Hi all and Wolfgang, I did some digging in the archives and found a response from Anna Schneebeli to almost the same question:
On Sat, 18 Oct 2003, Anna Schneebeli wrote: > > In fact, for the usual variational problems where H(curl)-conforming Nedelec > elements (those currently implemented in dealii) may be used for > FE-discretization, it does not make sense to impose a boundary condition on > the > normal component of the field - the normal trace of an H(curl)-function is not > well defined (in a suitable space of boundary functions), e.g. it can jump > across a boundary. > An appropriate Dirichlet boundary condition would be n x u = 0 , a Neumann > condition n x curl u = 0. > Anna. She continues to explain how to impose the tangential boundary conditions > As for imposing boundary conditions, the > VectorTools::interpolate_boundary_function > cannot (yet) be used for Nedelec elements. > The way I currenlty impose non-homogeneous Dirichlet bc in my Nedelec codes > is: > loop over all boundary edges e and approximate the value of the current dof > (which > is the line integral over e of some scalar function g, g being the bc for the > tangential component of u on e) by a (line) quadrature (actually, the > midpoint rule > is accurate enough). > Have a look at the attached patch for details ! > (Also included is a possible assembling routine for the model problem curl > curl u + > u = f in a 2D domain) > Hope it helps! > Anna This was helpful. Thanks to everyone and the archive. The nedelec element is in the polynomial space that I need, is discontinuous in the normal direction and continuous in the tangential directions. These are the conditions that I need to satisfy the inf-sup condition. That was the reason it was chosen. The Nedelec dofs seem to prohibit the boundary conditions from being enforced. I will place my follow-up question in a different e-mail. Thanks for your time. Dan On Thu, Jun 23, 2011 at 7:53 PM, Wolfgang Bangerth <[email protected]>wrote: > > 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. > > ------------------------------------------------------------------------- > Wolfgang Bangerth email: [email protected] > www: http://www.math.tamu.edu/~bangerth/ >
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