Project function

1

Hello all,

I want to confirm a quick question, in FEniCS, the project function is project the original function to the finite element space using the degree of freedom of that finite element space right?

Thanks

2

Yes. In particular, it is L^2 projection onto the finite element space. See the following code example:

from dolfin import *

# L^2 projection onto the space V:
def my_project(u,V):
    ut = TrialFunction(V)
    v = TestFunction(V)
    uh = Function(V)
    solve(inner(ut,v)*dx==inner(u,v)*dx,uh)
    return uh

# Compare with project function:
N = 10
mesh = UnitSquareMesh(N,N)
V = FunctionSpace(mesh,"CG",1)
x = SpatialCoordinate(mesh)
u = sin(x[0]*pi)*sin(2*x[1]*pi)
print(assemble(abs(my_project(u,V)
                   - project(u,V))*dx))
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3

Thank you, kamensky! But for example if you are using BDM or RT element, I guess the projection is not L^{2} projection right? It should make use of the degree of freedom of BDM or RT right?

Thanks

4

L^2 projection still makes sense for RT/BDM elements, as the corresponding spaces are subsets of L^2. You can verify this by modifying the last part of my example above:

V = FunctionSpace(mesh,"RT",1)
x = SpatialCoordinate(mesh)
u = as_vector(2*(sin(x[0]*pi)*sin(2*x[1]*pi),))
e = my_project(u,V) - project(u,V)
print(assemble(dot(e,e)*dx))
5

Hello Kamensky, thank you very much for the explanation in detail! For RT/BDM elements, are you aware of any projection using the degree of freedom of RT/BDM instead of the L^{2} projection?

Thanks

6

I’m not sure I understand what you’re asking. The result of L^2 projection \Pi^h: L^2\to V^h onto some finite-dimensional space V^h\subset L^2 (be it a CG space, an RT space, or some other subspace of L^2) will return a function in V^h, which will be defined by some vector of degrees of freedom for V^h. In that sense it is “using the degrees of freedom” of V^h, regardless of what type of space V^h is.

1 Like
7

Hello kamensky, thank you so much for all of your kind help and sorry for my late reply, what I am asking is actually that, you know for RT/BDM elements, they have their own degree of freedom, so if you have a H^{1} function, the degree of freedom from RT/BDM could be used to determine uniquely a function say in V_{h}. Yeah, I agree the L^{2} projection still makes sense in the RT/BDM spaces, but what I wonder is how to use the degree of freedom say of BDM/RT themselves to do the projection.