Nodal interpolant available?

Hey there,

The symbolic definition of (bi-) linearforms makes a lot of things easier.
But, I wonder whether one can also define a form including the nodal interpolant?

More specific: Consider the space of first order vector-valued H1 conforming elements, i.e., X = VectorH1(mesh, order=1).
The nodal interpolant I_h should now map a given (continuous) function f to a function f_h in the FE-space X, such that f(z) = f_h(z) for all nodes z in the mesh.

For two GridFunctions u and v in X, i can get the nodal interpolant of the cross-product denoted by I_h(u x v) quite easily (for example) by accessing the vectors of their nodal values.

But I struggle to use the nodal interpolant I_h in combination with test functions:
For example, is there a way to define a bilinear form like

<psi , curl( I_h(phi x v))>

in NgSolve?
Here again v denotes a GridFunction, while psi and phi denote a TrialFunction and a TestFunction, respectively. <.,.> denotes the L2 inner prduct.

Thanks in advance,
Best regards,
Carl

just a quick idea: can you introduce a new variable w = l_h(phi x v) in the system and use this for a second equation for w?
ngsolve.org/docu/latest/i-tutorials/unit...basis/dualbasis.html

Thank you Christopher for the suggestion - I will have a look into that.
Best, Carl

Hey there,

introducing a new variable as suggested by Christopher did the trick.
However, I used a mass-lumped integration rule rather than the dual basis you suggested.

For future reference I quickly describe the solution which worked for me,
in case somebody else runs into a similar problem.

Notation:
a x b denotes the cross-product of a and b.
<., .> denotes the L2-inner product;
<., .>_h denotes the mass-lumped inner-product (defined in the code below)

For given GridFunction eta in X, consider the problem of finding a GridFunction u (in X), such that

<u, phi> + <u, curl(I_h(phi x eta))> = <f, phi>

for all TestFunctions phi in X.

Introducing a new variable v = I_h(phi x eta) one arrives at a system on XX = FESpace([X, X]) :
Find (u, w) in XX, such that

<u, phi> - <eta x w, phi>_h = <f, phi>

<u, curl(v)> + <w, v>_h = 0

for all TestFunctions (phi, v) in XX.

Plugging in v = I_h(phi x eta), one sees that the original equation is satisfied.

The NGS-Python code snippet for defining the bilinear form looks like this


The resulting (larger) system, however, takes pretty long to be solved.
So make sure to additionally decouple the DOFs of the Lagrange multiplier w
by using XX = FESpace([X, Discontinuous(X)]) instead,
and static condensation as described at:
ngsolve.org/docu/latest/i-tutorials/unit...cond/staticcond.html

Thank you Joachim for your further help,
Best regards,
Carl