In this paper, an alternating positive semidefinite splitting (APSS) preconditioner is proposed for solving double saddle point problems. The corresponding APSS iteration method is proved to be convergent unconditionally. Moreover, to further improve its efficiency, a relaxed variant is established for the APSS preconditioner, which results in better spectral distribution and numerical performance. Numerical experiments with liquid crystal director models demonstrate the effectiveness of the APSS preconditioner and its relaxed variant when compared with other preconditioners. Comparison between the related numerical results shows that the proposed preconditioners are comparable with (though not really better than) the best existing preconditioners.

中文翻译：

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双鞍点问题的交替正半定分裂预处理器

本文提出了一种交替正半定分裂（APSS）预处理器来解决双鞍点问题。证明了相应的APSS迭代方法是无条件收敛的。此外，为了进一步提高效率，为APSS预调节器建立了一个宽松的变体，从而获得了更好的频谱分布和数值性能。液晶导向器模型的数值实验证明了与其他预处理器相比，APSS预处理器的有效性及其宽松的变体。相关数值结果之间的比较表明，所提出的预处理器可以与（尽管实际上并不比）最好的现有预处理器相媲美。