Interpolation function for vector finite element

Question

Concerning edge finite element (not nodal element), is there any explicit function (e.g. polynomial) to describe the interpolation within the element? I've read numerous literatures such as "Mixed finite element methods and application" by Boffi et al, and "FENICS manual (https://femtable.org/)". I could follow the concepts of mixed variational principles and Hilbert spaces: H(Div) and H(curl). However, I'm having a hard time implementing the element because the interpolatation functions and spaces are written in mathematical terms, which are rather jargons for me. For instance, $P_1\Lambda^1(\Delta_2)$, and $P_1\Lambda^2(\Delta_2)$ in exterior calculus.

It would be greatly appreciated if anyone suggest me a way to interprete these jargons, or some literatures to understand it. Thanks in advance.

Answer 1

There are a lot of less abstract frameworks for defining the finite element basis functions, and classical books among them like Ciarlet, Brezzi, Fortin, etc. As far as I know, the differential forms formalism is quite a new thing from the viewpoint of FEM discretizations and was developed not many years ago. Are you looking for something like in http://www.math.clemson.edu/~vjervin/papers/erv112.pdf ?

As for the formalism, most people refer to the works by Douglas Arnold with keywords "finite element exterior calculus".

Hope this could help

Answer 2

If you take the lowest-order 2d Raviart-Thomas element as an example, the degrees of freedom are located at the mid-point of edges, and represent the normal component of the vector field. As a consequence, if you want to define an interpolation operator $\vec\psi_h = I_h \vec f$ for some vector-valued (and continuous) function $\vec f$, then you need to evaluate $\vec f$ at the mid-points of edges, take the component of this vector normal to the edge, and that is the value of $\vec n \cdot \vec \psi_h$ at that mid-points of edges.

Similarly, if you had lowest-order Nedelec elements, it would be the tangential component that you had to evaluate.