Two block product-type preconditioners for double saddle point problems

Abstract

In this paper, a class of product-type (PT) preconditioners for generalized saddle point problems recently proposed in [N. Wang, J. Li, A class of preconditioners based on symmetric-triangular decomposition and matrix splitting for generalized saddle point problems, IMA J. Numer. Anal., (2023) 43, 2998–3025] are extended to solve the double saddle point problems arising from the modeling of liquid crystal directors. By combining augmented Lagrangian (AL) technique, two specific block PT preconditioners are developed, which are applied appropriately with the efficient conjugate gradient (CG) and conjugate residual (CR) methods although neither the preconditioners nor the double saddle point systems are symmetric positive definite (SPD). This is the biggest advantage and novelty of the proposed preconditioners. The proposed preconditioned CG (PCG) and preconditioned CR (PCR) methods actually belong to the categories of nonstandard inner product CG and nonstandard inner product CR methods, respectively. Moreover, the PCG and PCR algorithms and their convergence theorems are given. Theoretical and experimental analysis shows that the spectra of the preconditioned matrices are contained within real and positive intervals which are very sharp if the involved parameters are chosen appropriately. In addition, the practically useful values for parameters are easy to obtain. Numerical experiments are presented to illustrate the rapidity, effectiveness and numerical stability of the proposed preconditioners and show the advantages of the proposed preconditioners over the existing state-of-the-art preconditioners for double saddle point problems.