H(curl), extract the component of u, u is in H(curl)

Question

If u is in H(curl) in 3D, u is a 3D vector?
If so, how to extract the 3 components of u?
I tried the below method:

u.sub(o), u.sub(1), u.sub(2).

But it's not right.

I will appreciate your any help. Thanks!!!

Answer 1

Hi,

I know this problem (u.sub() does not work in Nedlec Space) very well. It results from the definition of the N1curl elements, you have only 6 DOFS per tetrahedral element which belong to magnitudes on the edges, the direction is stored implicitly. My good working workaround is just interpolating on a Lagrange-Space:

mesh = UnitCubeMesh(10, 10, 10)
V_ned = FunctionSpace(mesh, 'N1curl', 1)
V_cg = VectorFunctionSpace(mesh, 'CG', 1)

... solve e.g. "E_ned" on "V_ned" and interpolate afterwards

E_cg = interpolate(E_ned, V_cg)

Now you can use u.sub() to extract single components

E_x = E_cg.sub(0)
E_y = E_cg.sub(1) 
E_Z = E_cg_sub(2)

Hope it is still helpful. If you have further questions about Curl-problems, just write me a PM

Comments

Thank you very much! This idea is very good, but it cannot solve my problem, I think.
Actually, my question is how to implement the boundary condition of Maxwell equation, like
n x u = 0, i.e., curl u = 0, in 3D space.
In others example, like finite element function space V = H(curl, 0), somebody implements boundary condition by bc = DirichletBC(V, Constant(0,0,0), boundary). Is this right?

Answer 2

Hi, consider

from dolfin import *
import sys

n = int(sys.argv[1])
mesh = UnitSquareMesh(n, n)
# V = VectorFunctionSpace(mesh, 'CG', 1)
V = FunctionSpace(mesh, 'N1curl', 2)

f = Expression(('x[1]', '-x[0]'), degree=1)
f_e0 = Expression(('x[1]', '0'), degree=1)

u = interpolate(f, V)
e0 = interpolate(Constant((1, 0)), V)

u_e0 = (dot(u, e0)/dot(e0, e0))*e0
u_e0 = project(u_e0, V)

print errornorm(f_e0, u_e0, 'L2') 

Comments

Thanks for your help.

I don't understand your code.

How do I get the three components of u, if u is in H(curl) in 3D space?

Answer 3

How about just using u[0], u[1], u[2]? For instance:

from dolfin import *
mesh = UnitCubeMesh(2, 2, 2)
V = FunctionSpace(mesh, 'N1curl', 2)
v = Function(V)
print v[0]