How to understand the Graph Obtained by Nedelec Space

1

Dear Sir
I am not able to understand the Graph. I can explain even better
Here Domain is a unit square with two triangular elements, which means we have 5 edges( 5 unknows at each edge since we are using nedelec element ) and variational form is

K=assemble(inner(curl(u), curl(v))*dx+inner(u,v)*dx)
F=assemble(inner(f,v)*dx)

After solving the above variational form we get, u1, u2,u3,u4, u5 but when I plotted solution u I got only 4 vectors at each node, which I could not understand please help me with that.
full code is

from scipy import constants as S
import numpy as N
from dolfin import *
from mshr import*
import sys
import os

import matplotlib.pyplot as plt
mesh = UnitSquareMesh(1, 1)
V = FunctionSpace(mesh, "Nedelec 1st kind H(curl)", 1)
u = TrialFunction(V)
v = TestFunction(V)
f=Expression(("3-x[1]*x[1]", "3-x[0]*x[0]"), degree=2)
dm = V.dofmap()
#print(vertices(mesh))

def boundary_lr(x, on_boundary):
    return near(x[1],1)  and on_boundary
A_lr = Constant((2,2))
bc = DirichletBC(V, A_lr, boundary_lr)

K=assemble(inner(curl(u), curl(v))*dx+inner(u,v)*dx)
F=assemble(inner(f,v)*dx)

bc.apply(K, F)
u=Function(V)
solve(K, u.vector(), F)
print(u.vector()[:])

plot(mesh)
c=plot(u)
plt.colorbar(c)
plt.show()

And solution is
u=[2. 2. 2.29772978 3.61269561 2. 2.
3.63632356 1.97884064]

but plot is
Graph

My doubt is In graph we have only vectorial representation at node.

2

You need to use a custom plotter to plot Nedelec elements, as the dolfin.plot function interpolates any function to the mesh vertices. If you want to use an external plotter, I suggest you project/interpolate the solution to a suitable DG space, and use XDMFFile.write_checkpoint (see for instance Loading xdmf data back in - #4 by dokken) such that you can visualize it in Paraview.

3

Thank you sir,

I have tried what you said in above,

from scipy import constants as S
import numpy as N
from dolfin import *
from mshr import*
import sys
import os

import matplotlib.pyplot as plt
mesh = UnitSquareMesh(10, 10)
V = FunctionSpace(mesh, "Nedelec 1st kind H(curl)", 1)
#V = VectorFunctionSpace(mesh, "CG", 1)
u = TrialFunction(V)
v = TestFunction(V)
f=Expression(("3-x[1]*x[1]", "3-x[0]*x[0]"), degree=2)
dm = V.dofmap()
K=assemble(inner(curl(u), curl(v))*dx+inner(u,v)*dx)
F=assemble(inner(f,v)*dx)
u=Function(V)
solve(K, u.vector(), F)
print(u.vector()[:])

ufile = XDMFFile("u.xdmf")
ufile.write_checkpoint(u, "u_out", 0)

But in paraview its showing empty surface.

4

That is because you haven’t interpolated the solution into an appropriate DG space:
Replacing

with:

V_out = VectorFunctionSpace(mesh, "DG", 2)

ufile = XDMFFile("u.xdmf")
ufile.write_checkpoint(interpolate(u, V_out), "u_out", 0)

you get a visualization.