Least Squares Preconditioners for Stabilized Discretizations of the Navier–Stokes Equations

This paper introduces two stabilization schemes for the least squares commutator (LSC) preconditioner developed by Elman, Howle, Shadid, Shuttleworth, and Tuminaro [SIAM J. Sci. Comput., 27 (2006), pp. 1651–1668] for the incompressible Navier–Stokes equations. This preconditioning methodology is one of several choices that are effective for Navier–Stokes equations, and it has the advantage of being defined from strictly algebraic considerations. It has previously been limited in its applicability to div-stable discretizations of the Navier–Stokes equations. This paper shows how to extend the same methodology to stabilized low-order mixed finite element approximation methods.