Algebraic spectral analysis of the DSSR preconditioner

The dimension-wise splitting with selective relaxation (DSSR) is an efficient iterative method for solving linear systems arising from the discretization of incompressible Navier-Stokes equations. The unconditional convergence properties and optimal relaxation parameters are analyzed at the continuous level on model problems in Gander et al. (2016) [15] In this paper, we provide an algebraically spectral analysis of the DSSR preconditioner when applied to Krylov subspace methods. The analysis indicates that most eigenvalues of the preconditioned matrix are 1, and the corresponding eigenspace is nondeficient, which is an important favorable property for fast convergence of Krylov subspace iterative method. Finally, the preconditioner is explained from the deflation point of view, which provides a clue for developing hybrid direct iterative solvers.