A note on the block alternating splitting implicit iteration method for complex saddle‐point problems

Abstract

SummaryRecently, Bai has proposed a class of block alternating splitting implicit (BASI) iteration methods [Numer. Linear Algebra Appl. 19(6):914‐936] for the complex generalized saddle‐point problems. The author demonstrates the unconditional convergence of the methods when the (1,1) block of the saddle‐point coefficient matrix is positive definite. In this paper, we display that the BASI iteration method is convergent for wider problems, for instance, the (1,1) block is positive semidefinite, provided the splittings meet certain requirements. The theoretical result encourages us to explore more efficient iteration schemes for more general problems. We present new BASI iteration schemes for the algebraic linear systems from time‐harmonic eddy current models. The corresponding matrix splitting preconditioners are presented for the generalized minimal residual method. Experimental results show the feasibility and effectiveness of our new iteration methods and the corresponding preconditioned generalized minimal residual methods.