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**Conservative discretizations and parameter-robust preconditioners for Biot and multiple-network flux-based poroelasticity models.**
*(English)*
Zbl 1463.65372

Summary: The parameters in the governing system of partial differential equations of multiple-network poroelasticity models typically vary over several orders of magnitude, making its stable discretization and efficient solution a challenging task. In this paper, we prove the uniform Ladyzhenskaya-Babuška-Brezzi (LBB) condition and design uniformly stable discretizations and parameter-robust preconditioners for flux-based formulations of multiporosity/multipermeability systems. Novel parameter-matrix-dependent norms that provide the key for establishing uniform LBB stability of the continuous problem are introduced. As a result, the stability estimates presented here are uniform not only with respect to the Lamé parameter \(\lambda\) but also to all the other model parameters, such as the permeability coefficients \(K_i\); storage coefficients \(c_{p_i}\); network transfer coefficients \(\beta_{ij}\), \(i,j=1,\dots,n\); 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 LBB 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.

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

65F08 | Preconditioners for iterative methods |

65F10 | Iterative numerical methods for linear systems |

76S05 | Flows in porous media; filtration; seepage |

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |