Local and global facet normal for RT space


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11 weeks ago by
Cuiyu  
Hi, I want to understand how exactly the RT0 dofs are computed. Based on my understanding, the dof values are computed as the inner product of the value of the RT vector function (here is numer_flux) and  the global facet normal (here I assume it is facet.normal()). But obviously there is some misunderstanding since the following verification code does not always returns 0.  Any help is appropriated! Thank you!

Below is my code

from fenics import *

mesh = RectangleMesh(Point(-1,-1), Point(1,1), 2,2, 'right')
mesh.init()
W = FunctionSpace(mesh, "RT", 1)
f = Expression(('1','1'),degree=0)
numer_flux = project(f, W)
dofmap = W.dofmap()

for cell in cells(mesh):
for i, global_facet_index in enumerate(cell.entities(1)):
facet = Facet(mesh,global_facet_index)
f_normal = facet.normal().array()
local_facet_dof_index = dofmap.tabulate_facet_dofs(i)
global_facet_dof_index = dofmap.cell_dofs(cell.index())[local_facet_dof_index]
sigma_F = numer_flux(cell.midpoint())[0]*f_normal[0]
sigma_F += numer_flux(cell.midpoint())[1]*f_normal[1]
error = numer_flux.vector().get_local()[global_facet_dof_index] - sigma_F
print("facet_index=", global_facet_index, "error=",error)
Community: FEniCS Project

1 Answer


3
11 weeks ago by
Miguel  
I believe facet.normal() is not the global normal but the local normal. You cannot compare the element quantities calculated with the element local normal vectors (sigma_F) with the assembled vector components (numer_flux). The normal vectors shared by two elements must have the same orientation, and therefore, they do not coincide with the local element normal vectors (which are all pointing outside of the element). That explains the sign difference you are seeing. More info about normal orientation at https://arxiv.org/pdf/1205.3085.pdf
Thank you very much.  The resource is very helpful. The global normal is determined by the global indexes of vertices on the facet.
written 11 weeks ago by Cuiyu  
You are welcome. By the way, are you Cuiyu He who interned at the LLNL in 2016?
written 11 weeks ago by Miguel  
Yes, I am. The MFEM library developed by Tzanio's group in LLNL is also very powerful especially in quad-meshes. :)
written 11 weeks ago by Cuiyu  
Nice, I am Miguel Salazar, we met over that summer. Glad to see you here!
written 11 weeks ago by Miguel  
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