Splitting-based block preconditioning methods for block two-by-two matrices of real square blocks

Abstract

Recently, Bai proposed rotated block preconditioners for block two-by-two matrices of real square blocks. These rotated block preconditioners have the product form of a scaled orthogonal matrix and a block two-by-two triangular matrix. Theoretical and numerical results have shown the superiority of these rotated block preconditioners (see Bai (2013) [10]). In this paper, inspired by the efficiency and the special structure of the rotated block preconditioners, we establish a new equivalent linear system to the original linear system in which an orthogonal matrix arises. We construct block Jacobi and block Gauss–Seidel splitting iteration methods based on the coefficient matrix of the new linear system. The convergence of these splitting iterations is also demonstrated. Then, by utilizing the proposed block Jacobi and block Gauss–Seidel splittings, we put forward block preconditioners which are of the product form of a scaled orthogonal matrix and a block two-by-two diagonal or a block two-by-two triangular matrix. Spectral distributions of these preconditioned matrices and numerical experiments show that the proposed splitting-based block preconditioners can be quite competitive with the rotated block preconditioners when they are used to accelerate Krylov subspace iteration methods such as GMRES for solving the block two-by-two liner systems.