On Chebyshev accelerated iteration methods for two-by-two block linear systems

Abstract

In this paper, we consider applying the Chebyshev acceleration technique to improve the convergence behavior of some iteration methods for solving the two-by-two block linear systems with semidefinite sub-blocks. Practical problem independent approximations about the eigenvalue bounds of the preconditioned matrices facilitate parameter free implementations of the concerned Chebyshev accelerated methods. To analyze their convergence behaviors, we discuss in detail the spectral properties of the corresponding preconditioned matrices, especially their eigenvector distributions, which lead to almost problem independent iteration error bounds. Numerical experiments of the complex symmetric linear systems demonstrate the efficiency of the new Chebyshev acceleration methods compared with the recently published Chebyshev accelerated PMHSS method and some preconditioned Krylov subspace methods.