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

[Torquil Macdonald Sørensen]

Thanks Yves, I believe I am zeroing in on what I need due to the help from you 
guys. From the documentation you linked to it seems that I need to use the 
identity matrix for M.

So I copied and modified the Nedelec code, setting M to the identity instead, 
and now the mass matrix elements are correct for the few examples I have 
calculated by hand, including differing tetrahedron sizes. There are some 
sign-issues that I may or may not need to address, although I'm not yet sure if 
that will turn out to be important.

I will now study the RT0 code to see if I need to do something similar there to 
get the mass matrix I need for the Whitney 2-forms.

The reason I need to do all this is because I am going to do a simulation that 
involves calculation of an expression that contains the Whitney mass matrix, in 
both the 1-form and 2-form cases, for a tetrahedral mesh in 3D.

Torquil

On 24/09/10 12:58, Yves Renard wrote:
> The transformation of the basis function from the reference element to the
> real one is made via the definition of the matrix M in the method
> mat_trans(...) (line 1045 of getfem_fem.cc for RTO and line 1230 for Nedelec
> element). The role ofmatrix M is shortly explained here:
>
> http://download.gna.org/getfem/html/homepage/project/femdesc.html#finite-
> element-methods-description
>
> The rule is the following : For RT0 element, the degree of freedom is the
> normal component at the center of a face, for Nedelec one this is the
> tangential component. This means that the normal or tangential component of
> the corresponding shape functions is equal to 1.
>
> Yves.
>
>
>
> On vendredi 24 septembre 2010, Torquil Macdonald Sørensen wrote:
> > Thanks Ronan, that is very interesting. I checked the source file
> > getfem_fem.cc, around lines 1300-1320 where the computation of the Nedelec
> > basis functions for the reference element is done, but could find no such
> > division by an edge length.
> >
> > At that point in the source code I did not find any code that describes how
> > this is converted to the actual tetrahedron element (not the reference),
> > but I'm sure I'll find it somewhere :-)
> >
> > I'll also do some tests by multiplying the GetFEM edge basis mass matrix
> > elements by the corresponding edge lengths to see if I get the same results
> > as I get when manually calculating mass matrix elements from the
> > definitions of the Whitney 1-form basis that I mentioned in the first
> > email.
> >
> > Thanks again for that very useful info!
> >
> > Torquil
> >
> > On 23/09/10 20:23, Ronan Perrussel wrote:
> > > Hi Torquil,
> > >
> > > I have an element which is not an answer to your issue but it can help.
> > > I never used the RT0 elements but I used the Nedelec elements and the
> > > shape function associated to edge {ij} is :
> > > 1/length({ij})*(lambda_i grad lambda_j - lambda_j grad lambda_i).
> > >
> > > Maybe something similar is done for RT0 elements (for instance a dof
> > > associated to a face is not the flux through this face but the mean
> > > value of the normal component of the vector field on this face).
> > >
> > > Best regards,
> > > Ronan
> > >
> > > Le 23/09/2010 20:01, Torquil Macdonald Sørensen a écrit :
> > > > Hi!
> > > >
> > > > I'm having trouble with calculation of mass matrix elements on a 3d
> > > > tetrahedron
> > > > mesh. I have two different cases. In the first case I want to use
> > > > Whitney 1-form
> > > > edge elements, and in the second case Whitney 2-form face elements.
> > > > I'm given to
> > > > understand that these are the same as the NEDELEC and RT0 elements
> > > > offered by
> > > > GetFEM.
> > > >
> > > > The Whitney elements are defined as follows: For a point i, lambda_i
> > > > is the
> > > > usual affine function that is equal to 1 at i, and equal to zero at
> > > > neighbouring
> > > > points.
> > > >
> > > > For an edge {ij} going from point i to point j, the Whitney 1-form
> > > > element is
> > > >
> > > > lambda_ij := lambda_i grad lambda_j - lambda_j grad lambda_i
> > > >
> > > > The "2-form" element for a face {ijk} is
> > > >
> > > > lambda_{ijk} := 0.5*(lambda_i (grad lambda_j) x (grad lambda_k) + cyclic
> > > > permutations of {ijk}).
> > > >
> > > > When varying the "fineness"/lattice spacing of the mesh for a given
> > > > physical
> > > > domain [0,1]^3, these definitions lead to the following behaviour of
> > > > the mass
> > > > matrix components:
> > > >
> > > > m_ff' ~ 1/h
> > > >
> > > > m_ee' ~ h
> > > >
> > > > where m_ff' is a component of the mass matrix for the 2-form elements,
> > > > m_ee' is
> > > > the same for the 1-form elements, and h is the lattice spacing.
> > > >
> > > > However, in my case, the components m_ff' decreases when h decreases.
> > > > Does this
> > > > indicate a difference between the definition of the RT0 element
> > > > relative to the
> > > > Whitney 2-form element? If so, is it possible to get one from the
> > > > other by a
> > > > simple multiplication by h?
> > > >
> > > > Thanks
> > > > Torquil
> > > >
> > > >
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