SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page B984-B1007, January 2021.
The mechanical behavior of a poroelastic medium permeated by multiple interacting fluid networks can be described by a system of time-dependent partial differential equations known as the multiple-network poroelasticity (MPET) equations or multiporosity/multipermeability systems. These equations generalize Biot's equations, which describe the mechanics of the one network case. The efficient numerical solution of the MPET equations is challenging, in part due to the complexity of the system and in part due to the presence of interacting parameter regimes. In this paper, we present a new strategy for efficiently and robustly solving the MPET equations numerically. In particular, we discuss an approach to formulating finite element methods and associated preconditioners for the MPET equations based on simultaneous diagonalization of the element matrices. We demonstrate the technique for the multicompartment Darcy equations, with large exchange variability, and the MPET equations for a nearly incompressible medium with large exchange variability. The approach is based on designing transformations of variables that simultaneously diagonalize (by congruence) the equations' key operators and subsequently constructing parameter-robust block diagonal preconditioners for the transformed system. The proposed approach is supported by theoretical considerations as well as by numerical results.

中文翻译：

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多网络多孔弹性的同余参数鲁棒预处理

SIAM 科学计算杂志，第 43 卷，第 4 期，第 B984-B1007 页，2021 年 1 月。
被多个相互作用的流体网络渗透的多孔弹性介质的力学行为可以通过称为多网络多孔弹性 (MPET) 方程或多孔/多渗透系统的时间相关偏微分方程系统来描述。这些方程概括了 Biot 方程，后者描述了一个网络案例的机制。MPET 方程的有效数值求解具有挑战性，部分原因是系统的复杂性，部分原因是存在相互作用的参数机制。在本文中，我们提出了一种有效且稳健地数值求解 MPET 方程的新策略。特别是，我们讨论了一种基于元素矩阵的同时对角化为 MPET 方程制定有限元方法和相关预处理器的方法。我们演示了具有大交换变异性的多室达西方程的技术，以及用于具有大交换变异性的几乎不可压缩介质的 MPET 方程。该方法基于设计同时对角化（通过同余）方程的关键运算符的变量变换，然后为变换后的系统构造参数稳健的块对角预处理器。所提出的方法得到理论考虑以及数值结果的支持。以及具有大交换变异性的几乎不可压缩介质的 MPET 方程。该方法基于设计同时对角化（通过同余）方程的关键运算符的变量变换，然后为变换后的系统构造参数稳健的块对角预处理器。所提出的方法得到理论考虑以及数值结果的支持。以及具有大交换变异性的几乎不可压缩介质的 MPET 方程。该方法基于设计同时对角化（通过同余）方程的关键运算符的变量变换，然后为变换后的系统构造参数稳健的块对角预处理器。所提出的方法得到理论考虑以及数值结果的支持。