Ching L. Chang ¹, Suh-Yuh Yang ²

In this paper we analyze several first-order systems of Oseen-type equations that are obtained from the time-dependent incompressible Navier-Stokes equations after introducing the additional vorticity and possibly total pressure variables, time-discretizing the time derivative and linearizing the non-linear terms. We apply the [L-2, L-2, L-2] least-squares finite element scheme to approximate the solutions of these Oseen-type equations assuming homogeneous velocity boundary conditions. All of the associated least-squares energy functionals are defined to be the sum of squared L-2 norms of the residual equations over an appropriate products space. We first prove that the homogeneous least-squares functionals are coercive in the H-1 x L-2 x L-2 norm for the velocity, vorticity, and pressure, but only continuous in the H-1 x H-1 x H-1 norm for these variables. although equivalence between the homogeneous least-squares functionals and one of the above two product norms is not achieved, by using these a priori estimates and additional finite element analysis we are nevertheless able to prove that the least-squares method produces an optimal rate of convergence in the H-1 norm for velocity and suboptimal rate of convergence in the L-2 norm for vorticity and pressure. numerical experiments with various Reynolds numbers that support the theoretical error estimates are presented. In addition, numerical solutions to the time-dependent incompressible Navier-Stokes problem are given to demonstrate the accuracy of the semi-discrete [L-2, L-2, L-2] least-squares finite element approach.