# 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, 2021 at 16:24