Discontinuous Galerkin (DG) methods have proved to be well suited
for the construction of robust high-order numerical schemes on unstructured
and possibly non conforming grids for a variety of problems.
Their application to the incompressible Navier-Stokes (INS) equations
has also been recently considered, although the subject is far from being
fully explored. In this work we propose a new approach for the DG
numerical solution of the INS equations written in conservation form.
The inviscid numerical  fluxes both in the continuity and in the momentum
equation are computed using the values of velocity and pressure
provided by the (exact) solution of the Riemann problem associated
with a local artificial compressibility perturbation of the equations.
Unlike in most of the existing methods, artificial compressibility is here
introduced only at the interface  flux level, therefore resulting in a 
consistent
discretization of the INS equations irrespectively of the amount
of artificial compressibility introduced. The discretization of the viscous
term follows the well established DG scheme named BRMPS. The
performance and the accuracy of the method are demonstrated by computing
the Kovasznay  flow and the two-dimensional lid-driven cavity
 flow for a wide range of Reynolds numbers and for various degrees of
polynomial approximation. In addition, preliminary computations on
unsteady test problems are presented for both the INS and the incompressible
Euler equations. Finally, some theoretical issues regarding
the method applied to linearized problems are discussed.