On the velocity-vorticity-pressure least-squares finite element method for the stationary incompressible Oseen problem

In this paper, we study the L² least-squares finite element approximations to the Oseen problem for the stationary incompressible Navier-Stokes equations with the velocity boundary condition. The Oseen problem is first recast into the velocity-vorticity-pressure first-order system formulation by introducing the vorticity variable. We then derive some a priori estimates for the first-order system problem and identify the dependence of the estimates on the Reynolds number. Such a priori estimates play the crucial roles in the error analysis for least-squares approximations to the incompressible velocity-vorticity-pressure Oseen problem. It is proved that, with respect to the order of approximation for smooth exact solutions, the L² least-squares method exhibits an optimal rate of convergence in the H¹ norm for velocity and a suboptimal rate of convergence in the L² norm for vorticity and pressure. Numerical results that confirm this analysis are given. Furthermore, in order to maintain the coercivity and continuity of the homogeneous least-squares functional that are destroyed by large Reynolds numbers, a weighted least-squares energy functional is proposed and analyzed. Numerical experiments in two dimensions are presented, which demonstrate the effectiveness of the weighted least-squares approach. Finally, approximate solutions of the incompressible velocity-vorticity-pressure Navier-Stokes problem with various Reynolds numbers are also given by solving a sequence of Oseen problems arising from a Picard-type iteration scheme. Numerical evidences show that, except for large Reynolds numbers, the convergence rates of the weighted least-squares approximations for the Navier-Stokes problem are similar to that for the Oseen problem.