# Partially discontinuous nodal finite elements for $H(\operatorname{curl})$
and $H(\operatorname{div})$

Jun Hu and Kaibo Hu and Qian Zhang  LMAM and School of Mathematical Sciences,
Peking University, Beijing 100871, P. R. China<EMAIL_ADDRESS>Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe
Observatory Quarter, Oxford, OX2 6GG, UK<EMAIL_ADDRESS>Department
of Mathematical Sciences, Michigan Technological University, Houghton, MI
49931, USA<EMAIL_ADDRESS>
###### Abstract.

We investigate discretization of $H(\operatorname{curl})$ and
$H(\operatorname{div})$ in two and three space dimensions by partially
discontinuous nodal finite elements, i.e., vector-valued Lagrange finite
elements with discontinuity in certain directions. These spaces can be
implemented as a combination of continuous and discontinuous Lagrange elements
and fit in de Rham complexes. We construct well-conditioned nodal bases.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and existing results
    1. 2.1 Preliminaries
    2. 2.2 Existing results
  3. 3 $H(\operatorname{curl})$ nodal elements in three space dimensions
    1. 3.1 Motivation
    2. 3.2 Construction
  4. 4 $H(\operatorname{div})$ nodal elements in three space dimensions
  5. 5 Partially orthogonal nodal bases
    1. 5.1 Triangular $H(\operatorname{rot})$ element
    2. 5.2 Tetrahedral $H(\operatorname{curl})$ element on the Worsey-Farin split
    3. 5.3 Tetrahedral $H(\operatorname{div})$ elements
    4. 5.4 Condition number
  6. 6 Conclusions

##  1\. Introduction

The edge elements [33, 34] have achieved tremendous success in computational
electromagnetism. These elements fix the notorious spurious solutions in both
eigenvalue and source problems [3, 12, 41]. The key of this success is that
the Nédélec element fits in a finite element de Rham complex (c.f., [41]).
Even though discrete differential forms [5, 11, 25] have been a mature
solution for many problems in computational electromagnetism, there are still
interests in schemes with proper modifications of the vector-valued Lagrange
elements for both theoretical and practical reasons. A theoretical question is
what elements can solve the Maxwell equations without spurious solutions. In
[41] the authors showed that the vector-valued Lagrange element on the Powell-
Sabin mesh avoids spurious eigenmodes since the kernel of the
$\operatorname{rot}$ operator is characterized by the Powell-Sabin $C^{1}$
element. In a modern language, this means that the vector-valued Lagrange
space on this mesh fits in a smoother de Rham (Stokes) complex [10].
Nevertheless, the $C^{0}$ continuity may still lead to spurious solutions of
the source problem on L-shaped domains [3, Chapter 5]. Practically, it is
crucial to construct efficient bases for high order elements. This is always
an art since one has to balance condition numbers, sparsity, fast evaluation,
rotational symmetry, etc. See, e.g., [1, 30, 38, 42, 44, 46, 48, 49]. Nodal
elements may also be preferred due to the canonical implementation [8, 14].
Finite elements with extra vertex/edge smoothness (supersmoothness) also find
applications in solving the Hamilton-Jacobi-Bellman equations and may avoid
averaging in semi-Lagrange methods [43].

This paper investigates another variant of the vector-valued Lagrange finite
elements with the following desirable properties: 1) The elements allow
discontinuity (not $C^{0}$-conforming), such that spurious solutions arising
from regularity should not appear. 2) The elements fit in complexes. 3) There
exist Lagrange type nodal bases, and thus the construction (condition number,
orthogonality) can be based on results of scalar problems, c.f., [31]. The
idea of the construction is to allow discontinuity of certain components
across the faces of the elements. We will refer to these elements as partially
discontinuous nodal elements. In two space dimensions (2D), the Stenberg
element [40] satisfies all these conditions. The corresponding complex was
investigated in [16] (see also [22] for the cubical case). Nevertheless, the
three dimensional (3D) case was left open in [16] (see Section 2.2 for a
detailed review).

This paper aims to construct the 3D version of the partially discontinuous
nodal finite elements. In particular, we obtain two finite element de Rham
complexes: one contains an $H(\operatorname{div})$-conforming partially
discontinuous nodal element, and another contains an $H(\operatorname{curl})$
version (note that they cannot fit in the same complex). The
$H(\operatorname{div})$ element was constructed in [16]. To the best of our
knowledge, the complex constructed herein is new. Finite elements for
$H(\operatorname{curl})$ with a higher order supersmoothness can be found in
[16]. However, they do not admit Lagrange type bases. Complexes containing
vector-valued Lagrange elements on special meshes were constructed in [21,
23], primally in the context of fluid problems. Nevertheless, the $C^{0}$
continuity does not exclude spurious solutions for the Maxwell source
problems. In this paper, we obtain a new partially discontinuous nodal
$H(\operatorname{curl})$ finite element and the corresponding complex on the
Worsey-Farin split of tetrahedra.

Breaking some face degrees of freedom (e.g., treating the degrees of freedom
of the tangential components as interior ones by allowing multi-values on
neighboring elements) is equivalent to enriching face-based bubble functions.
The idea of using matrix-valued Lagrange spaces enriched with
$H(\operatorname{div})$ symmetric matrix bubbles led to partially
discontinuous nodal elements for the Hellinger-Reissner principle [26, 27,
28], see [15] for an extension of this idea.

The rest of this paper will be organized as follows. In Section 2, we
introduce notation and review existing results. In Section 3 we present a new
$H(\operatorname{curl})$ element and the corresponding complex. In Section 4,
we construct a discrete de Rham complex which contains the nodal
$H(\operatorname{div})$ elements [16]. In Section 5, we construct well-
conditioned bases for both 2D and 3D partially discontinuous nodal elemets. We
end the paper with concluding remarks.

##  2\. Preliminaries and existing results

###  2.1. Preliminaries

We assume that $\Omega$ is a polygonal or polyhedral domain. Given a mesh, we
will use $\mathcal{V}$ to denote the set of vertices, $\mathcal{E}$ for edges,
$\mathcal{F}$ for faces, and $\mathcal{T}$ for the 3D simplices. We use $V$,
$E$, $F$, and $T$ to denote the number of vertices, edges, faces, and
tetrahedra, respectively. From Euler’s formula, one has $V-E+F=1$ in 2D and
$V-E+F-T=1$ in 3D for contractible domains.

For a triangle $f$, we denote its Clough-Tocher split as
$f_{\operatorname{CT}}$, which is the collection of three triangles obtained
by connecting the vertices to an interior point of $f$. For a tetrahedron $K$
in $\mathcal{T}$, the Worsey-Farin split of $K$, denoted by
$K_{\operatorname{WF}}$, is obtained by adding a vertex to the interior of
$K$, connecting this vertex to the vertices of $K$; then adding an interior
vertex to each face of $K$, and connecting this point to the vertices of the
face and the interior point (see Figure 1). Therefore, each face is divided
into three triangles like the Clough-Tocher split. The set of the Worsy-Farin
splits of all the tetrahedra in $\mathcal{T}$ is referred to as the Worsey-
Farin mesh and denoted by $\mathcal{T}_{\operatorname{WF}}$. We refer to [21,
23, 32] for more details.

We use $\bm{\nu}_{f}$ and $\bm{\tau}_{f}$ to denote the unit normal and
tangential vectors of a simplex $f$, respectively. In 2D, the tangential and
normal directions of an edge are uniquely defined up to an orientation. For
edges in 3D there are one unit tangential and two linearly independent unit
normal directions, and for faces in 3D there are one unit normal and two
linearly independent unit tangential directions. We will write
$\bm{\tau}_{e}$, $\bm{\nu}_{e,i}$ and $\bm{\nu}_{f}$, $\bm{\tau}_{f,i}$,
$i=1,2$ for these cases, respectively.

In our discussions, $C^{r}(\mathcal{V})$ includes piecewise smooth functions
for which derivatives up to order $r$ coincide at vertices. Similarly we can
define $C^{r}(\mathcal{E})$ and $C^{r}(\mathcal{F})$, $C^{r}(\mathcal{T})$ for
the space of functions with $C^{r}$ continuity on the edges, faces, and 3D
cells, repectively.

In $d$D ($d=2,3$), we define the gradient operator
$\operatorname{grad}=\nabla$ and the $\operatorname{div}$ operator
$\operatorname{div}=\nabla\cdot$. In 3D, for
$\bm{u}=(u_{1},u_{2},u_{3})^{\mathrm{T}}$, we define

$\operatorname{curl}\bm{u}=(\partial_{x_{2}}u_{3}-\partial_{x_{3}}u_{2},\partial_{x_{3}}u_{1}-\partial_{x_{1}}u_{3},\partial_{x_{1}}u_{2}-\partial_{x_{2}}u_{1})^{\mathrm{T}}.$

In 2D, we define

$\operatorname{rot}\bm{u}=\partial_{x_{1}}u_{2}-\partial_{x_{2}}u_{1}\text{
for }\bm{u}=(u_{1},u_{2})^{\mathrm{T}}\text{ and
}\operatorname{curl}u=(\partial_{x_{2}}u,-\partial_{x_{1}}u)\text{ for a
scalar }u.$

We review some basic facts about the de Rham complex; further details can be
found, for instance, in [5, 13]. The de Rham complex consists of differential
forms and exterior derivatives. In 3D, the de Rham complex on $\Omega$ reads

(2.1)
${0}$${\mathbb{R}}$${H^{1}(\Omega)}$${H(\operatorname{curl};\Omega)}$${H(\operatorname{div};\Omega)}$${L^{2}(\Omega)}$${0,}$$\scriptstyle{\subset}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$

with
$\operatorname{curl}\operatorname{grad}=\operatorname{div}\operatorname{curl}=0$.
A complex is called exact at a space if the kernel of the subsequent operator
is identical to the range of the previous one. For example, we say that the
complex (2.1) is exact at $H(\operatorname{curl};\Omega)$ if
$\mathcal{N}(\operatorname{curl},H(\operatorname{curl}))=\mathcal{R}(\operatorname{grad},H^{1})$.
Here for a linear operator $d:X\to Y$, $\mathcal{N}(d,X)\subset X$ denotes the
kernel of $d$ and $\mathcal{R}(d,Y)\subset Y$ denotes the range. The complex
(2.1) is called exact if it is exact at all the spaces.

If there are finite element spaces $\Sigma_{h}\subset H^{1}(\Omega)$,
$V_{h}\subset H(\operatorname{curl};\Omega)$, $X_{h}\subset
H(\operatorname{div};\Omega)$, and $W_{h}\subset L^{2}(\Omega)$ such that

(2.2)
${0}$${\mathbb{R}}$${\Sigma_{h}}$${V_{h}}$${X_{h}}$${W_{h}}$${0}$$\scriptstyle{\subset}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$

is a complex, we say that (2.2) is a finite element subcomplex of (2.1). In
this case, a necessary condition for the exactness is the following dimension
condition

(2.3) $\displaystyle\dim(\mathbb{R})-\dim\Sigma_{h}+\dim V_{h}-\dim X_{h}+\dim
W_{h}=0.$

As an example, the Lagrange element space, Nédélec element space of the first
kind [33], Raviart-Thomas (RT) element space [33], and discontinuous element
space fit into a discrete de Rham complex. The Lagrange element space, Nédélec
element space of the second kind [34], Brezzi-Douglas-Marini (BDM) element
space [34], and discontinuous element space also fit into a discrete de Rham
complex. For a detailed discussion of these finite element spaces, see, e.g.,
[9].

We use $\mathcal{P}_{p}$ to denote the polynomial space of degree $p$ (a
nonnegative integer). Denote

$\operatorname{NED}_{p}^{1}=(\mathcal{P}_{p-1})^{3}+S_{p}\text{ with
}S_{p}=\\{q\in(\mathcal{P}_{p})^{3}:q\cdot\bm{x}=0\\}$

as the shape function spaces of the first-kind Nédélec element with polynomial
degree $p$, and

$\operatorname{RT}_{p}=(\mathcal{P}_{p-1})^{3}+\bm{x}\mathcal{P}_{p-1},$

as the shape function space of the RT element with the polynomial degree $p$.
The shape function space of the second-kind Nédélec element and the BDM
element with the polynomial degree $p$ are

$\operatorname{NED}_{p}^{2}=[\mathcal{P}_{p}]^{3}\text{ and
}\operatorname{BDM}_{p}=[\mathcal{P}_{p}]^{3},$

respectively. We denote the Nédélec finite element space of the first and
second kind with the polynomial degree $p$ on mesh $\mathcal{T}$ as
$\operatorname{NED}_{p}^{1}(\mathcal{T})$ and
$\operatorname{NED}_{p}^{2}(\mathcal{T})$, respectively.

In 2D, the de Rham complex reads

${0}$${\mathbb{R}}$${H^{1}(\Omega)}$${H(\operatorname{rot};\Omega)}$${L^{2}(\Omega)}$${0,}$$\scriptstyle{\subset}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{rot}}$

or

${0}$${\mathbb{R}}$${H^{1}(\Omega)}$${H(\operatorname{div};\Omega)}$${L^{2}(\Omega)}$${0.}$$\scriptstyle{\subset}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$

*(0, 0)(0, 1.5)(0.368, 0.748) *(0, 0)(0, 1.5)(-0.1, 0.6)  Figure 1. The Worsey-Farin split of a tetrahedron (relative to one face)

###  2.2. Existing results

In this section, we briefly review the constructions in 2D in [16] and some
questions that were left open there.

In 2D, the Stenberg $H(\operatorname{div})$ element can be obtained by
breaking the tangential component of a vector-valued Lagrange element on each
interior edge of the mesh, i.e., the tangential trace of the functions in the
Stenberg $H(\operatorname{div})$ element space from two sides of an edge can
be different. Therefore, the continuity of the Stenberg element is in between
the corresponding vector-valued $C^{0}$ Lagrange element and the discontinuous
element (see Figure 2). The Stenberg element fits in a discrete de Rham
complex in 2D, which begins with the Hermite element taking vertex derivatives
as degrees of freedom [16].

Figure 2. Continuous, partially discontinuous elements, and discontinuous.
Left: vector-valued Lagrange elements. Both the tangential and the normal
components are continuous across edges. Middle: partially discontinuous nodal
elements for $H(\operatorname{div})$ (Stenberg element). The normal component
is continuous while the tangential component is discontinuous in general.
Right: discontinuous elements. Both tangential and normal components are
discontinuous in general.

The 2D Stenberg $H(\operatorname{div})$ element can be extended to 3D (see
[16, Section 3.5] and $\mathcal{W}_{p+1}^{2}$ in (4.1)). However, the complex
containing it was not discussed in [16].

The construction in [16] contains two complexes in 3D: one begins with the
Hermite element, and another begins with a $C^{0}$ element with second order
supersmoothness at the vertices (formally, $C^{2}(\mathcal{V})$, although the
function is not $C^{2}$). The latter can be viewed as a 3D extension of the
Falk-Neilan complex for the Stokes problem [19] in terms of supersmoothness.
The $H(\operatorname{curl})$ element in the former complex does not admit
nodal basis, while the $H(\operatorname{curl})$ element in the latter complex
can be represented in terms of scalar Hermite bases. Nevertheless, it was left
open in [16] to construct a sequence containing an $H(\operatorname{curl})$
element with Lagrange bases. We solve these problems in this paper.

##  3\. $H(\operatorname{curl})$ nodal elements in three space dimensions

###  3.1. Motivation

A natural candidate for a partially discontinuous nodal
$H(\operatorname{curl})$ element is obtained by breaking the normal components
of the vector-valued Lagrange finite element on each face (at the same time,
the $C^{0}$ continuity at vertices and on edges is retained):

$Z_{h}^{p}:=\operatorname{NED}_{p}^{2}(\mathcal{T})\cap C^{0}(\mathcal{V})\cap
C^{0}(\mathcal{E}).$

This element is in between the $C^{0}$ vector-valued Lagrange element and the
Nédélec element in terms of continuity. Nevertheless, this element unlikely
fits in a finite element de Rham complex due to the following observation. In
most existing cases, the “restriction” (precise definition discussed in [17])
of a higher-dimensional finite element (complex) to a lower-dimensional cell
is a lower-dimensional finite element (complex). This holds for the standard
finite element differential forms [3], as well as all the examples in [16].
Therefore, to construct a complex in 3D, we want the restrictions of the
involved spaces and the degrees of freedom on each face to form a 2D complex
as well. Nevertheless, this is unlikely for the case of $Z_{h}^{p}$. Because
the restriction of $Z_{h}^{p}$ on each face is a 2D Lagrange element, which
unlikely fits in a de Rham complex, as reflected in the deficiency of
$Z_{h}^{p}$ (at least for low order cases) as a discretization for the Stokes
problem, c.f., [9, 24, 39].

Although the above observation already suggests that $Z_{h}^{p}$ may not be a
good candidate for an $H(\operatorname{curl})$ nodal element, one may still
ask if $Z_{h}^{p}$, as a space in between the vector-valued Lagrange element
(which produces spurious solutions) and the Nédélec element (which excludes
spurious solutions), suffers from spurious solutions for eigenvalue problems.
To the best of our knowledge, the emergence of spurious solutions for
eigenvalue problems is not predicted by theory in most cases. Therefore we
answer this question with numerical tests.

We apply the $H(\operatorname{rot})$ version of the Stenberg element in 2D
defined by

$Y_{h}^{p}:=\operatorname{NED}_{p}^{2}(\mathcal{T})\cap C^{0}(\mathcal{V}),$

and $Z_{h}^{p}$ in 3D to solve the following Maxwell eigenvalue problem on the
domain $\Omega=(0,\pi)^{d}$, $d=2,3$:

(3.1)
$\displaystyle\operatorname{curl}\operatorname{curl}\bm{u}=\lambda\bm{u}\
\text{in }\Omega,$ $\displaystyle\operatorname{div}\bm{u}=0\text{ in }\Omega,$
$\displaystyle\bm{u}\times\bm{n}=0\text{ on }\partial\Omega.$

In 2D, the eigenvalues of this problems are $m^{2}+n^{2}$ [3], and the
corresponding eigenvectors are

$\bm{u}(x_{1},x_{2})=\left(\begin{matrix}n\sin(mx_{1})\cos(nx_{2})\\\
-m\cos(mx_{1})\sin(nx_{2})\end{matrix}\right),$

where $m,n$ are nonnegative integers, not both 0. In 3D, the eigenvalues of
this problems are $m^{2}+n^{2}+l^{2}$, and the corresponding eigenvectors are

$\bm{u}(x_{1},x_{2},x_{3})=\left(\begin{matrix}(l-n)\cos(mx_{1})\sin(lx_{3})\sin(nx_{2})\\\
(m-l)\cos(nx_{2})\sin(mx_{1})\sin(lx_{3})\\\
(n-m)\cos(lx_{3})\sin(mx_{1})\sin(nx_{2})\end{matrix}\right),$

where $m,n,l$ are nonnegative integers, not all 0.

We first apply the Stenberg $H(\operatorname{rot})$ finite element space
$Y_{h}^{p}$ to solve the problem (3.1) in 2D. The numerical results are shown
in Table 1.

Table 1. Numerical eigenvalues for 2D (3.1) by the 2nd-order vector-valued Lagrange element and the Stenberg $H(\operatorname{rot})$ element  exact | 1 | 1 | 2 | 4 | 4 | 5 | 5 | 8 | 9 | 9
---|---|---|---|---|---|---|---|---|---|---
$H^{1}$ | 1.0000 | 1.0000 | 1.1375 | 2.0000 | 2.5668 | 3.1367 | 4.0000 | 4.0000 | 4.1947 | 5.0000
$H(\operatorname{rot})$ | 1.0000 | 1.0000 | 2.0000 | 4.0000 | 4.0000 | 5.0000 | 5.0000 | 8.0000 | 9.0001 | 9.0001

From the table, we see that spurious solutions do not appear. Therefore the
vector-valued Lagrange elements are remedied by breaking the normal degrees of
freedom on edges. Then we use $Z_{h}^{p}$ to solve the problem (3.1) for
$d=3$. Table 2 shows the numerical eigenvalues around 3. In this case,
spurious solutions occur. Therefore relaxing the normal continuity on faces
(equivalently, enriching face-based bubbles) on the vector-valued Lagrange
element is not enough to give a proper discretization of the Maxwell
equations.

Table 2. Numerical eigenvalues for 3D (3.1) by the 3rd-order vector-valued Lagrange elements with discontinuous normal components exact | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4
---|---|---|---|---|---|---|---|---|---|---
numerical | 2.9924 | 2.9960 | 2.9972 | 3.0000 | 3.0000 | 3.0042 | 3.0057 | 3.0059 | 3.0069 | 3.0158

Now both heuristic argument and numerical experiments inspire us to seek a
different construction other than $Z_{h}^{p}$. We cannot further relax the
vertex and edge continuity, which is required to ensure the existence of nodal
bases. Motivated by the heuristic argument on the face modes, now we plan to
construct an element such that its restriction on each face is a proper
velocity finite element for the Stokes problem. Recall that the
$\mathcal{P}_{2}$-$\mathcal{P}_{1}$ Scott-Vogelius pair on the Clough-Tocher
split of triangular meshes (see Figure 3) leads to a complex and is thus
stable [7, 36]. This motivates us to use the Worsey-Farin split of a
tetrahedron [32, 45], which, restricted on each face of the macro tetrahedron,
is a Clough-Tocher split.

$\operatorname{curl}$$\operatorname{div}$+3+3+3 Figure 3. Clough-Tocher
complex with discontinuous pressure.
The figure shows the lowest order case: the first space is piecewise cubic,
the second is continuous piecewise quadratic and the third is piecewise
linear.  Figure 4. The complex containing the 3D $H(\operatorname{curl})$
nodal element and its 2D restriction. For
$H(\operatorname{rot})$/$H(\operatorname{curl})$/$H(\operatorname{div})$
elements, a dot $\bullet$ represents the evaluation of the vector-valued
function, which counts as two/three degrees of freedom. In the 3D
$H(\operatorname{curl})$ element, the four red dashed lines indicate the
normal components associated with the four Lagrange points on the face. They
are counted as interior degrees of freedom and do not yield inter-element
continuity.

###  3.2. Construction

We consider a complex for $p\geq 3$ as follows:

(3.2)
${0}$${\mathbb{R}}$${\mathcal{V}_{p}^{0}(\mathcal{T})}$${\mathcal{V}_{p-1}^{1}(\mathcal{T})}$${\mathcal{V}_{p-2}^{2}(\mathcal{T}_{\operatorname{WF}})}$${\mathcal{V}_{p-3}^{3}(\mathcal{T}_{\operatorname{WF}})}$${0,}$$\scriptstyle{\subset}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$

where each space is constructed on the Worsey-Farin split. The Worsey-Farin
$C^{1}$ element [45] requires a mesh alignment, the subdivision point on each
face of the macro tetrahedron lies on the line connecting the subdivision
points of the two macro tetrahedra sharing the face. Nevertheless, such a
condition is not required in the construction below.

#### Space $\mathcal{V}_{p}^{0}(\mathcal{T})$.

The scalar space $\mathcal{V}_{p}^{0}(\mathcal{T})$ is a high order
generalization of the Worsey-Farin macroelement from [45]. The shape function
space on macroelement $K$ consists of $C^{1}$ piecewise polynomials of degree
$p$ on the Worsey-Farin split, i.e.

$\mathcal{V}_{p}^{0}(K)=\\{u\in
C^{1}(K):u|_{K_{i}}\in\mathcal{P}_{p}(K_{i}),\text{ for }K_{i}\in
K_{\operatorname{WF}}\text{ with }\cup_{i=1}^{4}K_{i}=K\\}.$

Across the macroelements, the functions in $\mathcal{V}_{p}^{0}(\mathcal{T})$
are of $C^{1}$ continuity at vertices and edges, and of $C^{0}$ continuity
across faces, that is

$\mathcal{V}_{p}^{0}(\mathcal{T})=\\{u\in H^{1}(\Omega):u\in
C^{1}(\mathcal{E})\cap C^{1}(\mathcal{V}),\
u|_{K}\in\mathcal{V}_{p}^{0}(K)\text{ for }K\in\mathcal{T}\\}.$

For the lowest order case ($p=3$, c.f., [32]), if the mesh alignment condition
holds, then the functions are also of $C^{1}$ continuity across the
macroelements. The subsequent discussions do not rely on such $C^{1}$
smoothness across macroelements, therefore the mesh alignments are not
required.

For $u\in\mathcal{V}_{p}^{0}(\mathcal{T})$, the degrees of freedom can be
given by

  * •

function value and the first order derivatives of $u$ at each vertex
$\bm{x}\in\mathcal{V}$,

$u(\bm{x}),\quad\partial_{i}u(\bm{x}),\quad i=1,2,3;$

  * •

moments of $u$ on each edge $e\in\mathcal{E}$,

$\int_{e}u\cdot q\,\mathrm{d}\bm{x},\quad q\in\mathcal{P}_{p-4}(e);$

  * •

moments of the normal derivatives of $u$ on each edge $e\in\mathcal{E}$,

$\int_{e}\partial_{\bm{\nu}_{e,1}}u\cdot
q_{1}\,\mathrm{d}\bm{x},\quad\int_{e}\partial_{\bm{\nu}_{e,2}}u\cdot
q_{2}\,\mathrm{d}\bm{x},\quad q_{1},q_{2}\in\mathcal{P}_{p-3}(e);$

  * •

moments of $\operatorname{grad}_{f}u$ on each face $f\in\mathcal{F}$,

$\int_{f}\operatorname{grad}_{f}u\cdot q\,\mathrm{d}\bm{x},\quad
q\in\operatorname{grad}B_{p}({f}_{\mathrm{CT}}),$

where the space $B_{p}(f_{\mathrm{CT}}):=\left\\{w\in
C^{1}(f):w|_{f_{i}}\in\mathcal{P}_{p}\text{ for all }f_{i}\in
f_{\operatorname{CT}}\text{ with }\right.$
$\left.\cup_{i=1}^{3}f_{i}=f\text{ and }u=\partial_{n}u=0\mbox{ on }\partial
f\right\\}$ consists of the bubble functions of the 2D Clough-Tocher element
on the face subdivision $f_{\mathrm{CT}}$, see the first element in Figure 3;

  * •

interior degrees of freedom on $K\in\mathcal{T}$,

$\int_{K}uq\,\mathrm{d}\bm{x},\quad q\in B_{p}({K}_{\mathrm{WF}}),$

where $B_{p}(K_{\mathrm{WF}}):=\left\\{w\in
C^{1}(K):w|_{K_{i}}\in\mathcal{P}_{p}\text{ for all }K_{i}\in
K_{\operatorname{WF}}\text{ with }\right.$
$\left.\cup_{i=1}^{4}K_{i}=K\text{ and }u=0,\operatorname{grad}u=0\mbox{ on
}\partial K\right\\}$.

The dimension of the shape function space $\mathcal{V}_{p}^{0}(K)$ has been
given in [37, Theorem 3.5] with a homological approach, that is,

(3.3) $\displaystyle{p+3\choose 3}+3{p-1\choose 3}+8{p\choose
3}=2(p^{3}-3p^{2}+5p-1).$

Note that the above degrees of freedom at vertices, edges, and faces are
linearly independent since they are a part of the degrees of freedom for
defining the $C^{1}$ element in [23]. Therefore the unisolvence holds and the
dimension of the bubble space (number of interior degrees of freedom) is equal
to the total dimension of (3.3) subtracting the number of degrees of freedom
at vertices, edges, and faces. The number of face degrees of freedom, i.e.,
the dimension of $B_{p}(f_{\mathrm{CT}})$ can also be derived in a similar way
based on the dimension formula of the 2D Clough-Tocher macroelement, c.f.,
[17, 37].

Therefore, we obtain the following dimension count of
$\mathcal{V}_{p}^{0}(\mathcal{T})$:

$4V+[(p-3)+2(p-2)]E+[3/2(p^{2}-p+2)-(6p-6)]F+2(p-1)(p-2)(p-3)T.$

######  Remark 1.

As a $C^{1}$ tetrahedral spline, the shape function space of
$\mathcal{V}_{p}^{0}(K)$ has been studied with the Bernstein-Bézier technique
(see [32] for example). In [32], this space is defined with the $C^{2}$
continuity at the interior subdivision point. Nevertheless, the $C^{2}$
condition is intrinsic supersmoothness [20], meaning that it can be removed
from the definition and the $C^{1}$ continuity would automatically imply the
$C^{2}$ condition at the subdivision point. We refer to [20] and the
references therein for more details on supersmoothnesses.

#### Space $\mathcal{V}_{p}^{1}(\mathcal{T})$.

The shape function space on macroelement $K$ is define as

$\mathcal{V}_{p}^{1}(K)=\\{\bm{u}\in
C^{0}(K):\bm{u}|_{K_{i}}\in\mathcal{P}_{p},\ \forall K_{i}\in
K_{\operatorname{WF}}\text{ with }\cup_{i=1}^{4}K_{i}=K\\}.$

Then the space $\mathcal{V}_{p}^{1}(\mathcal{T})$ is defined as

$\mathcal{V}_{p}^{1}(\mathcal{T})=\\{\bm{u}\in
H(\operatorname{curl};\Omega):\bm{u}\in C^{0}(\mathcal{E})\cap
C^{0}(\mathcal{V}),\ \bm{u}|_{K}\in\mathcal{V}_{p}^{1}(K),\forall
K\in\mathcal{T}\\}$

The degrees of freedom are given by

  * •

function value of each component $u_{i}$ of $\bm{u}$ at each vertex
$\bm{x}\in\mathcal{V}$,

$u_{i}(\bm{x}),\quad i=1,2,3;$

  * •

moments of each component $u_{i}$ of $\bm{u}$ on each edge $e\in\mathcal{E}$,

$\int_{e}u_{i}\cdot q\,\mathrm{d}\bm{x},\quad
q\in\mathcal{P}_{p-2}(e),i=1,2,3;$

  * •

moments of the tangential components of $\bm{u}$ on each face
$f\in\mathcal{F}$,

$\int_{f}(\bm{u}\times\bm{\nu}_{f})\cdot\bm{q}\,\mathrm{d}\bm{x},\quad
q\in\left[B^{0}_{p}(f_{\operatorname{CT}})\right]^{2},$

where $B_{p}^{0}(f_{\mathrm{CT}}):=\left\\{w\in
C^{0}(f):w|_{f_{i}}\in\mathcal{P}_{p}\text{ for all }f_{i}\in
f_{\operatorname{CT}}\text{ with }\cup_{i=1}^{3}f_{i}=f\right.$
$\left.\text{and }u=0\mbox{ on }\partial f\right\\}$ and
$\bm{u}\times\bm{\nu}_{f}$ is considered as a 2D vector field;

  * •

interior degrees of freedom.

The interior degrees of freedom consists of function values at
$3[1+8(p-1)+9(p-1)(p-2)+2(p-1)(p-2)(p-3)]=6p^{3}-9p^{2}+9p-3$ interior
Lagrange points of each $K\in\mathcal{T}$ and the values of the normal
component at $1+3(p-1)+3/2(p-1)(p-2)=(3p^{2}-3p+2)/2$ Lagrange points on each
face of $K$. Therefore, the dimension of $\mathcal{V}_{p}^{1}(\mathcal{T})$ is

$3V+3(p-1)E+(3p^{2}-3p+2)F+(6p^{3}-3p^{2}+3p+1)T.$

#### Space $\mathcal{V}_{p}^{2}(\mathcal{T}_{\operatorname{WF}})$.

From this step, the complex branches into the standard finite element de Rham
complex of discrete differential forms [5] on the Worsey-Farin mesh.

More precisely, the space
$\mathcal{V}_{p}^{2}(\mathcal{T}_{\operatorname{WF}})$ is the BDM element on
the Worsey-Farin mesh of the domain $\Omega$, i.e.,

$\mathcal{V}_{p}^{2}(\mathcal{T}_{\operatorname{WF}})=\\{u\in
H(\operatorname{div};\Omega):u|_{K_{i}}\in\mathcal{P}_{p},\text{ for }K_{i}\in
K_{\operatorname{WF}}\text{ and }\cup_{i=1}^{4}K_{i}=K\in\mathcal{T}\\}.$

Since each macro tetrahedron $K$ is divided into $12$ micro tetrahedra, there
are $18$ micro faces in the interior of $K$ and $3$ micro faces on each face
of $K$. As a result, it follows from the definition of the BDM element that
the dimension count of $\mathcal{V}_{p}^{2}(\mathcal{T}_{\operatorname{WF}})$
is

$3/2(p+2)(p+1)F+(9(p+1)(p+2)+6(p+2)(p+1)(p-1))T.$

#### Space $\mathcal{V}_{p}^{3}(\mathcal{T}_{\operatorname{WF}})$.

The space $\mathcal{V}_{p}^{3}(\mathcal{T}_{\operatorname{WF}})\subset
L^{2}(\Omega)$ consists of piecewise polynomials of degree $p$ on the Worsey-
Farin mesh $\mathcal{T}_{\operatorname{WF}}$ of $\Omega$. The dimension of
$\mathcal{V}_{p}^{3}(\mathcal{T}_{\operatorname{WF}})$ is $2(p+3)(p+2)(p+1)T$.

######  Lemma 1.

For $p\geq 3$, the sequence (3.2) is a complex and exact on contractible
domains.

###### Proof.

By construction, (3.2) is a complex. It is sufficient to show that (3.2) is
exact on contractible domains. Since the last two spaces
$\mathcal{V}_{p-2}^{2}(\mathcal{T}_{\operatorname{WF}})$ and
$\mathcal{V}_{p-3}^{3}(\mathcal{T}_{\operatorname{WF}})$ coincides with the
BDM and the DG elements in the standard finite element de Rham complexes on
the Worsey-Farin mesh [5], the operator $\operatorname{div}$ is onto. This
verifies the exactness at the space
$\mathcal{V}_{p-3}^{3}(\mathcal{T}_{\operatorname{WF}})$. The kernel of the
operator $\operatorname{grad}$ is constants, which shows the exactness at the
space $\mathcal{V}_{p}^{0}(\mathcal{T})$. By definition,

$\mathcal{V}_{p-1}^{1}(\mathcal{T})=\operatorname{NED}_{p-1}^{2}(\mathcal{T}_{\operatorname{WF}})\cap
C^{0}(\mathcal{V})\cap C^{0}(\mathcal{E})\cap C^{0}(\mathcal{T}),$

where $\operatorname{NED}_{p-1}^{2}(\mathcal{T}_{\operatorname{WF}})$ is the
second kind Nédélec finite element space on the Worsey-Farin mesh of $\Omega$.
Therefore, the kernel space of the operator $\operatorname{curl}$ is as
follows

(3.4)
$\displaystyle\mathcal{N}\left(\operatorname{curl},\mathcal{V}_{p-1}^{1}(\mathcal{T})\right)=\operatorname{grad}\left[\Sigma_{h}^{p}(\mathcal{T}_{\operatorname{WF}})\cap
C^{1}(\mathcal{V})\cap C^{1}(\mathcal{E})\cap C^{1}(\mathcal{T})\right],$

where $\Sigma_{h}^{p}(\mathcal{T}_{\operatorname{WF}})$ is the Lagrange
element space of degree $p$ on the Worsey-Farin mesh of $\Omega$. The space
$\Sigma_{h}^{p}(\mathcal{T}_{\operatorname{WF}})\cap C^{1}(\mathcal{V})\cap
C^{1}(\mathcal{E})\cap C^{1}(\mathcal{T})$ coincides with
$\mathcal{V}_{p}^{0}(\mathcal{T})$, which shows the exactness at
$\mathcal{V}_{p-1}^{1}(\mathcal{T})$. Then a dimension count in the sense of
(2.3) proves the exactness at
$\mathcal{V}_{p-2}^{2}(\mathcal{T}_{\operatorname{WF}})$.

∎

##  4\. $H(\operatorname{div})$ nodal elements in three space dimensions

In this section, we construct a discrete de Rham complex containing the
partially discontinuous $H(\operatorname{div})$ finite element from [16,
Section 3.5]. In particular, we will design a complex for $p\geq 2$ as follows

(4.1)
${0}$${\mathbb{R}}$${\mathcal{W}_{p+3}^{0}(\mathcal{T})}$${\mathcal{W}_{p+2}^{1}(\mathcal{T})}$${\mathcal{W}_{p+1}^{2}(\mathcal{T})}$${\mathcal{W}_{p}^{3}(\mathcal{T})}$${0.}$$\scriptstyle{\subset}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$

#### Space $\mathcal{W}^{0}_{p}(\mathcal{T})$.

For $p\geq 4$, the finite element space

$\mathcal{W}^{0}_{p}(\mathcal{T}):=\left\\{u\in H^{1}(\Omega):u\in
C^{2}(\mathcal{V})\cap
C^{1}(\mathcal{E}),\left.u\right|_{K}\in\mathcal{P}_{p}(K),~{}\forall
K\in\mathcal{T}\right\\},$

coincides with the element $\mathcal{P}_{2,p}\Lambda^{0}(\mathcal{T}^{3})$ in
[16], which is also identical to each component of the velocity element in the
discrete 3D Stokes complex by Neilan [35]. The space
$\mathcal{W}^{0}_{p}(\mathcal{T})$ admits enhanced $C^{2}$ and $C^{1}$
continuities at vertices and edges, respectively, but only $C^{0}$ continuity
across faces. Therefore no higher global continuity than $H^{1}$ is imposed.

The global dimension of $\mathcal{W}^{0}_{p}(\mathcal{T})$ on the mesh
$\mathcal{T}$ reads

${\dim}\mathcal{W}^{0}_{p}(\mathcal{T})=10V+\left[2(p-4)+(p-5)\right]E+{p-4\choose
2}F+{p-1\choose 3}T.$

#### Space $\mathcal{W}^{1}_{p}(\mathcal{T})$.

The finite element space

$\mathcal{W}^{1}_{p}(\mathcal{T}):=\left\\{\bm{u}\in
H(\operatorname{curl};\Omega):\bm{u}\in C^{1}(\mathcal{V})\cap
C^{0}(\mathcal{E}),\left.\bm{u}\right|_{K}\in\mathcal{P}_{p}(K),~{}\forall
K\in\mathcal{T}\right\\},$

consists of piecewise polynomials of degree less than or equal to $p$ for each
component, and the degrees of freedom for
$\bm{w}\in\mathcal{W}^{1}_{p}(\mathcal{T})$ can be given by

  * •

function values and the first order derivatives of $\bm{w}$ at each vertex
$\bm{x}\in\mathcal{V}$,

$\bm{w}(\bm{x}),\quad\partial_{i}\bm{w}(\bm{x}),~{}i=1,2,3;$

  * •

moments of $\bm{w}$ on each edge $e\in\mathcal{E}$,

$\int_{e}\bm{w}\cdot\bm{q}\,\mathrm{d}\bm{x},\quad\forall\bm{q}\in\left[\mathcal{P}_{p-4}(e)\right]^{3};$

  * •

moments of $\operatorname{curl}\bm{w}$ on each edge $e\in\mathcal{E}$,

$\quad\int_{e}(\operatorname{curl}\bm{w}\cdot\bm{\nu}_{e,1})\cdot
q_{1}\,\mathrm{d}\bm{x},\quad\int_{e}(\operatorname{curl}\bm{w}\cdot\bm{\nu}_{e,2})\cdot
q_{2}\,\mathrm{d}\bm{x},\quad\forall q_{1},q_{2}\in\mathcal{P}_{p-3}(e),$

where $\bm{\nu}_{e,1}$ and $\bm{\nu}_{e,2}$ are two linearly independent unit
normal vectors of edge $e$;

  * •

integrals on each face $f\in\mathcal{F}$,

$\int_{f}\left(\bm{w}\times\bm{\nu}_{f}\right)\cdot\operatorname{curl}_{f}\left(\psi_{f}\lambda_{f,1}^{2}\lambda_{f,2}^{2}\lambda_{f,3}^{2}\right)\mathrm{d}\bm{x},\quad\forall\psi_{f}\in\mathcal{P}_{p-5}(f),$
$\int_{f}(\operatorname{curl}\bm{w}\cdot\bm{\nu}_{f})\cdot
g\,\mathrm{d}\bm{x},\quad\forall g\in\mathcal{P}_{p-4}(f)/\mathbb{R},$

where $\lambda_{f,1}\lambda_{f,2}\lambda_{f,3}$ is a face bubble associate
with face $f$, $\bm{\nu}_{f}$ is the unit normal vector of $f$, and

$\mathcal{P}_{p-4}(f)/\mathbb{R}:=\left\\{q\in\mathcal{P}_{p-4}(f):\int_{f}q\,\mathrm{d}\bm{x}=0\right\\};$

  * •

interior degrees of freedom in each $K\in\mathcal{T}$,

(4.2)
$\displaystyle\int_{K}\bm{w}\cdot\bm{z}\,\mathrm{d}\bm{x},\quad\forall\bm{z}\in\operatorname{RT}_{p-2}(K).$

The dimension of $\mathcal{W}_{p}^{1}(\mathcal{T})$ is

$\displaystyle\dim\left(\mathcal{W}^{1}_{p}(\mathcal{T})\right)$
$\displaystyle=12V+[3(p-3)+2(p-2)]E+[1/2(p-2)(p-3)-1+1/2(p-3)(p-4)]F$
$\displaystyle\quad\quad+\left(\frac{1}{2}p^{3}-p^{2}-\frac{1}{2}p+1\right)T$
$\displaystyle=12V+(5p-13)E+(p^{2}-6p+8)F+\left(\frac{1}{2}p^{3}-p^{2}-\frac{1}{2}p+1\right)T.$

######  Theorem 1.

The degrees of freedom for $\mathcal{W}^{1}_{p}(\mathcal{T})$ are unisolvent
and $\mathcal{W}^{1}_{p}$ is $H(\operatorname{curl})$ conforming.

###### Proof.

On a tetrahedron, the number of the above degree of freedom is

$1/2\left(p^{3}+6p^{2}+11p+6\right).$

This coincides with the dimension of
$\left[\mathcal{P}_{p}\left(K\right)\right]^{3}$. Then it suffices to show
that if all the degrees of freedom vanish on a function
$\bm{w}\in\left[\mathcal{P}_{p}\left(K\right)\right]^{3}$, then $\bm{w}=0$ on
$K\in\mathcal{T}$.

Define ${\bm{w}_{f}}:=\bm{w}\times\bm{\nu}_{f}$ as a 2D vector field on $f$.
The first step is to show that $\bm{w}_{f}=0$ on each face $f$ of $K$.
Actually, from the degrees of freedom, we conclude that

  1. (1)

for each vertex $\bm{x}\subset f$, it holds that
$\bm{w}_{f}(\bm{x})=\partial_{\bm{\tau}_{f,i}}\bm{w}_{f}(\bm{x})=0,~{}i=1,2,$
where $\bm{\tau}_{f,i}$ is the $i$-th tangent vector on $f$;

  2. (2)

for each edge $e\subset f$, it holds that
$\int_{e}\bm{w}_{f}\cdot\bm{q}\,\mathrm{d}\bm{x}=\int_{e}\operatorname{div}_{f}\bm{w}_{f}\cdot
q_{0}\,\mathrm{d}\bm{x}=0,\
\forall\bm{q}\in\left[\mathcal{P}_{p-4}(e)\right]^{2}\text{ and
}q_{0}\in\mathcal{P}_{p-3}(e);$

  3. (3)

in the interior of $f$, it holds that
$\int_{f}\operatorname{curl}_{f}\left(\psi_{f}\lambda_{f,1}^{2}\lambda_{f,2}^{2}\lambda_{f,3}^{2}\right)\cdot\bm{w}_{f}\,\mathrm{d}\bm{x}=0,\
\forall\psi_{f}\in\mathcal{P}_{p-5}(f)$ and
$\int_{f}\operatorname{div}_{f}\bm{w}_{f}\cdot g\,\mathrm{d}\bm{x},\ \forall
g\in\mathcal{P}_{p-4}(f)/\mathbb{R}.$

The above functionals in (1) – (3) are the degrees of freedom of the shape
function space $\left[\mathcal{P}_{p}(f)\right]^{2}$, see [4].

To show $\bm{w}_{f}=0$, we first have
$\operatorname{div}_{f}\bm{w}_{f}\in\mathcal{P}_{p-1}(f)$ and the vanishing
degrees of freedom imply that $\operatorname{div}_{f}\bm{w}_{f}=0$ on $f$.
Then it follows that $\bm{w}_{f}=\operatorname{curl}_{f}\phi$ for some
$\phi\in\mathcal{P}_{p+1}(f)$. Without loss of generality, we assume that
$\phi$ vanishes at one of the vertices. Now it follows from (1) – (3) that the
following degrees of freedom vanish on $\phi$,

  1. (1)

for each vertex $x\in f$, it holds that
$\partial_{i}\phi(\bm{x})=\partial_{i}\partial_{j}\phi(\bm{x})=0\text{ for
}i,j=1,2$;

  2. (2)

for each edge $e\in\partial f$, it holds that
$\int_{e}(\operatorname{curl}_{f}\phi)\cdot q\mathrm{d}\bm{x},\ \forall
q\in[\mathcal{P}_{p-4}(e)]^{2}$;

  3. (3)

in the interior of $f$, it holds that

(4.3)
$\displaystyle\int_{f}\operatorname{curl}_{f}\phi\cdot\operatorname{curl}_{f}\left(\psi_{f}\lambda_{f,1}^{2}\lambda_{f,2}^{2}\lambda_{f,3}^{2}\right)\mathrm{d}\bm{x}=0,\
\forall\psi_{f}\in\mathcal{P}_{p-5}(f).$

This shows that $\operatorname{curl}_{f}\phi=0$ on all the edges of $f$, which
further implies that the tangential derivative of $\phi$ along each edge
vanishes. Since $\phi$ vanishes at one vertex of $f$, both $\phi$ and
$\operatorname{curl}_{f}\phi$ vanish on $\partial f$. Consequently, $\phi$ is
of the form $\phi=\psi_{f}\lambda_{f,1}^{2}\lambda_{f,2}^{2}\lambda_{f,3}^{2}$
for some polynomial $\psi_{f}\in\mathcal{P}_{p-5}(f)$, which together with
(4.3) yields $\psi_{f}=0$. This implies $\phi=0$, and further
$\bm{w}_{f}=\operatorname{curl}_{f}\phi=0$.

Then by the interior degrees of freedom (4.2), it follows from a similar
argument to [5] that $\bm{w}=0$. The proof for the unisolvence also shows that
$\mathcal{W}^{1}_{p}(\mathcal{T})$ is $H(\operatorname{curl})$ conforming. ∎

#### Space $\mathcal{W}^{2}_{p}(\mathcal{T})$.

The $H(\operatorname{div})$ element $\mathcal{W}^{2}_{p}(\mathcal{T})$ was
defined in [16]:

$\mathcal{W}^{2}_{p}(\mathcal{T}):=\left\\{\bm{u}\in
H(\operatorname{div}):\left.\bm{u}\right|_{K}\in\mathcal{P}_{p}(K),\forall
K\in\mathcal{T},\bm{u}\in C^{0}(\mathcal{V}),\bm{u}\cdot\bm{\nu}_{e,i}\in
C^{0}(\mathcal{E}),i=1,2\right\\},$

where $\bm{\nu}_{e,i},~{}i=1,2$ are two linearly independent unit normal
vectors of an edge $e$.

For $u\in\mathcal{W}_{p}^{2}(\mathcal{T})$, the degrees of freedom can be
given by

  * •

values of $\bm{u}$ at each vertex ${\bm{x}}\in\mathcal{V}$,

$\quad\bm{u}({\bm{x}});$

  * •

moments of two normal components of $\bm{u}$ on each edge $e\in\mathcal{E}$,

$\int_{e}(\bm{u}\cdot\bm{\nu}_{e,i})w\mathrm{d}\bm{x},\quad\forall
w\in\mathcal{P}_{p-2}(e),\ i=1,2;$

  * •

moments of the normal component of $\bm{u}$ on each face $f\in\mathcal{F}$,

$\int_{f}\left(\bm{u}\cdot\bm{\nu}_{f}\right){w}\mathrm{d}\bm{x},\quad\forall{w}\in\mathcal{P}_{p-3}(f);$

  * •

integrals over each tetrahedron $K\in\mathcal{T}$,

$\int_{K}\bm{u}\cdot\bm{v}\mathrm{d}\bm{x},\quad\forall\bm{v}\in\operatorname{NED}_{p-1}^{1}(K).$

The dimension of $\mathcal{W}^{2}_{p}(\mathcal{T})$ is

$\dim\mathcal{W}^{2}_{p}(\mathcal{T})=3V+2(p-1)E+1/2(p-1)(p-2)F+\left(\frac{1}{2}p^{3}-p^{2}+\frac{1}{2}p-1\right)T.$

######  Theorem 2.

The sequence (4.1) is a complex which is exact on contractible domains.

###### Proof.

To show that (4.1) is a complex, we only show that
$\operatorname{curl}\mathcal{W}_{p+1}^{1}(\mathcal{T})\subset\mathcal{W}_{p}^{2}(\mathcal{T})$.
This is a consequence of the identity

$\operatorname{curl}\bm{w}\cdot\bm{\nu}_{f}=\operatorname{div}_{f}(\bm{w}\times\bm{\nu}_{f})\text{
for }\bm{w}\in\mathcal{W}_{p+1}^{1}(\mathcal{T}),$

where $\bm{\nu}_{f}$ is the unit normal vector of $f$. Therefore the degrees
of freedom of $\mathcal{W}_{p+2}^{1}(\mathcal{T})$ imply the continuity of the
normal component of $\operatorname{curl}\bm{w}$, meaning that
$\operatorname{curl}\mathcal{W}_{p+1}^{1}(\mathcal{T})\subset\mathcal{W}_{p}^{2}(\mathcal{T})$.

As concluded in
[16]$,\operatorname{div}:\mathcal{W}_{p+1}^{2}(\mathcal{T})\mapsto\mathcal{W}_{p}^{3}(\mathcal{T})$
is onto. Moreover, since $\mathcal{W}_{p+2}^{1}(\mathcal{T})$ is of $C^{1}$
continuity at vertices and of $C^{0}$ continuity on edges, the preimage of
$\operatorname{grad}$ has to be $C^{2}$ at vertices and $C^{1}$ on edges.

Finally, a dimension count in the sense of (2.3) implies the exactness. ∎

Figure 5. The complex containing the 3D $H(\operatorname{div})$ nodal element
and its 2D restriction. For
$H(\operatorname{rot})$/$H(\operatorname{curl})$/$H(\operatorname{div})$
elements, a dot $\bullet$ represents the evaluation of the vector valued
function, which counts as two/three degrees of freedom. In the 3D
$H(\operatorname{div})$ element, the two red dashed lines indicate the two
tangential components associated with the Lagrange point on the face. They are
regarded as interior degrees of freedom, which do not contribute to inter-
element continuity.

######  Remark 2.

The rotation of the restriction of (4.1) to each face (the second row in
Figure 5) was discussed in [4] to interpret the Arnold-Winther elasticity
element with the Bernstein-Gelfand-Gelfand (BGG) construction [5, 6].

##  5\. Partially orthogonal nodal bases

To obtain practical bases for high order finite elements, one should take
several issues into consideration, including the condition number, sparsity,
symmetry and fast assembling. In particular, certain orthogonality is usually
incorporated in the bases. Bases for high order Lagrange elements are
relatively mature, see, e.g., [31]. In this section, we construct bases for
partially discontinuous nodal $H(\operatorname{curl})$ and
$H(\operatorname{div})$ elements in both 2D and 3D based on explicit formulas
of multi-variate orthogonal polynomials on simplices. We aim at orthogonality
in the $L^{2}$ sense. The reason to use the $L^{2}$ orthogonality has been
addressed in [29], see also [2, 47].

We choose proper Jacobi polynomials such that the bases associated to each
simplex are mutually orthogonal on the reference element. As in the case of
the $C^{0}$ elements, the $L^{2}$ condition numbers of the global mass and
stiffness matrices still grow with the polynomial degree if no proper
preconditioner is adopted .

To fix the notation, we use $J_{n}^{\alpha,\beta}(x)$ to denote the Jacobi
polynomial on $[-1,1]$ with weight
$\omega^{\alpha,\beta}:=(1-x)^{\alpha}(1+x)^{\beta}$, i.e.

$\int_{-1}^{1}J_{n}^{\alpha,\beta}(x)J_{n^{\prime}}^{\alpha,\beta}(x)(1-x)^{\alpha}(1+x)^{\beta}\,\mathrm{d}\bm{x}=\gamma_{n}^{\alpha,\beta}\delta_{nn^{\prime}},$

where
$\gamma_{n}^{\alpha,\beta}=\int_{-1}^{1}J_{n}^{\alpha,\beta}(x)^{2}(1-x)^{\alpha}(1+x)^{\beta}\,\mathrm{d}\bm{x}$.
By a simple transform $\xi=(x+1)/2$, we obtain Jacobi polynomials on the
reference interval $[0,1]$:

$\int_{0}^{1}J_{n}^{\alpha,\beta}(2\xi-1)J_{n^{\prime}}^{\alpha,\beta}(2\xi-1)(1-\xi)^{\alpha}\xi^{\beta}\,\mathrm{d}\xi=c_{n}^{\alpha,\beta}\delta_{nn^{\prime}},$

where

$c_{n}^{\alpha,\beta}=2^{-\alpha-\beta-1}\gamma_{n}^{\alpha,\beta}.$

We cite the explicit form of Jacobi polynomials on the reference simplex of
dimension $d$ (c.f. [18], Section 5.3):

######  Theorem 3.

The following polynomials form a group of mutually orthonormal bases on the
$d$-dimensional reference simplex with weight
$\lambda_{0}^{\alpha_{0}}\dots\lambda_{d}^{\alpha_{d}}$:

(5.1) $\displaystyle
J_{\bm{n}}^{\bm{\alpha}}(\lambda_{0},\dots,\lambda_{d-1}):=(c_{\bm{n}}^{\bm{\alpha}})^{-1}\prod_{j=0}^{d-1}\left(1-\sum_{i=0}^{j-1}\lambda_{i}\right)^{n_{j}}J_{n_{j}}^{a_{j},b_{j}}\left(\frac{2\lambda_{j}}{1-\sum_{i=0}^{j-1}\lambda_{i}}-1\right),$

where $\bm{n}=(n_{0},\dots,n_{d-1})$ and
$\bm{\alpha}=(\alpha_{0},\alpha_{1},\dots,\alpha_{d})$ are multi-indexes of
the polynomial degree and the weights, respectively,
$a_{j}=2\sum_{i=j+1}^{d-1}n_{i}+\sum_{i=j+1}^{d}\alpha_{i}+d-j-1$ and
$b_{j}=\alpha_{j}$. Here $c_{\bm{n}}^{\bm{\alpha}}$ is a scaling factor [46].
When $J_{n_{j}}^{a_{j},b_{j}}(\cdot)$ is scaled to be orthonormal, we have

$\left(c_{\bm{n}}^{\bm{\alpha}}\right)^{-2}=\Pi_{i=0}^{d-1}2^{a_{i-1}}.$

In 2D we obtain a group of mutually orthogonal bases as follows

(5.2) $\displaystyle
J_{\bm{n}}^{\bm{\alpha}}=(c_{\bm{n}}^{\bm{\alpha}})^{-1}J_{n_{0}}^{2n_{1}+\alpha_{1}+\alpha_{2}+1,\alpha_{0}}(2\lambda_{0}-1)J_{n_{1}}^{\alpha_{2},\alpha_{1}}\left(\frac{2\lambda_{1}}{1-\lambda_{0}}-1\right)(1-\lambda_{0})^{n_{1}}.$

From (5.2), we can obtain two groups of univariate polynomials which are
orthogonal on a triangle. Let $n_{1}=0$, we obtain the first group:

$cJ_{n_{0}}^{\alpha_{1}+\alpha_{2}+1,\alpha_{0}}(2\lambda_{0}-1),$

where $\lambda_{0}$ is considered as the variable. For the second, we let
$n_{0}=0$ and obtain

$cJ_{n_{1}}^{\alpha_{2},\alpha_{1}}\left(\frac{2\lambda_{1}}{1-\lambda_{0}}-1\right)(1-\lambda_{0})^{n_{1}},$

where $\lambda_{0}$ and $\lambda_{1}$ are considered as variables. Note that
$\lambda_{0}+\lambda_{1}+\lambda_{2}=1$ on a triangle, we get yet another
formula with variables $\lambda_{1}$ and $\lambda_{2}$:

(5.3) $\displaystyle
cJ_{n_{1}}^{\alpha_{2},\alpha_{1}}\left(\frac{\lambda_{1}-\lambda_{2}}{\lambda_{1}+\lambda_{2}}\right)(\lambda_{1}+\lambda_{2})^{n_{1}}.$

When $\alpha_{1}=\alpha_{2}=0$, (5.3) is the scaled Legendre polynomials used
in [49]. Due to the symmetry of variables, we will use (5.3) to build up edge
based bases for triangular elements.

The 3D version of (5.1) reads

$\displaystyle J_{\bm{n}}^{\bm{\alpha}}=(c_{\bm{n}}^{\bm{\alpha}})^{-1}$
$\displaystyle
J_{n_{0}}^{2(n_{1}+n_{2})+\alpha_{1}+\alpha_{2}+\alpha_{3}+2,\alpha_{0}}(2\lambda_{0}-1)J_{n_{1}}^{2n_{2}+\alpha_{2}+\alpha_{3}+1,\alpha_{1}}\left(\frac{2\lambda_{1}}{1-\lambda_{0}}-1\right)$
(5.4)
$\displaystyle\cdot(1-\lambda_{0})^{n_{1}}J_{n_{2}}^{\alpha_{3},\alpha_{2}}\left(\frac{2\lambda_{2}}{1-\lambda_{0}-\lambda_{1}}-1\right)(1-\lambda_{0}-\lambda_{1})^{n_{2}}.$

We can similarly consider its restriction to an edge or a face, which will be
used in the construction of edge and face bases for the tetrahedral elements.

###  5.1. Triangular $H(\operatorname{rot})$ element

  1. (1)

Vertex-based basis functions at $\bm{x}\in\mathcal{V}$:

$\bm{v}_{\bm{x},i}=\lambda_{\bm{x}}\bm{e}_{i},\quad i=1,2,$

where $\lambda_{\bm{x}}$ is the barycentric coordinates at $\bm{x}$ and
$\bm{e}_{i}$ is a 2D vector with the ith component 1 and the others 0.

  2. (2)

Edge-based tangential bases on $e\in\mathcal{E}$:

(5.5) $\displaystyle\bm{v}_{e,j}^{\tau}=\phi_{j}\bm{\tau}_{e},\quad
j=0,1,2,\cdots,\ \text{with
}\phi_{j}=J_{j}^{2,2}\left(\frac{\lambda_{1}-\lambda_{2}}{\lambda_{1}+\lambda_{2}}\right)(\lambda_{1}+\lambda_{2})^{j}\lambda_{1}\lambda_{2},$

where $\bm{\tau}_{e}$ is the tangential vector of the edge $e$ and
$\lambda_{1},\ \lambda_{2}$ are the two barycentric coordinates of the two
vertices of $e$.

  3. (3)

Edge-based normal bases on $e\in\mathcal{E}$:

$\bm{v}_{e,i,j}^{\nu}=\phi_{j}|_{f_{i}}\bm{\nu}_{e},\quad i=1,2,\quad
j=0,1,2,\cdots$

where $f_{1}$ and $f_{2}$ are the two elements sharing the edge $e$ and
$\bm{\nu}_{e}$ is the unit normal vector of $e$.

  4. (4)

Interior bases in $f\in\mathcal{T}$ (here, $\mathcal{T}$ is a triangular mesh
of a 2D domain):

$\bm{v}_{f,i}^{n_{0},n_{1}}=\phi_{n_{0},n_{1}}\bm{e}_{i},\quad i=1,2,$

where

$\phi_{n_{0},n_{1}}=(c_{\bm{n}}^{\bm{\alpha}})^{-1}J_{n_{0}}^{2n_{1}+5,2}(2\lambda_{0}-1)J_{n_{1}}^{2,2}\left(\frac{2\lambda_{1}}{1-\lambda_{0}}-1\right)(1-\lambda_{0})^{n_{1}}\lambda_{0}\lambda_{1}\lambda_{2},$

and $\lambda_{0},\lambda_{1},\lambda_{2}$ are the three barycentric
coordinates of the three vertices of $f$.

######  Remark 3.

The $\lambda_{1}\lambda_{2}$ factor appearing in (5.5) is such that the bases
decay to zero on the other two edges of the triangle. Then the $(2,2)$ index
in the Jacobi polynomials is such that the bases are $L^{2}$ mutually
orthogonal. A similar construction will also be used below for tetrahedral
elements.

###  5.2. Tetrahedral $H(\operatorname{curl})$ element on the Worsey-Farin
split

  1. (1)

Vertex-based basis functions at $\bm{x}\in\mathcal{V}$:

$\bm{v}_{\bm{x},i}=\lambda_{\bm{x}}\bm{e}_{i},\quad i=1,2,3,$

where $\lambda_{\bm{x}}$ is the barycentric coordinates at $\bm{x}$ and
$\bm{e}_{i}$ is a 3D vector with the ith component 1 and the others 0.

  2. (2)

Edge-based basis functions on $e\in\mathcal{E}$:

$\bm{v}_{e,j,i}=\phi_{j}\bm{e}_{i},\quad i=1,2,3,\quad j=0,1,2,\cdots.$

Here

$\phi_{j}=J_{j}^{2,2}\left(\frac{\lambda_{1}-\lambda_{2}}{\lambda_{1}+\lambda_{2}}\right)(\lambda_{1}+\lambda_{2})^{j}\lambda_{1}\lambda_{2},$

where $\lambda_{1}$ and $\lambda_{2}$ are the barycentric coordinates of the
two vertices of $e$.

  3. (3)

Face-based normal bases on $f\in\mathcal{F}$, which is shared by two
macroelements $K_{1}$ and $K_{2}$:

     * •

at the face subdivision point,

$\lambda|_{K_{i}}\bm{\nu}_{f},\quad i=1,2,$

where $\lambda|_{K_{i}}$ is the restriction of the hat function on one
macroelement (a piecewise linear function on each macroelement);

     * •

on a face-interior edge $e_{k}$ of $f$,

$\bm{v}_{j,i,k}=\phi_{j}^{(k)}|_{K_{i}}\bm{\nu}_{f},\quad i=1,2,\ \ k=1,2,3,\
\ j=0,1,2,\cdots,$

where

$\phi_{j}^{(k)}=J_{j}^{2,2}\left(\frac{\lambda_{1}^{(k)}-\lambda_{2}^{(k)}}{\lambda_{1}^{(k)}+\lambda_{2}^{(k)}}\right)(\lambda_{1}^{(k)}+\lambda_{2}^{(k)})^{j}\lambda_{1}^{(k)}\lambda_{2}^{(k)}$

with the barycentric coordinates $\lambda_{1}^{(k)}$ and $\lambda_{2}^{(k)}$
of the two vertices of $e_{k}$;

     * •

in the interior of a micro face $f_{j}$ of $f$,

$\bm{v}_{f,i,j}^{n_{1},n_{2}}=\left.\phi^{(j)}_{n_{1},n_{2}}\right|_{K_{i}}\bm{\nu}_{f},\quad
i=1,2,\ \ j=1,2,3,$ $\displaystyle\phi^{(j)}_{n_{1},n_{2}}=$ $\displaystyle
J_{n_{1}}^{2n_{2}+5,2}\left(\frac{\lambda_{1}^{(j)}-\lambda_{2}^{(j)}-\lambda_{3}^{(j)}}{\lambda_{1}^{(j)}+\lambda_{2}^{(j)}+\lambda_{3}^{(j)}}\right)\cdot(\lambda_{1}^{(j)}+\lambda_{2}^{(j)}+\lambda_{3}^{(j)})^{n_{1}}\cdot$
$\displaystyle
J_{n_{2}}^{2,2}\left(\frac{\lambda_{1}^{(j)}-\lambda_{2}^{(j)}}{\lambda_{1}^{(j)}+\lambda_{2}^{(j)}}\right)(\lambda_{1}^{(j)}+\lambda_{2}^{(j)})^{n_{2}}\lambda_{1}^{(j)}\lambda_{2}^{(j)}\lambda_{3}^{(j)},$

with barycentric coordinates $\lambda_{1}^{(j)}$, $\lambda_{2}^{(j)}$,
$\lambda_{3}^{(j)}$ of each micro face $f_{j}$ of $f$.

  4. (4)

Face-based tangential bases on $f\in\mathcal{F}$:

     * •

at the face subdivision point,

$\lambda\bm{\tau}_{f,i},\quad i=1,2,$

where $\lambda$ is the hat function at the face subdivision point and
$\bm{\tau}_{f,i},i=1,2,$ are the two linearly independent unit tangent vectors
of the face $f$;

     * •

on a face-interior edge $e_{k}$,

$\bm{v}_{i,j,k}=\phi_{j}^{(k)}\bm{\tau}_{f,i},\quad i=1,2,\ \ k=1,2,3\ \
j=0,1,2,\cdots$

     * •

in the interior of each micro face $f_{j}$ of $f$,

$\bm{v}_{f,i,j}^{n_{1},n_{2}}=\phi_{n_{1},n_{2}}^{(j)}\bm{\tau}_{f,i},\quad
i=1,2,\ \ j=1,2,3.$

  5. (5)

Interior bases:

     * •

at the subdivision point,

$\lambda_{Q}\bm{e}_{i},\quad i=1,2,3,$

where $\lambda_{Q}$ is the hat function at the interior subdivision point;

     * •

on an interior edge $e_{k}$,

$\bm{v}_{j,i,k}=\phi_{j}^{(k)}\bm{e}_{i},\ \ i=1,2,3,\ \ k=1,2,\cdots,8,\ \
j=0,1,2,\cdots$

     * •

in the interior of an interior micro face $f_{j}$,

$\bm{v}_{i,j}^{n_{1},n_{2}}=\phi_{n_{1},n_{2}}^{(j)}\bm{e}_{i},\quad i=1,2,3,\
\ j=1,2,\cdots,12,$

where

$\phi^{(j)}_{n_{1},n_{2}}=J_{n_{1}}^{2n_{2}+5,2}\left(\frac{\lambda_{1}^{(j)}-\lambda_{2}^{(j)}-\lambda_{3}^{(j)}}{\lambda_{1}^{(j)}+\lambda_{2}^{(j)}+\lambda_{3}^{(j)}}\right)\cdot(\lambda_{1}^{(j)}+\lambda_{2}^{(j)}+\lambda_{3}^{(j)})^{n_{1}}J_{n_{2}}^{2,2}\left(\frac{\lambda_{1}^{(j)}-\lambda_{2}^{(j)}}{\lambda_{1}^{(j)}+\lambda_{2}^{(j)}}\right)(\lambda_{1}^{(j)}+\lambda_{2}^{(j)})^{n_{2}}\lambda_{1}^{(j)}\lambda_{2}^{(j)}\lambda_{3}^{(j)},$

with $\lambda_{1}^{(j)}$, $\lambda_{2}^{(j)}$ and $\lambda_{3}^{(j)}$ the
barycentric coordinates associate with the interior face $f_{j}$;

     * •

in the interior of the micro tetrahedra $K_{j}$,

$\bm{v}_{i,j}^{n_{0},n_{1},n_{2}}=\phi_{n_{0},n_{1},n_{2}}^{(j)}\bm{e}_{i},\quad
i=1,2,3,\ \ j=1,2,\cdots,12,$

where

$\displaystyle\phi^{(j)}_{n_{0},n_{1},n_{2}}=J_{n_{0}}^{2(n_{1}+n_{2})+8,2}(2\lambda_{0}^{(j)}-1)J_{n_{1}}^{2n_{2}+5,2}\left(\frac{2\lambda_{1}^{(j)}}{1-\lambda_{0}^{(j)}}-1\right)(1-\lambda_{0}^{(j)})^{n_{1}}$
(5.6) $\displaystyle\cdot
J_{n_{2}}^{2,2}\left(\frac{2\lambda_{2}^{(j)}}{1-\lambda_{0}^{(j)}-\lambda_{1}^{(j)}}-1\right)(1-\lambda_{0}^{(j)}-\lambda_{1}^{(j)})^{n_{2}}\lambda_{0}^{(j)}\lambda_{1}^{(j)}\lambda_{2}^{(j)}\lambda_{3}^{(j)},$

and $\lambda_{i}^{(j)},i=0,1,2,3$ are the barycentric coordinates associate
with the micro tetrahedron $K_{j}$.

###  5.3. Tetrahedral $H(\operatorname{div})$ elements

  1. (1)

Vertex-based basis functions at $\bm{x}\in\mathcal{V}$:

$\bm{v}_{\bm{x},i}=\lambda_{\bm{x}}\bm{e}_{i},\quad i=1,2,3.$

  2. (2)

Edge-based tangential bases on $e\in\mathcal{E}$:

$\bm{v}_{e,i,j}^{\tau}=\phi_{j}|_{K_{i}}\bm{\tau}_{e},\ j=0,1,\cdots,$

where $\bm{\tau}_{e}$ is the tangent vector of $e$, $K_{i}$ takes $e$ as an
edge.

  3. (3)

Edge-based normal bases on $e\in\mathcal{E}$:

$\bm{v}_{e,i,j}^{\nu}=\phi_{j}\bm{\nu}_{e,i},\quad i=1,2,\ \ j=0,1,\cdots,$

where $\bm{\nu}_{e,i},i=1,2,$ are the two normal vectors of $e$.

  4. (4)

Face-based tangential bases on $f\in\mathcal{F}$:

$\bm{v}_{f,\tau,i}^{n_{1},n_{2}}=\phi_{n_{1},n_{2}}|_{K_{i}}\bm{\tau}_{{f},i},\quad
i=1,2,$

where $\bm{\tau}_{f,i},i=1,2,$ are the two tangent vectors of $f$, $K_{1}$ and
$K_{2}$ are the two elements sharing $f$. Moreover,

$\phi_{n_{1},n_{2}}=J_{n_{1}}^{2n_{2}+5,2}\left(\frac{\lambda_{1}-\lambda_{2}-\lambda_{3}}{\lambda_{1}+\lambda_{2}+\lambda_{3}}\right)\cdot(\lambda_{1}+\lambda_{2}+\lambda_{3})^{n_{1}}J_{n_{2}}^{2,2}\left(\frac{\lambda_{1}-\lambda_{2}}{\lambda_{1}+\lambda_{2}}\right)(\lambda_{1}+\lambda_{2})^{n_{2}}\lambda_{1}\lambda_{2}\lambda_{3},$

where $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ are the barycentric
coordinates associated with $f$;

  5. (5)

Face-based normal bases on $f\in\mathcal{F}$:

$\bm{v}_{f,\nu}^{n_{1},n_{2}}=\phi_{n_{1},n_{2}}\bm{\nu}_{f},$

where $\bm{\nu}_{f}$ is the normal vector of $f$.

  6. (6)

Interior basis functions in $K\in\mathcal{T}$:

$\bm{v}_{K,i}^{n_{0},n_{1},n_{2}}=\phi_{n_{0},n_{1},n_{2}}\bm{e}_{i},\quad
i=1,2,3,$

where $\phi_{n_{0},n_{1},n_{2}}$ is given by (5.6) without the superscript
$(j)$.

###  5.4. Condition number

We test the condition number of mass and quasi-stiffness matrices. The
condition number of the mass matrix $M$ is calculated by

$\kappa(A)=\frac{\lambda_{\max}}{\lambda_{\min}},$

where $\lambda_{\max}$, $\lambda_{\min}$ are the maximum and minimum
eigenvalues of the matrix $M$, respectively. For the quasi-stiffness matrix
$S$, only nonzero eigenvalues will be considered. We also consider the
normalized mass and quasi-stiffness matrices $\widetilde{M}$ and
$\widetilde{S}$, whose nonzero diagonal entries are scaled to 1. The condition
numbers of the four matrices are listed in Table 3. The condition numbers of
the normalized mass and quasi-stiffness matrices are $O(10^{4})$ and $O(10)$
when the $p$ reaches 9 and the condition number of $\tilde{S}$ is almost
constant.

Table 3. condition numbers of $M$, $\widetilde{M}$, $S$, and $\widetilde{S}$.

$p$ | $M$ | $\widetilde{M}$ | $S$ | $\widetilde{S}$
---|---|---|---|---
2 | 3.0345e+02 | 1.0121e+02 | 7.8806e+01 | 2.9527e+01
3 | 9.1025e+03 | 2.2944e+02 | 3.2677e+02 | 5.8554e+00
4 | 7.7549e+05 | 1.9106e+03 | 9.4503e+03 | 1.9563e+01
5 | 6.9908e+06 | 2.7049e+03 | 1.3638e+05 | 1.3648e+01
6 | 1.1922e+08 | 5.3600e+03 | 1.6570e+06 | 1.4868e+01
7 | 1.9673e+09 | 7.4625e+03 | 2.0611e+07 | 1.6779e+01
9 | 3.2159e+10 | 1.2387e+04 | 2.5840e+08 | 1.9111e+01

##  6\. Conclusions

This paper constructed partially discontinuous nodal finite elements for
$H(\operatorname{curl})$ and $H(\operatorname{div})$ by allowing discontinuity
in the tangential or normal directions of vector-valued Lagrange elements,
respectively. These elements can be implemented as a combination of standard
continuous and discontinuous elements. The bases appear to be well-conditioned
in the numerical tests, which shows the computational potential of these
finite elements. Other bases, local complete sequences [38] and $hp$\-
preconditioning can be investigated in future research.

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