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Crouzeix-Raviart elements on simplicial meshes in $d$ dimensions

Nis-Erik Bohne, Patrick Ciarlet, Stefan Sauter

TL;DR

This work advances CR non-conforming finite elements to general polynomial order and dimension on simplicial meshes by constructing explicit CR bases using orthogonal polynomials on simplices and face bubbles, and by proving a direct sum decomposition with conforming spaces. It introduces bidual DOFs and a local interpolation operator, enabling stable, local, and consistent approximations that inherit the conforming companion’s approximation properties. In two dimensions with odd $k$, it further develops edge-based DOFs and a practical quasi-interpolation operator; the paper also establishes nonexistence results for split DOFs in higher dimensions ($d\ge3$) and $k\neq1$, delineating the limits of local DOF constructions. Overall, the framework yields robust, high-order, non-conforming CR spaces with explicit bases and biorthogonal DOFs suitable for efficient numerical implementation and analysis.

Abstract

In this paper we introduce Crouzeix-Raviart elements of general polynomial order $k$ and spatial dimension $d\geq2$ for simplicial finite element meshes. We give explicit representations of the non-conforming basis functions and prove that the conforming companion space, i.e., the conforming finite element space of polynomial order $k$ is contained in the Crouzeix-Raviart space. We prove a direct sum decomposition of the Crouzeix-Raviart space into (a subspace of) the conforming companion space and the span of the non-conforming basis functions. Degrees of freedom are introduced which are bidual to the basis functions and give rise to the definition of a local approximation/interpolation operator. In two dimensions or for $k=1$, these freedoms can be split into simplex and $\left( d-1\right) $ dimensional facet integrals in such a way that, in a basis representation of Crouzeix-Raviart functions, all coefficients which belong to basis functions related to lower-dimensional faces in the mesh are determined by these facet integrals. It will also be shown that such a set of degrees of freedom does \textbf{not} exist in higher space dimension and $k>1$.

Crouzeix-Raviart elements on simplicial meshes in $d$ dimensions

TL;DR

This work advances CR non-conforming finite elements to general polynomial order and dimension on simplicial meshes by constructing explicit CR bases using orthogonal polynomials on simplices and face bubbles, and by proving a direct sum decomposition with conforming spaces. It introduces bidual DOFs and a local interpolation operator, enabling stable, local, and consistent approximations that inherit the conforming companion’s approximation properties. In two dimensions with odd , it further develops edge-based DOFs and a practical quasi-interpolation operator; the paper also establishes nonexistence results for split DOFs in higher dimensions () and , delineating the limits of local DOF constructions. Overall, the framework yields robust, high-order, non-conforming CR spaces with explicit bases and biorthogonal DOFs suitable for efficient numerical implementation and analysis.

Abstract

In this paper we introduce Crouzeix-Raviart elements of general polynomial order and spatial dimension for simplicial finite element meshes. We give explicit representations of the non-conforming basis functions and prove that the conforming companion space, i.e., the conforming finite element space of polynomial order is contained in the Crouzeix-Raviart space. We prove a direct sum decomposition of the Crouzeix-Raviart space into (a subspace of) the conforming companion space and the span of the non-conforming basis functions. Degrees of freedom are introduced which are bidual to the basis functions and give rise to the definition of a local approximation/interpolation operator. In two dimensions or for , these freedoms can be split into simplex and dimensional facet integrals in such a way that, in a basis representation of Crouzeix-Raviart functions, all coefficients which belong to basis functions related to lower-dimensional faces in the mesh are determined by these facet integrals. It will also be shown that such a set of degrees of freedom does \textbf{not} exist in higher space dimension and .
Paper Structure (15 sections, 18 theorems, 168 equations)

This paper contains 15 sections, 18 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Crouzeix-Raviart Finite Elements
  3. A direct sum decomposition of the canonical Crouzeix-Raviart space
  4. A basis for Crouzeix-Raviart spaces $\operatorname*{CR}_{k}\left( \mathcal{T}\right)$ and $\operatorname*{CR}_{k,0}\left( \mathcal{T}\right)$
  5. The simplicial complex of a simplicial finite element mesh
  6. The face bubble functions
  7. Orthogonal polynomials on the reference simplex
  8. Basis functions associated with simplex faces
  9. A basis for $\operatorname*{CR}_{k}\left( \mathcal{T}\right)$ and $\operatorname*{CR}_{k,0}\left( \mathcal{T}\right)$
  10. Degrees of freedom for the case $d\geq2$ and $k$ is odd
  11. Degrees of freedom and an approximation operator for the case $d=2$ and $k$ is odd
  12. Degrees of freedom
  13. An approximation operator
  14. Non-existence of split facet/simplex degrees of freedom for $d\geq3$ and $k\neq1$
  15. Proof of the determinant formula (\ref{['defformula']})

Key Result

Lemma 5

The functions $B_{k}^{\operatorname*{CR},F}$ and $B_{k}^{\operatorname*{CR},K}$ in Definition DefNCShapeFct belong to $\operatorname*{CR}\nolimits_{k}^{\max}\left( \mathcal{T}\right)$.

Theorems & Definitions (34)

  • Definition 2
  • Remark 3
  • Lemma 5
  • Definition 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Corollary 10
  • Lemma 11
  • Lemma 12
  • ...and 24 more