Re: [Getfem-users] RT0 and Nedelec vs. Whitney 1 and 2-forms

[Torquil Macdonald Sørensen]

On 25/09/10 08:48, Renard Yves wrote:
> Yes. If you set M to the identity, the value of the shape function will
> be intrinsic to the edge in the case of Nedelec element, so the
> connection to the degree of freedom of the neighbour element should be
> ok, except the sign. I think, you have at least to keep the selection of
> the direction which is made in the definition of the transformation
> matrix M.

I have discovered that I need to understand the meaning of the different "vtype" 
that can be assigned, but didn't find it in the documentation.

The Nedelec method uses vtype = VECTORIAL_DUAL_TYPE;

RT0 uses vtype = VECTORIAL_PRIMAL_TYPE;

A third type that seems to be possible is VECTORIAL_NOTRANSFORM_TYPE.

I noticed that even when using M = Identity (is_equiv = true), the mass matrix 
elements for the Nedelec case scaled correctly when the thetrahedron size was 
varied. This was unexpected , and must be due to the vtype setting?

If the tetrahedron side length is around h, the gradients of the hat functions 
are around 1/h. The Whitney edge elements are defined like

lambda_ij = lambda_i grad lambda_j - lambda_j grad lambda_i ~ 1/h

The correct mass matrix is therefore proportional to (1/h)^2 * h^3 ~ h.

I confirmed this behaviour for the modified Nedelec method (my unfinished 
attempt at Whitney 1-forms), which does not use any M at the moment (is_equiv = 
true).

So I'm guessing something is being taken care of "behind the scenes" due to the 
vtype setting?

On the other hand, with vtype = VECTORIAL_NOTRANSFORM_TYPE, the mass matrix 
element scaling was not correct, but instead ~ h^3 as expected when one does not 
take account the vector transformation.

When I understand this I need to understand how the cross product of gradients 
is to be treated for the Whitney 2-form case.

Torquil