Velocity-Correction Projection Methods for Incompressible Flows

Abstract

We introduce and study a new class of projection methods---namely, the velocity-correction methods in standard form and in rotational form---for solving the unsteady incompressible Navier--Stokes equations. We show that the rotational form provides improved error estimates in terms of the H1-norm for the velocity and of the L2-norm for the pressure. We also show that the class of fractional-step methods introduced in [S. A. Orsag, M. Israeli, and M. Deville, J. Sci. Comput., 1 (1986), pp. 75--111] and [K. E. Karniadakis, M. Israeli, and S. A. Orsag, J. Comput. Phys., 97 (1991), pp. 414--443] can be interpreted as the rotational form of our velocity-correction methods. Thus, to the best of our knowledge, our results provide the first rigorous proof of stability and convergence of the methods in those papers. We also emphasize that, contrary to those of the above groups, our formulations are set in the standard L2 setting, and consequently they can be easily implemented by means of any variational approximation techniques, in particular the finite element methods.