The generalized Uzawa-SHSS method for non-Hermitian saddle-point problems

Recently, Li and Wu (2015) proposed the single-step Hermitian and skew-Hermitian splitting (SHSS) method for solving the non-Hermitian positive definite linear systems. Based on the single-step Hermitian and skew-Hermitian splitting of the (1,1) part of the saddle-point coefficient matrix, a new Uzawa-type method is proposed for solving a class of saddle-point problems with non-Hermitian positive definite (1,1) parts. Convergence (Semi-convergence) properties of this new method for nonsingular (singular) are derived under suitable conditions. Numerical examples are implemented to confirm the theoretical results and verify that this new method is more feasibility and robustness than the new HSS-like (NHSS-like), the Uzawa-HSS and the parameterized Uzawa-skew-Hermitian triangular splitting (PU-STS) methods for solving both the nonsingular and the singular saddle-point problems with non-Hermitian positive definite and Hermitian dominant (1,1) parts.