ANALYSIS OF VELOCITY-FLUX FIRST-ORDER SYSTEM LEAST-SQUARES PRINCIPLESFOR THE NAVIER-STOKES EQUATIONS - PART I

This paper develops a least-squares approach to the solution of the in compressible Navier-Stokes equations in primitive variables. As with o ur earlier work on Stokes equations, we recast the Navier-Stokes equat ions as a first-order system by introducing a velocity-flux variable a nd associated curl and trace equations. We show that a least-squares p rinciple based on L-2 norms applied to this system yields optimal disc retization 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 sho wn to yield similar estimates, but now in the L-2 norm for velocity-fl ux and pressure. Although the H-1 least-squares principle does not all ow practical implementation, these results are critical to the analysi s 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 optim al multigrid convergence estimates for the algebraic system resulting from the L-2 norm approach.