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 Navier­Stokes equations in primitive variables. As with our earlier work on Stokes equations, we recast the Navier­Stokes 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 L² norms applied to this system yields optimal discretization error estimates in the H¹ 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 L² 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 L² norm approach. [ABSTRACT FROM AUTHOR]