This work is concerned with the iterative solution of systems of quasi-static multiple-network poroelasticity equations describing flow in elastic porous media that is permeated by single or multiple fluid networks. Here, the focus is on a three-field formulation of the problem in which the displacement field of the elastic matrix and, additionally, one velocity field and one pressure field for each of the [Formula: see text] fluid networks are the unknown physical quantities. Generalizing Biot’s model of consolidation, which is obtained for [Formula: see text], the MPET equations for [Formula: see text] exhibit a double saddle point structure. The proposed approach is based on a framework of augmenting and splitting this three-by-three block system in such a way that the resulting block Gauss–Seidel preconditioner defines a fully decoupled iterative scheme for the flux-, pressure-, and displacement fields. In this manner, one obtains an augmented Lagrangian Uzawa-type method, the analysis of which is the main contribution of this work. The parameter-robust uniform linear convergence of this fixed-point iteration is proved by showing that its rate of contraction is strictly less than one independent of all physical and discretization parameters. The theoretical results are confirmed by a series of numerical tests that compare the new fully decoupled scheme to the very popular partially decoupled fixed-stress split iterative method, which decouples only flow — the flux and pressure fields remain coupled in this case — from the mechanics problem. We further test the performance of the block-triangular preconditioner defining the new scheme when used to accelerate the generalized minimal residual method (GMRES) algorithm.

中文翻译：

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Biot 固结和多网络孔隙弹性模型中出现的双鞍点问题的参数稳健 Uzawa 型迭代方法

这项工作涉及准静态多网络孔隙弹性方程系统的迭代求解，该方程描述了由单个或多个流体网络渗透的弹性多孔介质中的流动。在这里，重点是问题的三场公式，其中弹性矩阵的位移场以及每个 [公式：见文本] 流体网络的一个速度场和一个压力场是未知的物理数量。推广 Biot 的固结模型，它是为 [公式：见文本] 获得的，[公式：见文本] 的 MPET 方程表现出双鞍点结构。所提出的方法基于一个扩展和拆分这个 3×3 块系统的框架，使得生成的块 Gauss-Seidel 预处理器为通量场、压力场和位移场定义了一个完全解耦的迭代方案。通过这种方式，人们获得了一种增强的拉格朗日 Uzawa 型方法，对其的分析是这项工作的主要贡献。该定点迭代的参数鲁棒均匀线性收敛性证明了它的收缩率严格小于一个独立于所有物理和离散化参数的速率。理论结果通过一系列数值试验得到证实，这些试验将新的完全解耦方案与非常流行的部分解耦固定应力拆分迭代方法进行了比较，它仅将流动（在这种情况下通量和压力场保持耦合）与力学问题解耦。我们进一步测试了定义新方案的块三角预条件器在用于加速广义最小残差法 (GMRES) 算法时的性能。