Do we still use vector potential in Maxwell's equations?

Question

In Maxwell's equations, since field $\mathbf{B}$ is divergence free, we can find vector potential $\mathbf{A}$ such that $\mathbf{B}=\nabla\times\mathbf{A}$. However, this identity holds only when the domain $\Omega$ is topologically trivial. If there is a 'hole' contained in the domain, then de Rham complex is not exact and the identity may not hold. Most of time, we deal with Maxwell's equations on non-trivial domains, especially in the industrial community. Do we have Maxwell's equation in potential formulation on these domains? Do we still deal with Maxwell's equations in potential formulation instead of field formulation? I also wonder what formulation of Maxwell's equations numerical mathematicians care about? (time-harmonic or evolution equation? field or potential?)