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.

翻译：本研究探索将超幂序列（一种常用于逼近Moore-Penrose逆的方法）与现有预处理技术相结合，以提升预处理器的效能。该思路与基于诺伊曼级数的多项式预处理密切相关。我们首先采用一种针对斯托克斯方程等几何保结构离散所导出鞍点系统的先进无矩阵预处理器。结果表明，引入超幂方法的多次迭代可增强预处理器的效果，从而显著降低模拟三维顶盖驱动腔内斯托克斯流所需的迭代次数和总求解时间。通过综合分析，我们评估了所提方案的稳定性、精度及数值计算成本。