Conservative discretizations and parameter-robust preconditioners for Biot and multiple-network flux-based poroelastic models [article]

Qinggou Hong and Johannes Kraus and Maria Lymbery and Fadi Philo
2018 arXiv   pre-print
The parameters in the governing system of partial differential equations of multicompartmental poroelastic models typically vary over several orders of magnitude making its stable discretization and efficient solution a challenging task. In this paper, inspired by the approach recently presented by Hong and Kraus [Parameter-robust stability of classical three-field formulation of Biot's consolidation model, ETNA (to appear)] for the Biot model, we prove the uniform stability, and design stable
more » ... isretizations and parameter-robust preconditioners for flux-based formulations of multiple-network poroelastic systems. Novel parameter-matrix-dependent norms that provide the key for establishing uniform inf-sup 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 λ, but also with respect to all the other model parameters such as permeability coefficients K_i, storage coefficients c_p_i, network transfer coefficients β_ij, i,j=1,...,n, the scale of the networks n and the time step size τ. Moreover, strongly mass conservative discretizations that meet the required conditions for parameter-robust 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.
arXiv:1806.00353v2 fatcat:zz3o7rgozrhlznqv5sdyyvmfg4