Discontinuous Galerkin method for the Oseen problem with mixed boundary conditions: a priori and aposteriori error analyses

We introduce and analyze a discontinuous Galerkin method for the Oseen equations in two dimension spaces. The boundary conditions are mixed and they are assumed to be of three different types:
the vorticity  and the normal component of the velocity are given on a first part of the boundary, the pressure and the tangential component of the velocity are given on a second part of the boundary and the Dirichlet condition is given on the remainder part . We establish a priori error estimates in the energy norm for the velocity and in the L2 norm for the pressure. An a posteriori error estimate is also carried out yielding optimal convergence rate. The analysis is based on rewriting the method in a non-consistent manner using lifting operators in the spirit of Arnold, Brezzi, Cockburn and Marini.