Polytopal templates for the formulation of semi-continuous vectorial nite elements of arbitrary order Adam Sky1and Ingo Muench2

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Polytopal templates for the formulation of semi-continuous vectorial

ﬁnite elements of arbitrary order

Adam Sky1and Ingo Muench2

December 29, 2022

Abstract

The Hilbert spaces H(curl) and H(div) are needed for variational problems formulated in the context of

the de Rham complex in order to guarantee well-posedness. Consequently, the construction of conforming

subspaces is a crucial step in the formulation of viable numerical solutions. Alternatively to the standard

deﬁnition of a ﬁnite element as per Ciarlet, given by the triplet of a domain, a polynomial space and degrees

of freedom, this work aims to introduce a novel, simple method of directly constructing semi-continuous

vectorial base functions on the reference element via polytopal templates and an underlying H1-conforming

polynomial subspace. The base functions are then mapped from the reference element to the element in

the physical domain via consistent Piola transformations. The method is deﬁned in such a way, that the

underlying H1-conforming subspace can be chosen independently, thus allowing for constructions of arbitrary

polynomial order. The base functions arise by multiplication of the basis with template vectors deﬁned for

each polytope of the reference element. We prove a unisolvent construction of N´ed´elec elements of the ﬁrst

and second type, Brezzi-Douglas-Marini elements, and Raviart-Thomas elements. An application for the

method is demonstrated with two examples in the relaxed micromorphic model.

Key words: polytopal templates, N´ed´elec elements, Brezzi-Douglas-Marini elements, Raviart-Thomas

elements, Piola transformations, relaxed micromorphic model.

1 Introduction

In many variational problems, well-posedness necessitates the use of either the H(curl) or H(div) Hilbert spaces.

Some classical examples are Maxwell’s equations [21–23] and mixed Poisson problems [7]. More recent examples

are the tangential-displacement-normal-normal-stress (TDNNS) method in elasticity [29, 34], and the relaxed

micromorphic model [26, 27, 30]. Other examples are curl based plasticity models [11, 15]. Commonly, the

application of the H(curl) or H(div) spaces arises in problems associated with the de Rham complex [13,32]. In

fact, as shown in [6], complexes are a powerful and general tool for mixed variational formulations. The latter

is also demonstrated in [31] and [8, 20, 33] where the elasticity complex and the div Div -complex are explored

in the context of mixed formulations of linear elasticity and the biharmonic equation, respectively.

Since analytical solutions to partial diﬀerential equations are rarely possible for general domain geometries

or boundary conditions, the application of numerical schemes with conforming subspaces is required. Unlike in

the classical Hilbert space H1, an element of the H(curl)-space is only required to be tangentially continuous

[41]. Analogously, elements of the H(div)-space are only required to be normal-continuous [41]. As such, the

formulation of H(curl)- and H(div)-conforming ﬁnite elements is more complex. The pioneering works [24]

and [25] introduced the N´ed´elec elements of the ﬁrst and second types, which represent polynomial subspaces

with the minimal regularity requirements of the H(curl)-space Np

II ⊂ N p

I⊂H(curl). In [9] and [35] the

authors introduced the Brezzi-Douglas-Marini and Raviart-Thomas elements, that allow to construct polynomial

subspaces for the H(div)-space BDMp⊂ RT p⊂H(div), such that the elements exhibit the minimal regularity

1Corresponding author: Adam Sky, Institute of Structural Mechanics, Statics and Dynamics, Technische Universit¨at Dortmund,

August-Schmidt-Str. 8, 44227 Dortmund, Germany, email: adam.sky@tu-dortmund.de

2Ingo Muench, Institute of Structural Mechanics, Statics and Dynamics, Technische Universit¨at Dortmund, August-Schmidt-Str.

8, 44227 Dortmund, Germany, email: ingo.muench@tu-dortmund.de

arXiv:2210.03525v2 [math.NA] 27 Dec 2022

needed in H(div). The elements are given in the classical element deﬁnition as per Ciarlet [10], and allow for

application on general grids.

An alternative methodology to construction of a basis directly on the grid is to build base functions on

the reference elements and map them to the physical elements on the grid by consistent transformations. The

construction of low order vectorial ﬁnite elements is demonstrated in [5,37–40]. The formulation of higher order

elements on the basis of Legendre polynomials can be found in [36,41,42]. Further, in [2,4], the authors present

a higher order construction based on Bernstein polynomials. We note that in general, the mapping alone does

not suﬃce in order to assert a consistent transformation and some additional algorithmic is required in order

to avoid the orientation problem [3, 5, 16, 39, 40, 42].

The aim of this work is to establish a method of deﬁning H(curl) and H(div) base functions on the reference

element, such that the underlying polynomial basis can be chosen independently. As such, the method allows

to directly construct conforming ﬁnite elements by using for example, Lagrange, Legendre, Jacobi or Bernstein

polynomials. This goal is achieved by deﬁning a template on the reference element, which can be subsequently

used in conjunction with an H1-conforming polynomial basis of one’s choice, in order to span a semi-continuous

ﬁnite element space. The template is composed of vector sets associated with the polytopes of the reference

element. Consequently, we dub the methodology ”polytopal templates”. In this work we consider subspaces for

the Hilbert spaces H(curl) and H(div).

This paper is structured as follows. First, we introduce the classical Hilbert spaces and their corresponding

diﬀerential and trace operators. Next, we derive two-dimensional polytopal templates for the construction of

N´ed´elec elements of the ﬁrst and second type, Brezzi-Douglas-Marini elements, and Raviart-Thomas elements on

the reference triangle. The methodology is subsequently utilized to derive polytopal templates on the reference

tetrahedron for N´ed´elec elements of the second type and Brezzi-Douglas-Marini elements. We demonstrate the

application of elements using the relaxed micromorphic model with one example in antiplane shear and one

three-dimensional example. Lastly, we present our conclusions and outlook.

The following deﬁnitions are employed throughout this work, see also Fig. 1:

•Vectors are indicated by bold letters. Non-bold letters represent scalars.

•In general, formulas are deﬁned using the Cartesian basis, where the base vectors are denoted by e1,e2

and e3.

•Three-dimensional domains in the physical space are denoted with V⊂R3. The corresponding reference

domain is given by Ω.

•Analogously, in two dimensions we employ A⊂R2for the physical domain and Γ for the reference domain.

•Curves on the physical domain are denoted by s, whereas curves in the reference domain by µ.

•The tangent and normal vectors in the physical domain are given by tand n, respectively. Their coun-

terparts in the reference domain are τfor tangent vectors and νfor normal vectors.

2 Hilbert spaces and trace operators

Hilbert spaces are the natural function spaces in the formulation of variational problems [23]. In preparation for

the construction of conforming subspaces we introduce the classical Hilbert spaces and their associated norms

H1(V) = {u∈L2(V)| ∇u∈[L2(V)]3},kuk2

H1(V)=kuk2

L2+k∇uk2

L2,(2.1a)

H(curl, V ) = {u∈[L2(V)]3|curl u∈[L2(V)]3},kuk2

H(curl,V )=kuk2

L2+kcurl uk2

L2,(2.1b)

H(div, V ) = {u∈[L2(V)]3|div u∈L2(V)},kuk2

H(div,V )=kuk2

L2+kdiv uk2

L2,(2.1c)

which are based on the Lebesgue space

L2(V) = {u:V→R| kukL2(V)<∞} ,kuk2

L2(V)=hu, uiL2(V)=ZV

u2dV . (2.2)

Ω

x: Ω →V

Figure 1: A domain Vwith Dirichlet and Neumann boundaries mapped from some reference domain Ω.

Note that on two-dimensional domains the diﬀerential operators are reduced to

∇u=u,x

u,y,div(Ru) = u2,x −u1,y ,R∇u=u,y

−u,x,R=0 1

−1 0,div u=u1,x +u2,y ,(2.3)

such that two curl operators are introduced: one for two-dimensional vectors div(R·), and one for scalars R∇(·).

On contractible domains the Hilbert spaces are connected by the exact de Rham sequence [6, 12, 13, 32], see

Fig. 2.

Each Hilbert space is associated with a corresponding trace operator [19]. The trace of the function u∈

H1(V) is deﬁned by the linear bounded operator

tr u=u∂V

∈H1/2(∂V ),∃c > 0 : ktr ukH1/2(∂V )≤ckukH1(V)∀u∈H1(V).(2.4)

In other words, the trace restricts the function to the boundary of the domain. The trace of the H(curl, V )

space is given by the tangential components on the boundary. On a surface the tangent vector is not unique,

and therefore, the tangential projection is deﬁned using the normal vector and the cross product

tr⊥

nu=n×u∂V

∈[H−1/2(∂V )]3,∃c > 0 : ktr⊥

nukH−1/2(∂V )≤ckukH(curl,V )∀u∈H(curl, V ).(2.5)

For two-dimensional domains A⊂R2the tangent vector is unique and the trace operator reduces to

trk

tu=ht,ui∂A

∈H−1/2(∂A).(2.6)

Lastly, the trace of the H(div, V ) space is deﬁned by the normal projection at the boundary

trk

nu=hn,ui∂V

∈H−1/2(∂V ),∃c > 0 : ktrk

nukH−1/2(∂V )≤ckukH(div,V )∀u∈H(div, V ).(2.7)

In this work we deﬁne the trace space via

H1/2(∂V ) = {v∈L2(∂V )| ∃ u∈H1(V) : tr u=v},(2.8)

and H−1/2(∂V ) is its dual. A thorough treatment of fractional Sobolev spaces is found in [14]. The trace

operators are used to deﬁne Hilbert spaces with boundary conditions. The exact de Rham sequence holds also

on Hilbert spaces with vanishing traces, see Fig. 3. Further, the trace operators allow to identify ﬁnite elements

of a speciﬁc Hilbert space via interface conditions. We state the interface theorem [28, 41] and apply it to the

construction of conforming subspaces.

Rid H1(V)∇H(curl, V )curl H(div, V )div L2(V)

Rid H1(A)∇H(curl, A)divRL2(A)

Rid H1(A)R∇H(div, A)div L2(A)

Figure 2: Classical de Rham exact sequences for three- and two-dimensional contractible domains. The range

of each operator is exactly the kernel of the next operator in the sequence.

Rid H1

0(V)∇H0(curl, V )curl H0(div, V )div L2

0(V)

Rid H1

0(A)∇H0(curl, A)divRL2

0(A)

Rid H1

0(A)R∇H0(div, A)div L2

0(A)

Figure 3: De Rham exact sequences for Hilbert spaces with vanishing traces. The Lebesgue zero-space is

characterized by functions with a vanishing integral over the domain.

Theorem 2.1 (Interface conditions)

A ﬁnite element space is a conforming subspace of a Hilbert space if and only if the jump of the trace of its

elements vanishes for all arbitrarily deﬁned interfaces Ξij =Vi∩Vj, i 6=jwhere V=Vi∪Vj⊂R3and Ξij ⊂R2

(and analogously for two-dimensional domains)

u∈H1(V)⇐⇒ [[tr u]] Ξij

= 0 ∀Ξij =Vi∩Vj,(2.9a)

u∈H(curl, V )⇐⇒ [[tr⊥

nu]] Ξij

= 0 ∀Ξij =Vi∩Vj,(2.9b)

u∈H(div, V )⇐⇒ [[trk

nu]] Ξij

= 0 ∀Ξij =Vi∩Vj.(2.9c)

In the following sections the construction of arbitrary order H(curl)- and H(div)-conforming subspaces is

presented. The construction is based on a polytopal association of base functions.

Deﬁnition 2.1 (Polytopal base functions)

Each base function is associated with its respective polytope and the underlying Hilbert space as follows:

1. A vertex base function has a vanishing trace on all other vertices and non-neighbouring edges and faces.

2. An edge base function has a vanishing trace on all other edges and non-neighbouring faces.

3. A face base function has a vanishing trace on all other faces.

4. A cell base function has a vanishing trace on the entire boundary of the element.

The deﬁnition is general and the respective trace may change according to the corresponding Hilbert space.

v1v3

e12

e13

e23

c123

Figure 4: Decomposition of the unit triangle into vertices, edges and the cell.

3 Two-dimensional templates

This section is dedicated to the introduction of polytopal templates on the reference triangle

Γ = {(ξ, η)∈[0,1]2|ξ+η≤1}.(3.1)

To that end, the triangle is decomposed into its base polytopes given by its vertices {v1, v2, v3}, its edges

{e12, e13, e23}, and its interior cell c123, see Fig. 4. Further, each polytope is associated with base functions

belonging to an H1-conforming subspace Up(Γ) with dim Up(Γ) = dim Pp(Γ) = (p+ 2)(p+ 1)/2.

Deﬁnition 3.1 (Triangle Up(Γ)-polytopal spaces)

Each polytope is associated with a space of base functions as follows:

•Each vertex is associated with the space of its respective base function Vp

i. As such, there are three spaces

in total i∈ {1,2,3}and each one is of dimension one, dim Vp

i= 1 ∀i∈ {1,2,3}.

•For each edge there exists a space of edge functions Ep

jwith the multi-index j∈ J ={(1,2),(1,3),(2,3)}.

The dimension of each edge space is given by dim Ep

j=p−1.

•Lastly, the cell is equipped with the space of cell base functions Cp

123 with dim Cp

123 = (p−2)(p−1)/2.

The association with a respective polytope reﬂects Deﬁnition 2.1 with the trace operator for H 1-spaces.

A depiction of vertex, edge, and cell base functions is given in Fig. 5. Clearly, this is the standard deﬁnition

of base functions for approximations in H1. Common examples of such bases are Lagrange, Legendre and

Bernstein.

In the following we deﬁne templates on the reference triangle, such that their multiplications with corre-

sponding base functions from Up(Γ) generate vectorial base functions for either H(curl) or H(div).

3.1 N´ed´elec II

In order to construct a template for the N´ed´elec element of the second type [25] we consider the decomposition

of the reference triangle in Fig. 4. On the ﬁrst vertex v1we deﬁne a vector with a projection of one on the

tangent vector of the ﬁrst edge e12 and a zero projection on the second edge e13. Next we deﬁne a vector with a

projection of one on the tangent vector of the ﬁrst edge e12. Further, we construct a normal vector on the ﬁrst

edge e12. Lastly, we deﬁne two unit vectors in the cell. The remaining vectors for their respective polytopes are

computed by mapping the triangle c123 to various permutations of cijk on the unit domain via covariant Piola

transformations (see Appendix A) and adjusting the sign to ensure a positive projection on the tangent vector,

see Fig. 6. The complete template is depicted in Fig. 7.

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Polytopaltemplatesfortheformulationofsemi-continuousvectorialniteelementsofarbitraryorderAdamSky1andIngoMuench2December29,2022AbstractTheHilbertspacesH(curl)andH(div)areneededforvariationalproblemsformulatedinthecontextofthedeRhamcomplexinordertoguaranteewell-posedness.Consequently,theconstructiono...

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Polytopal templates for the formulation of semi-continuous vectorial nite elements of arbitrary order Adam Sky1and Ingo Muench2

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