A relaxed deteriorated PSS preconditioner for nonsymmetric saddle point problems from the steady Navier–Stokes equation

For nonsymmetric saddle point problems arising from the steady Navier–Stokes equations, Pan, Ng and Bai presented a deteriorated positive-definite and skew-Hermitian splitting (DPSS) preconditioner (Pan et al., 2006) to accelerate the convergence rates of the Krylov subspace iteration methods such as GMRES. In this paper, the unconditional convergence property of the DPSS iteration method is proved and a relaxed DPSS (RDPSS) preconditioner is proposed. The RDPSS preconditioner is much closer to the coefficient matrix than the DPSS preconditioner in certain norm, which straightforwardly results in an RDPSS iteration method. The convergence conditions of the RDPSS iteration are analyzed and the optimal parameter, which minimizes the spectral radius of the RDPSS iteration matrix, is derived. Using the RDPSS preconditioner to accelerate some Krylov subspace methods (like GMRES) is also studied. The eigenproperty of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained. Finally, numerical experiments of a model Navier–Stokes equation are presented to illustrate the efficiency of the RDPSS preconditioner.