals and enrichments. Numerical experiments demonstrate that the enriched elements achieve lower -norm errors than the standard CR element, with superior performance on meshes with more elements, illustrating practical gains in accuracy for elliptic PDE approximations. This enrichment framework offers a robust and systematic way to enhance nonconforming FEMs without increasing the computational complexity beyond polynomial-based enrichment.

Abstract

In this paper, we introduce two one-parameter families of quadratic polynomial enrichments designed to enhance the accuracy of the classical Crouzeix--Raviart finite element. These enrichments are realized by using weighted line integrals as enriched linear functionals and quadratic polynomial functions as enrichment functions. To validate the effectiveness of our approach, we conduct numerical experiments that confirm the improvement achieved by the proposed method.
Paper Structure (4 sections, 11 theorems, 108 equations, 2 figures, 6 tables)

This paper contains 4 sections, 11 theorems, 108 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. A first one-parameter family of enrichment of the Crouzeix--Raviart finite element
  3. A second one-parameter family of enrichment of the Crouzeix--Raviart finite element
  4. Numerical experiments

Key Result

Theorem 2.1

The basis functions $\varphi_i,\phi_i$, $i=1,2,3,$ of $\mathbb{P}_2(T)$ associated to the finite element $AF3$ have the following expressions

Figures (2)

  • Figure 1: Delaunay triangulation of $N=722$ (left), $N=6962$ (center) and $N=19602$ (right) tringles.
  • Figure 2: Delaunay triangulation of $N=306$ (left), $N=2650$ (center) and $N=23576$ (right) tringles with no angle smaller than ${20}^{\circ}$.

Theorems & Definitions (14)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Remark 3.1
  • Lemma 3.2
  • ...and 4 more