Conservative discretizations and parameter-robust preconditioners for Biot and multiple-network flux-based poroelastic models

TL;DR

This paper develops stable discretizations and preconditioners for multicompartmental poroelastic models with parameters spanning multiple orders of magnitude, ensuring uniform stability and efficient solutions.

Contribution

It introduces novel parameter-matrix-dependent norms and discretizations that achieve parameter-robust stability for flux-based poroelastic systems, extending previous work on Biot models.

Findings

Proved uniform stability with respect to all model parameters.

Designed discretizations meeting stability and mass conservation criteria.

Confirmed theoretical results through numerical experiments.

Abstract

The parameters in the governing system of partial differential equations of multicompartmental poroelastic models typically vary over several orders of magnitude making its stable discretization and efficient solution a challenging task. In this paper, inspired by the approach recently presented by Hong and Kraus~[Parameter-robust stability of classical three-field formulation of Biot's consolidation model, ETNA (to appear)] for the Biot model, we prove the uniform stability, and design stable disretizations and parameter-robust preconditioners for flux-based formulations of multiple-network poroelastic systems. Novel parameter-matrix-dependent norms that provide the key for establishing uniform inf-sup stability of the continuous problem are introduced. As a result, the stability estimates presented here are uniform not only with respect to the Lam\'e parameter $\lambda$, but also with respect to all the other model parameters such as permeability coefficients $K_i$, storage coefficients $c_{p_i}$, network transfer coefficients $\beta_{ij}, i,j=1,\cdots,n$, the scale of the networks $n$ and the time step size $\tau$. Moreover, strongly mass conservative discretizations that meet the required conditions for parameter-robust stability are suggested and corresponding optimal error estimates proved. The transfer of the canonical (norm-equivalent) operator preconditioners from the continuous to the discrete level lays the foundation for optimal and fully robust iterative solution methods. The theoretical results are confirmed in numerical experiments that are motivated by practical applications.