Construction of finite element differential forms based on deRham sequences

Question

Hi, mathoverflow, I am currently working on finite element exterior calculus, and right now my concern is the construction of some certain kind of "conforming" basis functions for a finite element space. Guess most of you guys are working on pure math, hence before I asked my questions I would like to briefly introduce what this is.

For simplicity consider a 3D simply-connected polyhedral domain $\Omega$ which has a triangulation consisting of tetrahedra, we have the following deRham sequence: $$ H^1_0(\Omega) \xrightarrow{\nabla} H_0(\mathbf{curl}) \xrightarrow{\nabla\times} H_0(\mathrm{div})\xrightarrow{\nabla\cdot} L^2(\Omega) $$ And its discrete counterpart reads: $$ P^1 \xrightarrow{\nabla} \mathcal{Nd}^0 \xrightarrow{\nabla\times} \mathcal{RT}^0 \xrightarrow{\nabla\cdot} P^0_{-1} $$ in which each space is the conforming finite element space of its continuous counterpart, $P^1$ is the continuous piecewise degree one polynomial on this triangulation, within each tetrahedron, it consists four basis functions in the form of $\lambda_i$ of which the degrees of freedom associated with the four vertices $V_j$ of this tetrahedron, $\lambda_i(V_j) = \delta_{ij}$ is the barycentric coordinate of each vertices. $\mathcal{Nd}^0$ is the lowest order Nédélec elements, it basis functions are in the form of $\lambda_i\nabla\lambda_j - \lambda_j\nabla\lambda_i$ for each of the six edges $\overrightarrow{V_i V_j}$. $\mathcal{RT}^0$ is the lowest order Raviart-Thomas element, the basis functions are $\lambda_i \nabla\lambda_j\times \nabla \lambda_k + \lambda_j \nabla\lambda_k\times \nabla \lambda_i + \lambda_j \nabla\lambda_i\times \nabla \lambda_j$ for the face $V_i V_j V_k$. Lastly $P^1_{-1}$ is the constant on each tetrahedra with no continuity across the inter-element faces.

Use the construction of $\mathcal{Nd}^0$ as an example, the basis functions are obtained by Whitney 1-form interpolation, we would like to construct 1-form such that: $$ \int_{V_k V_l} \boldsymbol{\varphi}_{ij} d\boldsymbol{l}= \begin{cases} 1 & \text{if $i=k$, $j=l$}\newline -1 & \text{if $i=l$, $j=k$}\newline 0 & \text{otherwise} \end{cases} $$

and what we get would be $\boldsymbol{\varphi}_{ij} =\lambda_i\nabla\lambda_j - \lambda_j\nabla\lambda_i $ which is the lowest order Whitney form.

Now my question is: Instead of using the standard interpolation technique to construction these element, can we directly use the deRham sequence to construct the finite element basis functions? Say we know that $R(\nabla) = \nabla H^1_0= \ker(\mathbf{curl})$, hence we have $$ H_0(\mathbf{curl}) = R(\nabla) \oplus R(\nabla)^{\perp} $$ where the polar set $R(\nabla)^{\perp}$ is isomorphic to $R(\mathbf{curl})$. And this decomposition also holds for discrete spaces: $$ \mathcal{Nd}^0 = \nabla P^1 \oplus W $$ where $W$ is isomorphic to the range of $\mathbf{curl}$-operator into the space $\mathcal{RT}^0$, since $R(\mathbf{curl}) = \ker(\mathrm{div})$, now if we cook up a finite element space for divergence free functions, we could know what $W$ is.

However, if following this route, we wouldn't have something nice like Whitney 1-form: $\lambda_i\nabla\lambda_j - \lambda_j\nabla\lambda_i$, so where does the construction of discrete 1-form on a tetrahedron falls into above deRham sequence framework? or rather to say, how does this deRham cohomology framework affect or guide our construction of the discrete k-forms basis on a simplex?

Answer 1

As robot suggested, a good place to start would be the work of Arnold, Falk, and Winther -- particularly their 2006 paper in Acta Numerica and 2010 paper in Bulletin of the AMS. (Both available at http://ima.umn.edu/~arnold/publications.html.)

One key insight of their work is that, to get a stable discretization, we need two properties. First, the discrete complex needs to be a subcomplex, in the sense that the subspace inclusions commute with the differentials. Second, there needs to be a bounded projection mapping the continuous complex to the discrete complex, and this projection must also commute with the differentials. In $\mathbb{R}^n$, there are two main families of piecewise-polynomial differential forms, which Arnold et al. call $\mathcal{P}_r$ and $\mathcal{P}_r^-$, where $r$ denotes the degree of the polynomials. (The lowest-order Whitney forms are $\mathcal{P}_1^-$.) They show that these spaces can be used, systematically, to construct subcomplexes that satisfy the two stability conditions.

This defines the key relationship between the choice of basis functions and the de Rham complex: the discrete inclusion and projection functions must "respect the structure" of the de Rham complex, in the sense of commuting with the differentials, otherwise one is not guaranteed stability. As you suggest, there are of course many other ways of constructing interpolants, and as long as they satisfy the conditions above, you are guaranteed a stable discretization. However, piecewise-polynomials are especially useful for computation, and their approximation properties are well understood (which is part of the reason they are used in classical finite element methods, not to mention FEEC).