A general class of shift-splitting preconditioners for non-Hermitian saddle point problems with applications to time-harmonic eddy current models

Abstract

Based on a general splitting of the (1,1) leading block matrix, we first construct a general class of shift-splitting (GCSS) preconditioners for non-Hermitian saddle point problems. Convergence conditions of the corresponding matrix splitting iteration methods and preconditioning properties of the GCSS preconditioned saddle point matrices are analyzed. Then the GCSS preconditioner is specifically applied to the non-Hermitian saddle point problems arising from the finite element discretizations of the hybrid formulations of the time-harmonic eddy current models. With suitable choices of the splittings, the new GCSS preconditioners are easier to implement and have faster convergence rates than the existing shift-splitting preconditioner and its modified variant. Two numerical examples are presented to verify the theoretical results and show effectiveness of the new proposed preconditioners.