A block positive-semidefinite splitting preconditioner for generalized saddle point linear systems

Abstract

In this paper, we propose a new block positive-semidefinite splitting (BPS) preconditioner for a class of generalized saddle point linear systems. The new BPS preconditioner is based on two positive-semidefinite splittings of the generalized saddle point matrix, resulting in an unconditional convergent fixed-point iteration method. Theoretical results show that all eigenvalues of the BPS preconditioned matrix are clustered at only two points as the iteration parameter is close to zero. Two numerical examples arising from the mixed finite element discretization of the linearized Navier–Stokes equation and the meshfree discretization of the piezoelectric structure equation are used to illustrate the effectiveness of the new preconditioner.