On generalized local Hermitian and skew-Hermitian splitting iterative method for block two-by-two linear systems

For large sparse saddle point problems whose (1, 2) and (2, 1)-blocks are the transpose of each other, Zhu studied a generalized local Hermitian and skew-Hermitian splitting (GLHSS) iterative method (see Zhu (2012)). In this paper, the GLHSS method is extended to the block 2×22×2 linear system, which allows that the (1, 2)-block is not equal to the transpose of the (2, 1)-block or the (2, 2)-block is non-zero. With different choices of the parameter matrices, the existing methods are included and the new algorithms for solving the block 2×22×2 linear system are obtained. The conditions for guaranteeing the convergence of the new iterative method are studied and the correctness of the existing corollaries are questioned. Numerical experiments are provided to show that the proposed method is feasible and effective, and it is not necessary to introduce some parameter matrices in some cases.