Bilinear Form to find the inner product of a scalar trial function and a vector field test function?

[TheBEllis]

Hi MFEM team,
I was wondering if there exists a bilinear form integrator whose operator looks like so,
$(u, \nabla \cdot \vec{v})$.
Where the function $v$ is from the test space of a vector field which is just multiple copies of a scalar field. I don't think this BLF integrator currently exists within MFEM, but I might be looking in the wrong place. Any help is very appreciated:)

[mlstowell]

Hi, @TheBEllis,

I believe we have two different bilinear form integrators which are equivalent to this although neither is written in quite that manner. The first is called GradientIntegrator and it implements $(\lambda\nabla u, \vec{v})\approx-\left(u, \nabla\cdot(\lambda \vec{v})\right)$. The second is called VectorDivergenceIntegrator and it implements $(\lambda\nabla\cdot\vec{u},v)$ which is equivalent to the transpose of $(\lambda u,\nabla\cdot\vec{v})$. One main difference is the location of the optional $\lambda$ coefficient. If you have a non-constant coefficient then you must be careful to pick the correct integrator. If your coefficient is spatially constant either of these will do but the VectorDivergenceIntegrator may be the better choice. This is because the GradientIntegrator involves an integration-by-parts which gives rise to a boundary integral. Such boundary integrals can be used to enforce Neumann boundary conditions but with a rectangular operator such as this its meaning is less clear.

Best wishes,
Mark

[TheBEllis]

Hi Mark,
Thanks for a brilliant run through on the options available, I will take a look and see if I can get something working:)
Kind regards,
Bill