Definition of the term "De Rham map" I am a PhD student working in the field of numerical simulation. In several papers, the term "De Rham map" pops up (for instance in the very good thesis by Jérôme Bonelle : https://tel.archives-ouvertes.fr/tel-01116527v2/document ).
I am unsure about the definition of this term. I have come to believe that this term generally means "operation that have continuous objects correspond to discrete ones" i.e. "means of defining the actual values of the degrees of freedom of a discrete object from a continuous object", but I gradually suspect that it might rather mean "result of the integration of a cochain on a differentiable manifold". Of course, the two notions coincide in the litterature I have come accross.
So my question is : What is the definition of the term "De Rham map" ?
Regards,
 A: In the context of approximating second order boundary value problems we invented the name "de Rham map" with A. Bossavit in paper T. Tarhasaari, L. Kettunen, A. Bossavit: Some realizations of a discrete Hodge operator: A reinterpretation of finite element techniques. IEEE Trans. Magn., Vol 35(3), 1999, pp. 1494-97.
Intuitively, the idea is as follows. Finite element and finite difference kind of methods provide one with approximative solutions of boundary value problems in finite dimensional spaces. To build such approximations one needs a map from the (infinite dimensional) linear spaces of fields to some finite dimensional spaces in which the fields are approximated. 
Having a "mesh", that is, a cellular complex, and a field, say p-form $f$, the de Rham map sends $f$ to integrals of $f$ on p-chains (which in turn are formal sums of p-cells of the complex/mesh). Or, alternatively, a simplified view, the de Rham map sends f to an array of integrals of $f$ on the (oriented) p-cells of the complex. 
For example, the de Rham map sends the 2-form magnetic flux $b$ to an array of real numbers representing fluxes $\int_c b$ on the 2-cells $c$ (such as triangles) of the complex/mesh. In terms of classical language, if magnetic flux density ${\bf B}$ is considered as a (smooth) vector field, pair $({\bf B}, c)$ is mapped to $\int_c {\bf B}\cdot {\bf n}\, {\rm da}$.
Formally, the definition is something like: Let $F^p$ be the space of differential forms on manifold $\Omega$, $C^p$ the space of $p$-cochains in cellular complex $K$, and $K$ is a cellular tessellation of $\Omega$. Then, map $\mathcal{C}$ from $F^p$ to $C^p$ is the de Rham map, if for all $p$-chains $c\in K$ and smooth $p$-forms $f\in F^p$ map $\mathcal{C}$ satisfies $$\mathcal{C}f: C^p \rightarrow \mathbb{R}, \quad c\mapsto \int\limits_c f\,.$$
