dolfin team mailing list archive

[Marie Rognes <meg@xxxxxxxxxxx>]

Evan Lezar wrote:
> On Tue, Aug 19, 2008 at 5:04 PM, Marie Rognes <meg@xxxxxxxxxxx 
> <mailto:meg@xxxxxxxxxxx>> wrote:
>
>     Evan Lezar wrote:
>
>
>
>         I do have one quick question regarding the extension to the
>         Nedelec basis functions of higher degree.  Now as soon as I
>         increase the degree of the finite elements (to say 1), then
>         the number of basis functions increases, which is expected.
>          However, I am not 100% sure why the dimension of the space is
>         8 for a degree of 1 - is there some documentation regarding
>         the basis functions.
>
>
>     Are you in 2 or 3D?
>
>
> I am working in 2D.  I just realized what the issue is.  The Nedelec 
> basis functions that are implemented by FFC are of the first kind and 
> thus have a reduced order gradient subspace - this is what it says in 
> the FFC manual at least.

Correct.  (The second kind Nedelecs are not implemented, yet.)

> Also, the degree specified is the degree of the gradient subspace with 
> the rotational subspace being one degree higher - and thus explaining 
> the 8 basis functions (6 edge-based and 2 face based) over a triangle.

The numbering for the Raviart-Thomas and the first kind Nedelecs can be 
a bit confusing. However, I think the idea is that the numbering 
reflects the approximation properties. The k'th degree normally gives 
k+1'th order convergence.


> This does not resolve the question regarding the heirarchical nature 
> of the basis functions though.
>  
>
>
>         I have done some work in the past on sets of higher degree
>         basis functons (mostly those of Webb (1999)), and as far as I
>         can remember there should be 6 basisfunctions of degree 1.
>
>
>     In 3D, there are 6 basis functions in the lowest order case. We
>     start the numbering for the Nedelecs at 0.
>
>     In 2D, there are 3 basis functions in the lowest order case
>     (degree = 0) and as far as I remember 8 for the next.
>
>
>         Then, the Nedelec set of basis functions are heirarchical and
>         thus a set of a given degree should include the basis
>         functions of a lower degree - the reason that I bring this up
>         is that if I simply increase the degree of the finite element
>         in the code sample that you supplied (as well as adjust the
>         dimension of the space accordingly) then the basis functions
>         plotted for degree 0 are not present in the 8 basis functions
>         that are plotted.

The Nedelec bases constructed by FIAT corresponds to the original ones 
suggested by Nedelec and are not hierarchical.


>          In addition, the normal component along an edge seems to be
>         quadratic.  I would expect at most a linear variation in the
>         normal component. 


You mean tangential component here, right?


--
Marie E. Rognes
Ph.D Fellow, 
Centre of Mathematics for Applications, 
University of Oslo
http://folk.uio.no/meg