In this paper we introduce Crouzeix-Raviart elements of general polynomial order k and spatial dimension <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$d\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$k=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, these degrees of feedom can be split into simplex and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\left( d-1\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mfenced> </math></EquationSource> </InlineEquation> dimensional facet integrals in such a way that, in a basis representation of Crouzeix-Raviart functions, all coefficients which correspond to basis functions related to lower-dimensional faces in the mesh are determined by these facet integrals. It will also be shown that such a set of degrees of freedom does not exist in higher space dimension and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.

Crouzeix-Raviart elements on simplicial meshes in d dimensions

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发表日期：

2026年2月27日

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Nis-Erik Bohne,
Patrick Ciarlet Jr.,
Stefan Sauter

摘要

In this paper we introduce Crouzeix-Raviart elements of general polynomial order k and spatial dimension $d\ge 2$ $d \geq 2$ 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 $k=1$ $k = 1$ , these degrees of feedom can be split into simplex and $\left( d-1\right) $ $(d - 1)$ dimensional facet integrals in such a way that, in a basis representation of Crouzeix-Raviart functions, all coefficients which correspond 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 not exist in higher space dimension and $k>1$ $k > 1$ .