Difference between Function on RT and VectorFunction on CR?

Question

Can someone explain to me the the difference between the function spaces

V = VectorFunctionSpace(mesh, 'CR', 1)

and

V = FunctionSpace(mesh, 'RT', 1)

(with mesh a 3-D mesh)?

Looking at the fenics book, pg 119, it looks as if a RT finite element is defined by four vectors (one on each face). To me this seems to be equivalent to defining three scalar values on each face, i.e. using sets of 3 CR elements.

On the other hand, the dimension for RT in 3-D is listed as 4 (just like CR) rather than 3x4=12. But if I'm just setting one scalar value per face, how is that different from using CR? Does the interpretation of the scalar as a normal component of some vector really make a difference?

(I don't have a particular problem in mind, this is just a question of general understanding).

Comments

First: Your question is not very FEniCS-specific.

Second: For what kind of problem do you consider these elements? You realize that the (Crouzeix-Raviart)^d space has d times the number of degrees of freedom as the Raviart-Thomas space (here, d=3)?

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I don't have a specific problem. Just a question of general understanding. I'll try clarify the question.

Answer 1

RT has one dof per face and it is the face-normal vector.
A VectorFunction of CR will have 3 dofs per face
and they are in x,y, and z direction.
RT is suitable for H(div) problems, because
the dofs are normal vectors. CR for H^1 problems, but
it is non-conforming.

CR will not give you a good approximation of H(div) problems
and RT will not give a good approximation of H^1 problems, cf.
eg. http://dr.ntu.edu.sg/bitstream/handle/10220/4594/Mardal-Tai-Winther-RFE-02.pdf

Answer 2

Does the interpretation of the scalar as a normal component of some vector really make a difference?

Yes, it makes a difference. Definition of DOF functionals is inseparable part of a finite element definition (in Ciarlet's sense). Respective DOF functionals of those two spaces are distinct, so the spaces are different.