Periodic Table of the Finite Elements

Background

The table presents the primary spaces of finite elements for the discretization of the fundamental operators of vector calculus: the gradient, curl, and divergence. A finite element space is a space of piecewise polynomial functions on a domain determined by: (1) a mesh of the domain into polyhedral cells called elements, (2) a finite dimensional space of polynomial functions on each element called the shape functions, and (3) a unisolvent set of functionals on the shape functions of each element called degrees of freedom (DOFs), each DOF being associated to a (generalized) face of the element, and specifying a quantity which takes a single value for all elements sharing the face. The element diagrams depict the DOFs and their association to faces.

The spaces $\mP_r^-\Lambda^k$ and $\mP_r\Lambda^k$ depicted on the left half of the table are the two primary families of finite element spaces for meshes of simplices, and the spaces $\mQ_r^-\Lambda^k$ and $\mS_r\Lambda^k$ on the right side are for meshes of cubes or boxes. Each is defined in any dimension $n\ge 1$, for each value of the polynomial degree $r\ge 1$, and each value of $0 \leq k \leq n$. The parameter $k$ refers to the operator: the spaces belong to the domain of the $k$th exterior derivative. Thus for $k=0$, the spaces discretize the Sobolev space $H^1$, the domain of the gradient operator; for $k=1$, they discretize $H(\curl)$, the domain of the curl; for $k=n-1$ they discretize $H(\div)$, the domain of the divergence; and for $k=n$, they discretize $L^2$.

The spaces $\mP_r^-\Lambda^0$ and $\mP_r\Lambda^0$, which coincide, are the earliest finite elements, going back in the case $r=1$ of linear elements to Courant,¹ and collectively referred to as the Lagrange elements. The spaces $\mP_{r+1}^-\Lambda^n$ and $\mP_r\Lambda^n$, which also coincide, are the discontinuous Galerkin elements, consisting of piecewise polynomials with no interelement continuity imposed, first introduced by Reed and Hill.² The space $\mP_r^-\Lambda^1$ in 2 dimensions was introduced by Raviart and Thomas³ and generalized to the 3-dimensional spaces $\mP_r^-\Lambda^1$ and $\mP_r^-\Lambda^2$ by Nédélec,⁴ while $\mP_r\Lambda^1$ is due to Brezzi, Douglas, and Marini⁵ in 2 dimensions, its generalization to 3 dimensions again due to Nédélec.⁶ The unified treatment and notation of the $\mP_r^-\Lambda^k$ and $\mP_r\Lambda^k$ families is due to Arnold, Falk, and Winther as part of finite element exterior calculus,⁷ extending the earlier work of Hiptmair for the $\mP_r^-\Lambda^k$ family.⁸ The space $\mP_1^-\Lambda^k$ is the span of the elementary forms introduced by Whitney.⁹

The family $\mQ_r^-\Lambda^k$ of cubical elements can be derived from the 1-dimensional Lagrange and discontinuous Galerkin elements by a tensor product construction detailed by Arnold, Boffi, and Bonizzoni,¹⁰ but for the most part were presented individually along with the corresponding simplicial elements in the papers mentioned. The second cubical family $\mS_r\Lambda^k$ is due to Arnold and Awanou.¹¹

The finite elements in this table have been implemented as part of the FEniCS Project^{12, 13, 14} Each may be referenced by the Unified Form Language (UFL)¹⁵ by giving its family, shape and degree, with the family as shown on the table. For example, the space $\P{3}{1}{3}$ may be referred in UFL as:

FiniteElement("N2E", tetrahedron, 3)

Alternatively, the elements may be accessed in a uniform fashion as:

FiniteElement("P-", shape, r, k)
FiniteElement("P", shape, r, k)
FiniteElement("Q-", shape, r, k)
FiniteElement("S", shape, r, k)

for $\mP_r^-\Lambda^k$, $\mP_r\Lambda^k$, $\mQ_r^-\Lambda^k$, and $\mS_r\Lambda^k$, respectively.

The $\mP_r^-\Lambda^k$ family

The shape function space for $\mP_r^-\Lambda^k$ is $\mP_{r-1}\Lambda^k+\kappa\mP_{r-1}\Lambda^{k+1}$ where $\kappa$ is the Koszul differential.⁷ It includes the full polynomial space $\mP_{r-1}\Lambda^k$, is included in $\mP_r\Lambda^k$, and has dimension

$$\dim \mP_r^-\Lambda^k(\Delta_n)=\binom{r+n}{r+k}\binom{r+k-1}{k}.$$

The degrees of freedom are given on faces $f$ of dimension $d\ge k$ by moments of the trace weighted by a full polynomial space:

$$u\mapsto \int_f (\tr_f u)\wedge q, \quad q\in\mP_{r+k-d-1}\Lambda^{d-k}(f).$$

The spaces with constant degree $r$ form a complex:

$$\mP_r^-\Lambda^0\xrightarrow{\mathrm d}\mP_r^-\Lambda^1\xrightarrow{\mathrm d}\cdots\xrightarrow{\mathrm d}\mP_r^-\Lambda^n.$$

Dimensions of the $\mP_r^-\Lambda^k(\Delta_n)$ spaces
n	k	r = 1	2	3	4	5	6	7
1	0	2	3	4	5	6	7	8
	1	1	2	3	4	5	6	7
2	0	3	6	10	15	21	28	36
	1	3	8	15	24	35	48	63
	2	1	3	6	10	15	21	28
3	0	4	10	20	35	56	84	120
	1	6	20	45	84	140	216	315
	2	4	15	36	70	120	189	280
	3	1	4	10	20	35	56	84
4	0	5	15	35	70	126	210	330
	1	10	40	105	224	420	720	1155
	2	10	45	126	280	540	945	1540
	3	5	24	70	160	315	560	924
	4	1	5	15	35	70	126	210

The $\mP_r\Lambda^k$ family

The shape function space for $\mP_r\Lambda^k$ consists of all differential $k$-forms with polynomial coefficients of degree at most $r$, and has dimension

$$\dim \mP_r\Lambda^k(\Delta_n)=\binom{r+n}{r+k}\binom{r+k}{k}.$$

The degrees of freedom are given on faces $f$ of dimension $d\ge k$ by moments of the trace weighted by a $\mP_r^-$ space:

$$u\mapsto \int_f (\tr_f u)\wedge q, \quad q\in\mP^-_{r+k-d}\Lambda^{d-k}(f).$$

The spaces with decreasing degree $r$ form a complex:

$$\mP_r\Lambda^0\xrightarrow{\mathrm d}\mP_{r-1}\Lambda^1\xrightarrow{\mathrm d}\cdots\xrightarrow{\mathrm d}\mP_{r-n}\Lambda^n.$$

Dimensions of the $\mP_r\Lambda^k(\Delta_n)$ spaces
n	k	r = 1	2	3	4	5	6	7
1	0	2	3	4	5	6	7	8
	1	2	3	4	5	6	7	8
2	0	3	6	10	15	21	28	36
	1	6	12	20	30	42	56	72
	2	3	6	10	15	21	28	36
3	0	4	10	20	35	56	84	120
	1	12	30	60	105	168	252	360
	2	12	30	60	105	168	252	360
	3	4	10	20	35	56	84	120
4	0	5	15	35	70	126	210	330
	1	20	60	140	280	504	840	1320
	2	30	90	210	420	756	1260	1980
	3	20	60	140	280	504	840	1320
	4	5	15	35	70	126	210	330

The $\mQ_r^-\Lambda^k$ family

This family is constructed from the complex of 1-dimensional finite elements using a tensor product construction.¹⁰ The shape function space on the unit cube $\square_n=I^n$ is given by

$$\mQ_r^-\Lambda^k(\square_n) = \bigoplus_{\sigma\in \Sigma(k,n)}\left[\bigotimes_{i=1}^n \mP_{r-\delta_{i,\sigma}}(I)\right]\, \mathrm{d}x^{\sigma_1}\wedge\cdots\wedge \mathrm{d}x^{\sigma_k}, $$

where $\Sigma(k,n)$ denotes the increasing maps $\{1,\ldots,k\}\to\{1,\ldots,n\}$. Its dimension is $\dim \mQ_r^-\Lambda^k(\square_n) =\binom{n}{k}r^k(r+1)^{n-k}. $ The degrees of freedom are

$$u \mapsto \int_f (\tr_f u)\wedge q, \quad q\in \mQ_{r-1}^-\Lambda^{d-k}(f).$$

The spaces with constant degree $r$ form a complex:

$$\mQ_r^-\Lambda^0 \xrightarrow{\mathrm d} \mQ_r^-\Lambda^1 \xrightarrow{\mathrm d} \cdots \xrightarrow{\mathrm d} \mQ_r^-\Lambda^n.$$

Dimensions of the $\mQ_r^-\Lambda^k(\square_n)$ spaces
n	k	r = 1	2	3	4	5	6	7
1	0	2	3	4	5	6	7	8
	1	1	2	3	4	5	6	7
2	0	4	9	16	25	36	49	64
	1	4	12	24	40	60	84	112
	2	1	4	9	16	25	36	49
3	0	8	27	64	125	216	343	512
	1	12	54	144	300	540	882	1344
	2	6	36	108	240	450	756	1176
	3	1	8	27	64	125	216	343
4	0	16	81	256	625	1296	2401	4096
	1	32	216	768	2000	4320	8232	14336
	2	24	216	864	2400	5400	10584	18816
	3	8	96	432	1280	3000	6048	10976
	4	1	16	81	256	625	1296	2401

The $\mS_r\Lambda^k$ family

The shape function space for $\mS_r\Lambda^k$ is given by

$$\mP_r\Lambda^k \oplus \bigoplus_{\ell\ge 1}[\kappa\mH_{r+\ell-1,\ell}\Lambda^{k+1} \oplus \mathrm{d}\kappa\mH_{r+\ell,\ell}\Lambda^k], $$

where $\mH_{r,\ell}\Lambda^k$ consists of homogeneous polynomial $k$-forms of degree $r$ which are linear and undifferentiated in at least $\ell$ variables.¹¹ Its dimension is $\dim \mS_r\Lambda^k(\square_n) = \sum_{d\ge k} 2^{n-d}\binom{n}{d} \binom{r-d+2k}{d}\binom{d}{k}.$ The degrees of freedom are

$$u\mapsto \int_f (\operatorname{tr}_f u)\wedge q, \quad q\in \mP_{r-2(d-k)}\Lambda^{d-k}(f).$$