Abstract: We further analyze the solution of a class of block two-by-two linear systems. Instead of using the preconditioned GMRES iteration methods, we propose a new approximation of the Schur complement based on the special structure of this kind of block two-by-two matrix, and construct a practical restrictive preconditioner accordingly. Subsequently, we propose a practical restrictively preconditioned conjugate gradient (RPCG) method to solve this class of linear systems. The convergence property of the practical RPCG method is similar to the RPCG method. Last, numerical experiments show that this method is more efficient than some classical preconditioned Krylov subspace iteration methods.

Key words: Block two-by-two matrix, Conjugate gradient method, Restrictive preconditioner