A class of new extended shift-splitting preconditioners for saddle point problems

Abstract

In this paper, a class of new extended shift-splitting (NESS) preconditioners containing t > 0 are proposed for saddle point problems, which contain the existing shift-splitting-based preconditioners as special cases. The convergence properties of the NESS iteration and the spectral distribution of the NESS preconditioned matrix are investigated. We give an estimated bound for the real eigenvalues of the NESS preconditioned matrix whose left end is positive and show that the non-real eigenvalues of the NESS preconditioned matrix are located in an intersection of two rings, particularly, these non-real eigenvalues are located in an intersection of two rings and one circle if t ≥ 1 2 . Moreover, we show that under appropriate conditions our proposed eigenvalue bounds are tighter than the corresponding bounds given in Ren et al. (2017). Using Fourier analysis, we derive an optimized parameter t ∗ independent of the viscosity ν for the continuous version of the NESS preconditioned GMRES method for the 2D Stokes equation. Moreover, we find that the NESS preconditioned GMRES method with a constant multiple of the optimized parameter t ∗ is effective and robust to solve 2D Stokes problems in practice. Numerical experiments are used to verify our theoretical results.