In this paper, we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems. One type of the preconditioners are based on the nested (or recursive) Schur complement, the other is based on an additive type Schur complement after permuting the original saddle point systems. We analyze different preconditioners incorporating the exact Schur complements. We show that some of them will lead to positively stable preconditioned systems if proper signs are selected in front of the Schur complements. These positive-stable preconditioners outperform other preconditioners if the Schur complements are further approximated inexactly. Numerical experiments for a 3-field formulation of the Biot model are provided to verify our predictions.

翻译：本文考虑利用Schur补设计二重与块三对角鞍点问题的预处理子。其中一类预处理子基于嵌套（或递归）Schur补，另一类则基于对原始鞍点系统进行置换后的加性Schur补。我们分析了整合精确Schur补的不同预处理方法。研究表明，若在Schur补前选择适当符号，部分预处理方法将产生正定稳定的预处理系统。当对Schur补进行进一步非精确逼近时，这类正稳定预处理子优于其他方法。基于Biot模型三场公式的数值实验验证了我们的理论预测。