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From: | Torquil Macdonald Sørensen |
Subject: | Re: [Getfem-users] RT0 and Nedelec vs. Whitney 1 and 2-forms |
Date: | Sat, 25 Sep 2010 17:40:30 +0200 |
User-agent: | Mozilla/5.0 (X11; U; Linux x86_64; en-US; rv:1.9.1.12) Gecko/20100917 Icedove/3.0.8 |
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
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