# Parameter-robust stability of classical three-field formulation of Biot's consolidation model

## TL;DR

This paper establishes parameter-robust stability for a classical three-field formulation of Biot's consolidation model, enabling the development of effective preconditioners and stable discretizations that ensure mass conservation.

## Contribution

It introduces parameter-dependent norms that prove full stability of the model uniformly across all parameters, including the Lamé parameter, facilitating robust numerical methods.

## Key findings

- Proves uniform inf-sup stability with respect to all model parameters.
- Constructs a block diagonal preconditioner based on the stability analysis.
- Designs stable discretizations with optimal error estimates and mass conservation.

## Abstract

This paper is devoted to the stability analysis of a classical three-field formulation of Biot's consolidation model where the unknown variables are the displacements, fluid flux (Darcy velocity), and pore pressure. Specific parameter-dependent norms provide the key in establishing the full parameter-robust inf-sup stability of the continuous problem. Therefore, stability results presented here are uniform not only with respect to the Lam\'e parameter $\lambda$, but also with respect to all the other model parameters. This allows for the construction of a uniform block diagonal preconditioner within the framework of operator preconditioning. Stable discretizations that meet the required conditions for full robustness and guarantee mass conservation, both locally and globally, are discussed and corresponding optimal error estimates proven.