We're having problems with getting higher order Lagrange converge in
3D and I think we've tracked it down to FFC/DOLFIN and FIAT not
agreeing on the numbering of edges on the reference tetrahedron.
Here's the numbering that FIAT seems to use (listed as the incident
vertices of each edge):
e0 = (1, 2)
e1 = (0, 2)
e2 = (0, 1)
e3 = (0, 3)
e4 = (1, 3)
e5 = (2, 3)
The UFC convention wants the edges sorted by their non-incident
vertices so we would like to have
e0 = (2, 3) non-incident to (0, 1)
e1 = (1, 3) non-incident to (0, 2)
e2 = (1, 2) non-incident to (0, 3)
e3 = (0, 3) non-incident to (1, 2)
e4 = (0, 2) non-incident to (1, 3)
e5 = (0, 1) non-incident to (2, 3)
The ordering convention is described in detail in the FFC, DOLFIN and
UFC manuals and it's based on what Arnold, Falk, Winther use for
numbering sub simplices in arbitrary dimensions.
The basic principle is that for each entity, you look at its
*non-incident* vertices and sort the entities lexicographically by
those (sorted) tuples of non-incident vertices. So the first edge
of a tetrahedron would then be the edge that is non-incident to
vertices 0 and 1. This is pretty neat since it also specialized to the
familiar numbering of edges in 2D and faces in 3D, so that edge (or
face) i is the edge (or face) opposite to vertex i.
Would it be possible to change the edge numbering for tets in FIAT to
follow this convention?
(Ideally, I'd also like to have the vertices at (0, 0, 0) - (1, 0, 0)
- (0, 1, 0) - (0, 0, 1) rather than (-1, -1, 0) etc but we already
handle this in FFC by scaling all the basis functions and
derivatives.)
/Anders
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