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

TL;DR

This paper introduces a parameter-robust Uzawa-type iterative method for solving complex double saddle point systems in multi-network poroelasticity models, ensuring uniform convergence regardless of physical or discretization parameters.

Contribution

It develops a fully decoupled, augmented Lagrangian Uzawa-type iterative scheme with proven uniform linear convergence for double saddle point problems in MPET models, extending Biot's model.

Findings

The method achieves parameter-robust convergence independent of physical parameters.

Numerical tests show the scheme outperforms existing partially decoupled methods.

The block preconditioner effectively accelerates GMRES iterations.

Abstract

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 Biot's model of consolidation, which is obtained for $n=1$, the MPET equations for $n\ge1$ 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.