# product of linear functionals as BilinearForm

Hello guys,

I have questions: in my finite element formulation I would like to add the following bilinear form,
$$\int_{\Omega}p\ dx *\int_{\Omega}q\ dx$$
where p is the trial function in scalar space Q and q is the test function in scalar space Q.
I didn’t find a way to use
SymbolicBFI()
to formulate this term.
Would you please give a hint about how this term can be implemented?

Best,
Qi

Hi Qi,

a product of linear functionals as bilinearform cannot be achieved directly this way. One reason is that the resulting matrix would be completely dense.
However, you can use a mixed formulation by introducing a new unknown representing one integral

$$\mu = \int_{\Omega}p dx,$$
where the additional unknown is a single scalar.

This can be implemented in NGSolve via

[code]
Q = H1(mesh, order=1)
N = NumberSpace(mesh)
fes = Q*N
(p,mu), (q,lam) = fes.TnT()

a = BilinearForm(fes, symmetric=True)
a += (p - mu)lamdx
a += muqdx[/code]
Then you obtain a sparse matrix, where only one row and column is completely filled.

Best,
Michael

I see. Thank you very much, Micheal.