  In this paper we propose a new spatially high order accurate semi-implicit
discontinuous Galerkin (DG) method for the solution of the two dimensional
incompressible Navier-Stokes equations on staggered unstructured curved meshes.
While the discrete pressure is defined on the primal grid, the discrete
velocity vector field is defined on an edge-based dual grid. The flexibility of
high order DG methods on curved unstructured meshes allows to discretize even
complex physical domains on rather coarse grids.
  Formal substitution of the discrete momentum equation into the discrete
continuity equation yields one sparse block four-diagonal linear equation
system for only one scalar unknown, namely the pressure. The method is
computationally efficient, since the resulting system is not only very sparse
but also symmetric and positive definite for appropriate boundary conditions.
Furthermore, all the volume and surface integrals needed by the scheme
presented in this paper depend only on the geometry and the polynomial degree
of the basis and test functions and can therefore be precomputed and stored in
a preprocessor stage, which leads to savings in terms of computational effort
for the time evolution part. In this way also the extension to a fully curved
isoparametric approach becomes natural and affects only the preprocessing step.
The method is validated for polynomial degrees up to $p=3$ by solving some
typical numerical test problems and comparing the numerical results with
available analytical solutions or other numerical and experimental reference
data.
