Modified SIMPLE preconditioners for saddle point problems from steady incompressible Navier–Stokes equations

Abstract

Based on the semi-implicit method for pressure linked equations (SIMPLE) preconditioner, which is presented by the authors C.-G Li and C. Vuik in Li and Vuik (2004), we propose a class of preconditioners named modified SIMPLE (MS) preconditioner for incompressible flow. The convergence property of the corresponding MS iteration is discussed and proved to be convergent under a mild condition. The distribution of eigenvalues of the preconditioned matrix is analyzed, indicating that more intensive ones can be derived than the REHSS preconditioner as well as the HSS preconditioner. Moreover, the preconditioned system is solved by a Krylov subspace method like restarted GMRES. Finally, some numerical comparisons on test problems from finite element discretization of the Navier–Stokes problem, are made, showing a great advantage of the MS preconditioner over some of the existing ones.