SIMPLE-like preconditioners for saddle point problems from the steady Navier–Stokes equations

The semi implicit method for pressure linked equations (SIMPLE) preconditioner is further generalized to a class of SIMPLE-like (SL) preconditioners for solving saddle point problems from the steady Navier–Stokes equations. The SL preconditioners can be also viewed as a generalization of the relaxed deteriorated PSS (RDPSS) preconditioner proposed by Cao et al. (2015). Convergence analysis of the corresponding SL iteration is presented and the optimal iteration parameter is obtained by minimizing the spectral radius of the SL iteration matrix. Moreover, Krylov subspace acceleration of the SL preconditioning is studied. The SL preconditioned saddle point matrix is analyzed. Results about eigenvalue and eigenvector distributions and the minimal polynomial are derived. Numerical experiments from “leaky” two dimensional lid-driven cavity problems are given, which show the advantages of the SL preconditioned GMRES over the DPSS and RDPSS preconditioned ones for solving saddle point problems.