Parameter-robust stability of classical three-field formulation of Biot's consolidation model
For researchers in computational geomechanics and poroelasticity, this work provides a theoretical foundation for robust numerical methods that are insensitive to parameter variations, addressing a known bottleneck in preconditioner design.
This paper proves parameter-robust inf-sup stability for the continuous three-field formulation of Biot's consolidation model, uniform with respect to all model parameters including the Lamé parameter, enabling construction of a uniform block diagonal preconditioner. Stable discretizations with local and global mass conservation are discussed and optimal error estimates are proven.
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 Lamé parameter $λ$, 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.