Splitting-based Block Preconditioning Methods For Block Two-by-Two Matrices Of Real Square Blocks

Many practical problems from scientific computing and engineering applications re-quire the solution of a block two-by-two linear system. Thus, solving block two-by-two linear system has attracted much more attention and lots of efficient solvers could be found in the literature. Of which the the preconditioned Krylov subspace methods are the most important solvers. In this paper, we establish a new equivalent linear system to the original linear system by an orthogonal matrix. 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 splitting preconditioners. Spectral distributions of these preconditioned matrices and numerical experiments show that the proposed splitting-based block preconditioners can be quite competitive with the existed preconditioners when they are used to acceler-ate Krylov subspace iteration methods such as GMRES for solving the block two-by-two liner systems.