Preconditioned iterative method for nonsymmetric saddle point linear systems

Abstract

In this paper, a new preconditioned iterative method is presented to solve a class of nonsymmetric nonsingular or singular saddle point problems. The implementation of the proposed preconditioned Krylov subspace method avoids solving inverse of Schur complement and only needs to solve one linear sub-system at each step, which implies that it may save considerable costs. Theoretical convergence analysis, including the bounds of eigenvalues and eigenvectors, the degree of the minimal polynomial of the preconditioned matrix, are discussed in details. Moreover, a novel algebraic estimation technique for finding a practical iteration parameter is presented, which is very effective and practical even for large scale problems. At last, some numerical examples are carried, showing that the theoretical results are valid and convincing.