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

[Yves Renard]

On samedi 25 septembre 2010, Torquil Macdonald Sørensen wrote:
> 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?

Yes. A vectorial element will be at least tranformed such that either the 
normal or the tangent vector to the edges will be preserved.
vtype = VECTORIAL_PRIMAL_TYPE means that we consider a vectorial field. The 
shape function are multiplied to the left by the gradient of the 
transformation. With vtype = VECTORIAL_DUAL_TYPE we suppose that the vector 
field react like  a gradient and have to be multiplied to the left by the 
inverse of the transpose of the transformation gradient.

You are right, this is missing in the documentation and should be added ... !


> 
> 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


-- 

  Yves Renard (address@hidden)       tel : (33) 04.72.43.87.08
  Pole de Mathematiques, INSA-Lyon             fax : (33) 04.72.43.85.29
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  69621 Villeurbanne Cedex, FRANCE
  http://math.univ-lyon1.fr/~renard

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