The convergence theory for the restricted version of the overlapping Schur complement preconditioner

The restricted version of the overlapping Schur complement (SchurRAS) preconditioner was introduced by Li and Saad (2006) for the solution of linear system Ax=b, and numerical results have shown that the SchurRAS method outperforms the restricted additive Schwarz (RAS) method both in terms of iteration count and CPU time. In this paper, based on meticulous derivation, we give an algebraic representation of the SchurRAS preconditioner, and prove that the SchurRAS method is convergent under the condition that A is an M-matrix and it converges faster than the RAS method.