Analysis of Velocity-Flux First-Order System Least-Squares Principles for the Navier--Stokes Equations: Part I

This paper develops a least-squares approach to the solution of the incompressible Navier--Stokes equations in primitive variables. As with our earlier work on Stokes equations, we recast the Navier--Stokes equations as a first-order system by introducing a velocity-flux variable and associated curl and trace equations. We show that a least-squares principle based on L 2 norms applied to this system yields optimal discretization error estimates in the H 1 norm in each variable, including the velocity flux. An analogous principle based on the use of an H -1 norm for the reduced system (with no curl or trace constraints) is shown to yield similar estimates, but now in the L 2 norm for velocity-flux and pressure. Although the H -1 least-squares principle does not allow practical implementation, these results are critical to the analysis of a practical least-squares method for the reduced system based on a discrete equivalent of the negative norm. A practical method of this type is the subject of a companion paper. Finally, we establish optimal multigrid convergence estimates for the algebraic system resulting from the L 2 norm approach.