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

Qingguo Hong and Johannes Kraus

### 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, the stability results presented here are uniform not only with respect to the Lamé 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 strongly, i.e., pointwise, are discussed and corresponding optimal error estimates proved.

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### Key words

Biot's consolidation model, three-field formulation, parameter-robust stability, conservative discretizations, uniform preconditioners, optimal error estimates

### AMS subject classifications

65F10, 65N20, 65N30

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