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Parameter-robust stability of classical three-field formulation of Biot’s consolidation model. (English) Zbl 1398.65046

Summary: 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.

MSC:

65F10 Iterative numerical methods for linear systems
65N20 Numerical methods for ill-posed problems for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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