I formulated a physical problem into a vector poisson equation like $$ -\nabla^2\vec{A} = \vec{J}$$ and now want to derive the weak form of it. My first approach was to rewrite the Laplacian operator just like I would for scalar fields, as i think all those operations are also valid for vector fields (partially resulting in tensor fields): $$-\nabla\cdot\left(\nabla \vec{A} \right)=J$$ First question: Is this a valid formulation?
The normal way forward is to multiply with test function vector field $\vec{v}$ and integrate it over the domain:
$$\int_\Omega -\nabla\cdot\left(\nabla \vec{A} \right)\cdot \vec{v} \:d \Omega=\int_\Omega J\cdot \vec{v} \:d \Omega$$
But than I struggle with the integration by parts formulation including tensor fields. I came up with $$\nabla \cdot \left(\nabla\vec{A}\cdot\vec{v}\right) = \left(\nabla\cdot\left(\nabla \vec{A} \right)\right)\cdot\vec{v} \:+ \left(\nabla \vec{A} \right)\cdot\left(\nabla\cdot\vec{v}\right)$$ Is the left hand side of this term valid, a mjultiplication of tensorfield and vector field? Also on the last term in the equation, there is a multiplication of tensorfield and a scalar field, is that valid?
And if they are valid, how would I go on to the weak form with the substituted result?
$$\int_\Omega \nabla \cdot \left(\nabla\vec{A}\cdot\vec{v}\right) \:d \Omega - \int_\Omega \left(\nabla \vec{A} \right)\cdot\left(\nabla\cdot\vec{v}\right) \:d \Omega=\int_\Omega J\cdot \vec{v} \:d \Omega$$
I can't imagine how to apply the divergence theorem on this function, which normally is the next step.