L2 space nodal values

General
1

Hi,

I was wondering if there was a way to switch between nodal values and moments regarding the degrees of freedom of an L2 space. I tried to follow what is written in Dual basis functions without success.

As of now if I run

from ngsolve import *
import matplotlib.pyplot as plt

mesh = Mesh(unit_square.GenerateMesh(maxh = 0.1))
fes = L2(mesh, order = 1)
gfu = GridFunction(fes)
gfu.Set(1)

then gfu.vec.data has zeros and ones as it should be. How can I get the values at the corresponding Lagrange nodes, so to have all ones for order=1?

Thanks a lot for your help

2

You can use

fes = Discontinuous(H1(mesh, order=1))
3

Oh I see, thanks a lot that is exactly what I was looking for! I now wrote a little script to pass from one version to the other:

from ngsolve import *
from ngsolve.webgui import Draw
from netgen.geom2d import unit_square

mesh = Mesh(unit_square.GenerateMesh(maxh=0.2))

fesl2 = L2(mesh, order=1) 
fesdh1 = Discontinuous(H1(mesh, order=1, dgjumps = True)) 

gful2 = GridFunction(fesl2)
gful2.Set(1)
gfudh1 = GridFunction(fesdh1)

def transform_funct(fesl2, fesdh1, gful2, gfudh1):
    amixed = BilinearForm(trialspace=fesl2, testspace=fesdh1)
    adh1 = BilinearForm(fesdh1)

    u,v = fesdh1.TnT()
    vdual = v.Operator('dual')
    uh1 = fesl2.TrialFunction()

    dS = dx(element_boundary=True)
    adh1 += u*vdual * dx + u*vdual * dS
    adh1.Assemble()

    amixed += uh1*vdual*dx + uh1*vdual*dS
    amixed.Assemble()

    # build transformation operator:
    transform = adh1.mat.Inverse() @ amixed.mat
    gfudh1.vec.data = transform * gful2.vec

transform_funct(fesl2, fesdh1, gful2, gfudh1)

print(gfudh1.vec.data)

Draw (gful2)
Draw (gfudh1)

But I have noticed that:

  • This leads to an empty matrix for adh1. It seems like using v instead of vdual does the job
  • If I use L2(mesh, order=1, dgjumps = True) and try to do the inverse conversion then the software just crashes

Any idea on why is it happening?

Thanks,
Alessandro

4

For H1, the dual basis functions contain co-dimension 2 (and codim 3 for 3D) functionals, see the dual basis function tutorial.

There is a builtin functionality for that purpose, with a lot of options:

transform = ConvertOperator(fesL2, fesH1, geom_free=True)
gfH1.vec.data = transform * gfL2.vec
5

I see, thank you very much for the clarification :slight_smile: