Daniel,
> I am trying to combine the previous e-mails into one. I have attached
> 3 slides from a slideshow for the MHD equations and the linear form.
> The Nedelec element used is for the current J.
If I see this right on page 3, then the only derivatives on the variable J is
the divergence operator. For the divergence operator to make sense, you need
continuity of the *normal* component of a finite element across element faces
(something the Raviart-Thomas element provides but the Nedelec element does
not) while you don't care about continuity of the *tangential* component
(something the Nedelec element provides but not the RT element).
This is also reflected in the fact that you impose boundary conditions for n.J
(which again requires continuity of the normal component), not for n \times J
(which would require continuity of the tangential component). What motivates
your choice of elements?
> From reading through Anna Schneebeli's paper, I understand that
> the shape functions don't really have support points. Thus
> setting boundary conditions for Nedelec elements would
> be different from settting the boundary conditions for Lagrangian elements.
They do have a sort of support points, but these support points are for vector
components tangential to the face these points lie on. So they're no help for
what you want here.
> I was wondering how boundary conditions could be set for a variable
> associated with a Nedelec element. In the other e-mail you mentioned
> Baerbel Janssen working with Hermite-type elements. I was thinking
> about a simple linear Hermite element on an interval, where the
> degrees of freedom are defined using the value and
> the derivative at the midpoint. How would someone set the boundary
> conditions for this element, if it has only the midpoint as a "support"
> point.
That's difficult. People use Hermite elements for the biharmonic equation for
which you need boundary conditions for both the value and the derivative, and
that's exactly the degrees of freedom the element defines.
> Finally, letting x = "generalized support point", I was wondering if it
> would make any sense to put in a constraint matrix
> a form of "J(x) . n(x) = sum_{k=1} J_{k} phi_{k}(x) . n(x) =
> boundary_value".
> Where k runs through the Nedelec shape functions phi_{k}, and J_{k}
> are the solution coefficients. Thanks for everyone's time.
Let's first figure out whether the Nedelec element is really what you want :-)
Cheers
W.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.tamu.edu/~bangerth/
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