# 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 Thomas^{3} 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 Marini^{5} 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:

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

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:

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

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:

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

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

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

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

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

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 | 4 | 8 | 12 | 17 | 23 | 30 | 38 |

1 | 8 | 14 | 22 | 32 | 44 | 58 | 74 | |

2 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | |

3 | 0 | 8 | 20 | 32 | 50 | 74 | 105 | 144 |

1 | 24 | 48 | 84 | 135 | 204 | 294 | 408 | |

2 | 18 | 39 | 72 | 120 | 186 | 273 | 384 | |

3 | 4 | 10 | 20 | 35 | 56 | 84 | 120 | |

4 | 0 | 16 | 48 | 80 | 136 | 216 | 328 | 480 |

1 | 64 | 144 | 272 | 472 | 768 | 1188 | 1764 | |

2 | 72 | 168 | 336 | 606 | 1014 | 1602 | 2418 | |

3 | 32 | 84 | 180 | 340 | 588 | 952 | 1464 | |

4 | 5 | 15 | 35 | 70 | 126 | 210 | 330 |

## Further reading

More details are provided in the SIAM News article Periodic Table of the Finite Elements.

## References

- R. Courant, Bulletin of the American Mathematical Society 49, 1943.
- W. H. Reed and T. R. Hill, Los Alamos report LA-UR-73-479, 1973.
- P. A. Raviart and J. M. Thomas, Lecture Notes in Mathematics 606, Springer, 1977.
- J. C. Nédélec, Numerische Mathematik 35, 1980.
- F. Brezzi, J. Douglas Jr., and L. D. Marini, Numerische Mathematik 47, 1985.
- J. C. Nédélec, Numerische Mathematik 50, 1986.
- D. N. Arnold, R.S. Falk, and R. Winther, Acta Numerica 15, 2006.
- R. Hiptmair, Mathematics of Computation 68, 1999.
- H. Whitney, Geometric Integration Theory, 1957.
- D. N. Arnold, D. Boffi, and F. Bonizzoni, Numerische Mathematik, 2014.
- D. N. Arnold and G. Awanou, Mathematics of Computation, 2014.
- A. Logg, K.-A. Mardal, and G.N. Wells (eds.), Automated Solution of Differential Equations by the Finite Element Method, Springer, 2012.
- R. C. Kirby, ACM Transactions on Mathematical Software 30, 2004.
- A. Logg and G. N. Wells, ACM Transactions on Mathematical Software 37, 2010.
- M. Alnæs, A. Logg, K. B. Ølgaard, M. E. Rognes, and G. N. Wells, ACM Transactions on Mathematical Software 40, 2014.