Raviart-Thomas elements global definition and compact support

Question

As per the suggestion by Christian in the comments here, as part of my continuing quest to understand the Raviart-Thomas (RT) elements I'd like to know how exactly the RT elements are defined globally, and in particular how they have compact support.

For RT0 on the reference square, one of the basis functions is $\mathbf{\phi}(\mathbf{x}) = \frac{1}{4}\langle 1 + x, 0\rangle^T$. This function is only dependent on $x$ so it is non-zero over all elements above and below the reference square. Since the RT are $H$(div) conforming, I suppose there is no need to enforce continuity in the solution, or the basis functions. As I understand it, this means that we could simply set $\mathbf{\phi}$ to be $\mathbf{0}$ outside some domain.

As a concrete example, given the edge numbering below, (I assume there is one basis functions per edge for RK0, but this may be wrong) what basis functions are non-zero over the middle element (A)?

As a separate question, for Langrangian elements of order $k$ we choose the finite dimensional subset of $H^1$ to be the set of all piece-wise continuous polynomials of order $k$. For the RT elements of order $k$ we take the subspace of $H$(div):

$$P_{k+1,k} \times P_{k,k+1}$$ as defined in the answer to my last question. Does this space have a name?

Answer 1

In your example, the basis functions corresponding to the four edges 9,12,13,16. (For $k=1$, it would be two basis functions per edge, plus the four basis functions corresponding to the interior degrees of freedom (DOF).)

The reason is that the normal trace of an interpolant from the local polynomial space over one of the edges is uniquely determined by the corresponding edge DOFs (this is part of the reason the local polynomial space is defined the way it is). Since the global Raviart-Thomas interpolant should be $H(div)$ conforming, it must have continuous normal trace across this edge, and hence you cannot choose these DOFs differently for the adjoining elements (otherwise, you'd introduce a discontinuity). This means that the corresponding basis functions cannot have different coefficients, so they cannot be linearly independent and hence not really separate basis functions. You therefore have to

1. identify local DOFs for the same edge with one global degree of freedom (for $k>0$, you of course still have to keep the moments separate, i.e., identify the first DOF of element $K_1$ with the first DOF of element $K_2$ etc.) and

2. "glue together" all corresponding local basis functions to a single global basis function.

(By the way, this is exactly the same way you'd work with $H^1$ conforming elements; in particular, for the implementation you only need to do 1.).

To your separate question: The local Raviart-Thomas space of degree $k$ is usually just called $RT_k(K)$ (with $K$ usually denoting the reference element). The global Raviart-Thomas space $$\{v\in H(div):v|_K \in RT_k(K) \text{ for all }K\in \mathcal{T}_h\}$$ over a tesselation $\cal{T}_h$ of your domain is usually just denoted by $RT_k$ (or $RT_k(\mathcal{T}_h)$, if this should be emphasized).