Parameterized approximate block LU preconditioners for generalized saddle point problems

In this paper, we are concerned with the iteration solution of generalized saddle point problems. Based on the exact block LU factorization of the coefficient matrix, we construct a class of parameterized approximate block LU factorization preconditioners, which rely on suitable approximations of the Schur complement of the (1,1) block of the coefficient matrix. Convergence of the corresponding iteration methods is analyzed and the optimal iteration parameters minimizing the spectral radii are deduced. Algebraic characteristics of the related preconditioned matrices are discussed, including eigenvalue and eigenvector distributions and upper bounds for degree of the minimal polynomial. The established results extend those of the approximate factorization and variants of the Hermitian and skew-Hermitian splitting and positive and skew-Hermitian splitting preconditioners for saddle point problems. Numerical experiments are demonstrated to illustrate the efficiency of the new preconditioners.