Test functions of Raviart-Thomas elements?

The test functions of general finite elements are like interpolation functions (if my understanding is correct). But how about test functions of Raviart-Thomas elements?

Let's raise the $$RT0$$ element as an example. The test functions of the three edges are

$$\phi_1 = \sqrt{2}\left(\hat{x_1} \quad \hat{x_2}\right)^{T}, \phi_2 = \left(-1+\hat{x_1} \quad \hat{x_2}\right)^{T}, \text{ and } \phi_3 = \left(\hat{x_1} \quad -1+\hat{x_2}\right)^{T}$$,

where $$\hat{x_1} \text{ and } \hat{x_2}$$ are coodinates of a reference triangular element (as shown in the figure below). We can calculate the $$\hat{x_1} \text{ and } \hat{x_2}$$ based on the coordinates of original elements, i.e. $${x_1} \text{ and } {x_2}$$.

But, what is the physical meaning of the test functions of RT element? I think they are not like interpolation functions. They just kind of link to the normal vectors of edges?

• I don't have an answer to your question, but I would like to say that people who like the FEM tend to like interpolatory basis function and sometimes also use interpolatory quadrature methods, too. They can make life really easy, but they can also lead to instabilities. Thus the zoo of other non-interpolatory bases. Nov 22 '21 at 16:24

Why not try drawing the basis?

You use Raviart-Thomas basis to approximate a vector field so a quiver plot makes sense. This basis function (one out of three) is associated with the left edge. It is a kind of flux towards the left edge which is orthogonal to the normals of the other edges and magnitude is zero on the opposite vertex.

In the global basis you combine two of these on the neighboring elements so that the arrows are pointing to the same direction. The global basis does not look very smooth because only the normal component is continuous. However, neither does the standard piecewise linear basis look very smooth.

I'd argue that RT0 basis are interpolants, but for functions that are observed/measured in terms of flux across facets (as opposed to functions that are measured by point sampling). Each RT0 function has unit flux across one particular facet, zero across the others, and smooth behavior over the interior. Similarly, the Nedelec basis interpolates functions that are measured in terms of circulations along edges.

In all cases you can think of your DoF's as observations/measurements of your field over some particular bit of geometry (vertex/line/facet), and it's the role of the basis function to reconstruct/interpolate that field at other/non-observed points (eg points on the interior).