On the preconditioned APSS iterative method for singular coupled saddle point problems

Abstract

Salkuyeh et al. (D.K. Salkuyeh, H. Aslani, Z.Z. Liang, An alternating positive semi-definite splitting preconditioner for the three-by-three block saddle point problems, Math. Commun. 26 (2021) 177-195) recently established an alternating positive semi-definite splitting (APSS) method for nonsymmetric block three-by-three nonsingular saddle point problems arising from the Picard iteration method for a class of mixed finite element scheme. In this work, we analyse the semi-convergence of the APSS method for solving a class of nonsymmetric block three-by-three singular saddle point problems. The APSS induced preconditioner is applied to improve the semi-convergence rate of the flexible GMRES (FGMRES) method. Numerical results are designated to support the theoretical results. These results show that the served preconditioner is efficient compared with FGMRES without a preconditioner.