  We propose a novel Structure-Preserving Discontinuous Galerkin (SPDG)
operator that recovers at the discrete level the algebraic property related to
the divergence of the curl of a vector field, which is typically referred to as
div-curl problem. A staggered Cartesian grid is adopted in 3D, where the vector
field is naturally defined at the corners of the control volume, while its curl
is evaluated as a cell-centered quantity. Firstly, the curl operator is
rewritten as the divergence of a tensor, hence allowing compatible finite
difference schemes to be devised and to be proven to mimic the algebraic
div-curl property. Successively, a high order DG divergence operator is built
upon integration by parts, so that the structure-preserving finite difference
div-curl operator is exactly retrieved for first order discretizations. We
further demonstrate that the novel SPDG schemes are capable of obtaining a zero
div-curl identity with machine precision from second up to sixth order
accuracy. In a second part, we show the applicability of these SPDG methods by
solving the incompressible Navier-Stokes equations written in vortex-stream
formulation. This hyperbolic system deals with divergence-free involutions
related to the velocity and vorticity field as well as to the stream function,
thus it provides an ideal setting for the validation of the novel schemes. A
compatible discretization of the numerical viscosity is also proposed in order
to maintain the structure-preserving property of the div-curl DG operators even
in the presence of artificial or physical dissipative terms. Finally, to
overcome the time step restriction dictated by the viscous sub-system,
Implicit-Explicit (IMEX) Runge-Kutta time stepping techniques are tailored to
handle the SPDG framework.
