On parameterized block symmetric positive definite preconditioners for a class of block three-by-three saddle point problems

Abstract

In this paper, we consider some preconditioning techniques for a class of block three-by-three saddle point problems, which arise from a coupled diffuse element-finite element technique for transient coupled-field analysis and some other applications. Firstly, we propose an exact parameterized block symmetric positive definite preconditioner for solving the block three-by-three saddle point problems, and by analyzing the spectrum of the corresponding preconditioned matrix, we get that it has at most four distinct eigenvalues. Secondly, for the needs of practical applications, we also propose the inexact version of the above preconditioner for solving the block three-by-three saddle point problems, and through the analysis of the spectral properties of the corresponding preconditioned matrix, we obtain the bounds of its eigenvalues. In addition, we also study the choices of the (approximately) optimal parameters in the above inexact preconditioner. Finally, numerical experiments are performed to demonstrate the effectiveness of our proposed inexact preconditioner compared with the existing block preconditioners studied recently for the block three-by-three saddle point problems.