  This study explores the integration of the hyper-power sequence, a method
commonly employed for approximating the Moore-Penrose inverse, to enhance the
effectiveness of an existing preconditioner. The approach is closely related to
polynomial preconditioning based on Neumann series. We commence with a
state-of-the-art matrix-free preconditioner designed for the saddle point
system derived from isogeometric structure-preserving discretization of the
Stokes equations. Our results demonstrate that incorporating multiple
iterations of the hyper-power method enhances the effectiveness of the
preconditioner, leading to a substantial reduction in both iteration counts and
overall solution time for simulating Stokes flow within a 3D lid-driven cavity.
Through a comprehensive analysis, we assess the stability, accuracy, and
numerical cost associated with the proposed scheme.
