As indicated in the documentation
<http://dealii.org/developer/doxygen/deal.II/classFE__DGP.html>, the finite
elements FE_DGP and FE_FaceP are not based on nodal interpolation (as eg.
FE_DGQ and FE_FaceQ), but on projection. Apparently, this prevents these
elements from being used in combination with a hp::DoFHandler, since there
seems to be no easy way (or even no way?) to implement the functions
hp_*_identities for FE_FaceP (which is needed to support hp) when no nodal
interpolation (ie. no unit_support_points) is available.
I would really like to use the FE_DGP/FE_FaceP combination in a hp FEM
computation, as this requires fewer DoFs than the FE_DGQ/FE_FaceQ couple.
Does anyone have an idea how to fix this problem? I imagine that either:
- one could implement hp_*_identities, somehow, based on the existing code
with projection, or
- one could implement a different version of FE_FaceP with nodal interpolation
but i don't really know whether these approaches could work and where to start.
The interpolation property is not actually necessary. What is necessary for
using things in the hp framework is that the space on one side is *larger*
than the one on the other side, and consequently that you can constrain one to
the other by a set of linear constraints. That is what
make_hanging_node_constraints() does.
The hp_*_identities actually do a subset of that, namely identify degrees of
freedom on one side with those on the other side. For interpolation, that is
simple. But it doesn't have to be. Think of shape functions
1, x, x^2
along an edge (-1,1), for example. These are moments, not interpolatory. Now
on a neighboring cell, you may have
1, x, x^2, x^3
and for continuity, you'll have to have that on that cell, the coefficient for
the last shape function is zero and the coefficients for the first three equal
the same coefficients for the three shape functions on the first of the two
cells. This identity is what the hp_*_identities() functions record. In other
words, for that face, you should get the identities
(0,0) // DoF 0 on one side equals DoF 0 on the other side
(1,1)
(2,2)
and later on, make_hanging_node_constraints() would set U_3 on the second
cell's face to zero.
Does that make sense?
Best
W.
--
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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