How to set up finite element space for 1d periodic mesh correctly?

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1

Hallo!

I’m trying to define a finite element space on 1d periodic mesh with the following simple codes:

########################
from ngsolve import *
from ngsolve.meshes import Make1DMesh

n = 3
mymesh = Make1DMesh(n, periodic = True)

fes = Periodic(H1(mymesh,order=1))
u,v = fes.TnT()

a = BilinearForm(fes)
a += uvds

a.Assemble()
print(a.mat)
########################

When I ran these codes, it said “used dof inconsistency
(silence this warning by setting BilinearForm(…check_unused=False) )”
with “print(a.mat)” to be:

Row 0: 0: 2 1: 0 2: 0
Row 1: 0: 0 1: 0 2: 0
Row 2: 0: 0 1: 0 2: 0
Row 3:

this mass matrix is definitely not correct for periodic setting.

Does anyone know what’s wrong in the codes? Thank you very much!

767×506 36.8 KB

2

Hi,

with
uvds
you are integrating only over the boundary of the mesh. To obtain the mass matrix you have to use dx instead of ds.

Best,
Michael

3

Thank you Michael! I’m new to NGsolve. May I ask another simple question?

After I changed ds to dx, “print(fes.ndof)” is 4 instead of 3 (actual nodes number). Moreover if I set the gridfunction below

gfu = GridFunction(fes)
gfu.Set(cos(2pix))

the first entry does not coincide with the 4th entry. So it seems to break the periodic condition. Do you have any idea?

4

Hi,

the behavior is correct. If you plot fes.FreeDofs() you observe that the last dof is “not-free”, it depends on the first one due to periodicity. If you draw the function you get a periodic function

from ngsolve.internal import visoptions
from ngsolve.internal import viewoptions
visoptions.showsurfacesolution=True
viewoptions.drawedges=1
visoptions.deformation=1
Draw(gfu, mymesh, “gfu”)

With fes = Compress(Periodic(H1(mymesh,order=1))) the last dof is compressed out and you have only 3 dofs left.

Best,
Michael