Symmetric Bases for Finite Element Exterior Calculus Spaces

Abstract

In 2006, Arnold, Falk, and Winther developed finite element exterior calculus, using the language of differential forms to generalize the Lagrange, Raviart–Thomas, Brezzi–Douglas–Marini, and Nédélec finite element spaces for simplicial triangulations. In a recent paper, Licht asks whether, on a single simplex, one can construct bases for these spaces that are invariant with respect to permuting the vertices of the simplex. For scalar fields, standard bases all have this symmetry property, but for vector fields, this question is more complicated: Such invariant bases may or may not exist, depending on the polynomial degree of the element. In dimensions two and three, Licht constructs such invariant bases for certain values of the polynomial degree r, and he conjectures that his list is complete, that is, that no such basis exists for other values of r. In this paper, we show that Licht’s conjecture is true in dimension two. However, in dimension three, we show that Licht’s ideas can be extended to give invariant bases for many more values of r; we then show that this new larger list is complete. Along the way, we develop a more general framework for the geometric decomposition ideas of Arnold, Falk, and Winther.

Appendix A: Consistent Extension Operators

In this appendix, we show that the extension operators discussed in [3] yield valid choices of complements $W_d$ as discussed in Sect. 3, and so, in this context, the notion of geometric decomposition in Sect. 3 matches the one in [3]. However, we also show that our notion of geometric decomposition is more general (and hence weaker). For our results, our simpler and more general setup suffices, but we expect that other work in finite element exterior calculus may require the more stringent requirements on the extension maps in [3].

In our notation, we take the following definition from [3, Section 4].

Definition A.1

For faces $K\subseteq F\subseteq T^n$, an operator $E_{K, F}:V(K)\rightarrow V(F)$ is an extension operator if ${{\,\textrm{tr}\,}}_{F, K}E_{K, F}\,\alpha =\alpha$ for all $\alpha \in V(K)$.

A family of such extension operators $\{E_{F,F'}\mid F\subseteq F'\subseteq T^n\}$ is consistent if the diagram

commutes for any subfaces F and $F'$ of a face H, where $K=F\cap F'$.

Given a consistent family of extension operators, we can construct corresponding complements $W_d$ as in Definition 3.5.

Proposition A.2

Given a consistent family of extension operators, let

$$\begin{aligned} W_d:=\bigoplus _{\begin{array}{c} F\subseteq T^n\\ \dim F=d \end{array}}E_{F, T^n}\mathring{V}(F). \end{aligned}$$

Then, $V_d=W_d\oplus V_{d+1}$, where the $V_d$ are defined in Definition 3.3.

Proof

First, the direct sum notation in the definition of $W_d$ implicitly assumes that $E_{F, T^n}\mathring{V}(F)\cap E_{F', T^n}\mathring{V}(F')=0$; this is a consequence of [3, Equation 4.9]. Moreover, by [3, Lemma 4.1], if $F'$ and F are d-dimensional faces and $F'\ne F$, then the composition ${{\,\textrm{tr}\,}}_{T^n, F'}E_{F, T^n}$ is the zero map on $\mathring{V}(F)$. On the other hand, if $F'=F$, then the composition ${{\,\textrm{tr}\,}}_{T^n, F'}E_{F, T^n}$ is the identity map. Consequently, for any $\beta \in W_d$ written as

$$\begin{aligned} \beta =\bigoplus _{\begin{array}{c} F\subseteq T^n\\ \dim F=d \end{array}}E_{F, T^n}\beta _F, \end{aligned}$$

where $\beta _F\in \mathring{V}(F)$, we have

$$\begin{aligned} {{\,\textrm{tr}\,}}_{T^n, F'}\beta =\beta _{F'} \end{aligned}$$

(5)

for any d-dimensional face $F'$. From Eq. (5), it is easy to see that $W_d\cap V_{d+1}=0$. Indeed, with $\beta$ as above, if $\beta \in V_{d+1}$ then by definition ${{\,\textrm{tr}\,}}_{T^n, F'}\beta =0$ for any d-dimensional face $F'$, so $\beta _{F'}=0$ for all d-dimensional $F'$, and so $\beta =0$.

Now, let $\alpha \in V_d$; we aim to write $\alpha =\beta +\gamma$ where $\beta \in W_d$ and $\gamma \in V_{d+1}$. By definition of $V_d$, we know that $\alpha$ vanishes on any $(d-1)$-dimensional face of $T^n$, and hence, ${{\,\textrm{tr}\,}}_{T^n, F}\alpha \in \mathring{V}(F)$ for any d-dimensional face F of $T^n$. Thus, we can let

$$\begin{aligned} \beta =\bigoplus _{\begin{array}{c} F\subseteq T^n\\ \dim F=d \end{array}}E_{F, T^n}{{\,\textrm{tr}\,}}_{T^n, F}\alpha \in W_d. \end{aligned}$$

By Eq. (5), if $F'$ is a d-dimensional face, then

$$\begin{aligned} {{\,\textrm{tr}\,}}_{T^n, F'}\beta ={{\,\textrm{tr}\,}}_{T^n, F'}\alpha . \end{aligned}$$

Thus, if we let $\gamma =\alpha -\beta$, then ${{\,\textrm{tr}\,}}_{T^n, F'}\gamma =0$ for any d-dimensional face $F'$, so $\gamma \in V_{d+1}$ by definition. $\square$

Remark A.3

The decomposition $V=W_0\oplus \cdots \oplus W_n$ in Proposition 3.6 then exactly yields the decomposition $V=\bigoplus _{F\subseteq T^n}E_{F, T^n}\mathring{V}(F)$ given in [3, Equation 4.9]. Moreover, by Eq. (5), the geometric decomposition map ${\mathcal {D}}$ in Definition 3.8 is just the natural map $\bigoplus _{F\subseteq T^n}E_{F, T^n}\mathring{V}(F)\rightarrow \bigoplus _{F\subseteq T^n}\mathring{V}(F)$.

Remark A.4

Only some of the converse holds. Given complements $W_d$, we obtain an injective geometric decomposition map ${\mathcal {D}}$ as in Definition 3.8. If this map is furthermore an isomorphism, then we have extension maps $E_{F, T^n}:\mathring{V}(F)\rightarrow W_d\subseteq V$ as discussed in Remark 3.13. As discussed in Corollary 3.15, these extension maps enjoy several properties. In particular, if $\alpha \in E_{F, T^n}\mathring{V}(F)$, then its restriction ${{\,\textrm{tr}\,}}_{T^n, F'}\alpha$ to any other d-dimensional face $F'\ne F$ is zero. However, [3, Lemma 4.1] makes a stronger claim that ${{\,\textrm{tr}\,}}_{T^n, F'}\alpha$ is zero for any face $F'$ (of any dimension) not containing F; this may fail if $F'$ has higher dimension than F.

Indeed, in the case where $V={\mathcal {P}}_2\Lambda ^0(T^2)$ discussed in Example 3.4, the choice of $W_0$ is quite flexible. The natural choice is $W_0={{\,\textrm{span}\,}}\{\lambda _0^2,\lambda _1^2,\lambda _2^2\}$, but, even if we require invariance with respect to the $S_3$ symmetry, nothing stops us from choosing

$$\begin{aligned} W_0={{\,\textrm{span}\,}}\{\lambda _0^2+17\lambda _1\lambda _2,\lambda _1^2+17\lambda _2\lambda _0,\lambda _2^2+17\lambda _0\lambda _1\}. \end{aligned}$$

With this latter choice, if F is the vertex 0 and $F'$ is the edge 12, one can check that $E_{F, T^2}(1)=\lambda _0^2+17\lambda _1\lambda _2$ and ${{\,\textrm{tr}\,}}_{T^2, F'}(\lambda _0^2+17\lambda _1\lambda _2)=17\lambda _1\lambda _2\ne 0$.

1.1 $S_{n+1}$-Invariance

In the context where V is $S_{n+1}$-invariant, we want $W_d$ to be $S_{n+1}$-invariant as well. In light of the construction in Proposition A.2, it suffices to require that the extension operators respect this action, in the sense that the diagram

commutes, where $F'=S_\pi F$, analogously to diagram (4) in Sect. 3.2. There is no reason to expect this property to be true in general, but it holds for extension operators defined by “natural” properties, including the extension operators constructed in [3] for the ${\mathcal {P}}_r\Lambda ^k(T^n)$ and ${\mathcal {P}}_r^-\Lambda ^k(T^n)$ spaces. We roughly sketch the arguments below; see also [10, Section 7].

There are actually two sets of extension operators constructed in [3]. The first, denoted with the symbols $F^{k, r}_{F, T^n}$ and $F^{k, r,-}_{F, T^n}$ in [3, Section 5], is defined in terms of degrees of freedom: The degrees of freedom of $F^{k, r, (-)}_{F, T^n}\alpha$ must match those of $\alpha$ on all subfaces $K\subseteq F$ and be zero on all other faces of $T^n$; this uniquely determines the extension. One then checks that if $\alpha '\in {\mathcal {P}}_r^{(-)}\Lambda ^k(F')$, then the condition that the degrees of freedom of $F^{k, r, (-)}_{F', T^n}\alpha '$ match those of $\alpha '$ implies that the degrees of freedom of $S_\pi ^*F^{k, r, (-)}_{F', T^n}\alpha '$ match those of $S_\pi ^*\alpha '$, and likewise for the degrees of freedom required to be zero. (Intuitively, rotating $T^n$ “rotates” the degrees of freedom.) Since this condition uniquely determines the extension, we conclude that $S_\pi ^*F^{k, r, (-)}_{F', T^n}\alpha '$ is indeed the extension of $S_\pi ^*\alpha '$, as desired.

The reasoning is similar for the extension operators $E^{k, r}_{F, T^n}$ and $E^{k, r, -}_{F, T^n}$ defined in [3, Sections 7 and 8]. In [3, Theorems 7.4 and 8.4], the authors show that $E^{k, r, (-)}_{F, T^n}\alpha$ is the unique extension of $\alpha$ that satisfies a certain vanishing condition on the face $F^*$ that is “opposite” F. As above, one checks that if $\alpha '\in {\mathcal {P}}_r^{(-)}\Lambda ^k(F')$, then this vanishing condition for $E^{k, r, (-)}_{F', T^n}\alpha '$ on $F'^*$ implies the corresponding vanishing condition for $S_\pi ^*E^{k, r, (-)}_{F', T^n}\alpha '$ with respect to $F^*$. Since this condition uniquely determines the extension, we conclude that $S_\pi ^*E^{k, r, (-)}_{F', T^n}\alpha '$ is indeed the extension of $S_\pi ^*\alpha$.