sign of DOFs for Raviart-Thomas elements

Question

My question is whether one can relate the DOFs of a Raviart-Thomas element to the FacetNormal of the mesh.

Specifically, if u belongs to the lowest order RT space, FunctionSpace(mesh, 'RT', 1), in two dimensions, then the degree of freedom of u associated to an edge e is the integral of u.nn on the edge where nn is one of the unit normals to e. If n = FacetNormal(mesh) in FEniCS, for an interior edge e it may happen that nn = n('+') or nn may equal n('-') and for a boundary edge it may happen that nn equals n or nn equals -n on e. For a construction we are doing we need to know which is the case. Is there a good way to do this?

It seems from experimentation that nn is always obtained by rotating the unit tangent along the edge clockwise, if the tangent is chosen to point from the lower indexed to higher indexed vertex on the edge. Can anyone confirm that this is indeed always the case? And, if so, is it safe to count on it remaining the case in the future?

Comments

Hi, I believe that the implementation follows this paper by Marie Rognes. Your observations about the edge tangent and clockwise rotation agree with eq. (2.11) and fig (5.1) therein.

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Thanks very much for the reference, which is indeed very pertinent. The first two paragraphs of section 5.1, in particular, confirm my suppositions above.

Answer 1

Answer from Miro: I believe that the implementation follows this paper by Marie Rognes. Your observations about the edge tangent and clockwise rotation agree with eq. (2.11) and fig (5.1) therein.