Global accelerated Hermitian and skew–Hermitian splitting preconditioner for the solution of discrete Stokes problems

Abstract

In fluid mechanics, numerous applications necessitate solving a sequence of linear systems. These systems typically arise from discretizing the Stokes equations using mixed-finite element methods. The matrices that result from this process often exhibit a saddle point structure, which makes the iterative solution of preconditioned linear systems challenging. To effectively solve the large scale and ill-conditioned linear systems, it is necessary to implement efficient linear solvers. We present a global approach, specifically the preconditioned global conjugate gradient (PGCG), aimed at enhancing the performance of preconditioned Hermitian and skew–Hermitian splitting preconditioner $\left(P_{PHSS}\right)$ and accelerated Hermitian and skew–Hermitian splitting preconditioner $\left(P_{AHSS}\right)$. The new preconditioners can be utilized to expedite the convergence of the generalized minimal residual (GMRES) method. We evaluate the effectiveness of the preconditioned iterative methods by considering the Central Processing Unit ($\mathrm{CPU}$) times and numbers of the $P$GMRES iterations. The numerical results indicate that incorporating $P$GCG method improves the performance of the preconditioners $P_{PHSS}$ and $P_{AHSS}$.