[dune-pdelab] dim < dimWorld ?

[Dan Popovic]

Dear Bernd, 

I have a 1D-treelike grid embedded in 3D space, such
that dim = 1 and dimW = 3, and stumbled across the same
problem. The first problem was actually how to realize the
inverse ('jacobianInverseTransposed') of a non-quadratic
matrix. The solution was to implement the pseudo-inverse

http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse

of the (dimw x dim)-Jacobian. In my case (1D embedded in 3D)
this was fairly easy though I do not remember the details
any more (I use PDELab very seldom...) and did not look at the
general case dim > 1. Still I think the starting point sould be
the pseudo-inverse.

Hope that helps a little bit. Kind regards,
Dan



Am Mittwoch, den 09.05.2012, 17:53 +0200 schrieb Bernd Flemisch:
> Dear DUNE-PDELab,
> 
> some people here including me are wondering what are your current 
> thoughts on performing calculations with dim < dimWorld? Has someone 
> already used PDELab for this or plans to do so?
> 
> Looking, e.g., at localoperator/convectiondiffusion.hh, line 310ff, I find
> 
> const Dune::FieldMatrix<DF,dimw,dim> jac = 
> eg.geometry().jacobianInverseTransposed(it->position());
> std::vector<Dune::FieldVector<RF,dim> > gradphi(lfsu.size());
> for (size_type i=0; i<lfsu.size(); i++)
> {
>    gradphi[i] = 0.0;
>    jac.umv(js[i][0],gradphi[i]);
> }
> 
> This will not work for dimw != dim, since jac is dimWorld x dim, and 
> should map the dim-vector js[i][0] to a dimWorld-vector. However gradphi 
> is a dim-vector.
> 
> One could fix this by declaring gradphi as a dimWorld-vector, but I 
> don't think that this is what we want to do. We want to solve an 
> equation like
> div_s D_s grad_s u_s = q_s
> where all _s are surface quantities. The vectors and tensors have a 
> dimWorld coordinate representation, but I think that one would want to 
> use the dim representations.
> 
> So either jacobianInverseTransposed should map to a dim-coordinate 
> vector, or one should have a method for projecting 
> dimWorld-representations to dim-representations. What do you think?
> 
> Thank you for your thoughts. Kind regards
> Bernd
>