Parameter-robust Uzawa-type iterative methods for double saddle point problems arising in Biots consolidation and multiple-network poroelasticity models

This work is concerned with the iterative solution of systems of quasi-static multiple-network poroelasticity (MPET) 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 $n ge 1$ fluid networks are the unknown physical quantities. Generalizing Biots model of consolidation, which is obtained for $n=1$, the MPET equations for $nge1$ 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 GMRES algorithm.