Analysis of [H−1,L2,L2] first-order system least squares for the incompressible Oseen type equations

This paper is devoted to the error analysis of least-squares finite element approximations to the stationary incompressible Oseen type equations with the homogeneous velocity boundary condition. With the vorticity as a new dependent variable, we consider two first-order system problems for the Oseen type equations in the velocity-vorticity-pressure and the velocity-vorticity-Bernoulli pressure formulations. The least-squares functional is defined in terms of the sum of the squared H−1 and L2 norms of the residual equations over a suitable product function space. The well-posedness of the proposed least-squares variational problem is shown. We then analyze the case where the H−1 norm in the least-squares functional is replaced by a discrete functional to make the computation feasible. Optimal error estimates for all unknowns are derived.