A mixed finite element method for nearly incompressible multiple-network poroelasticity. In this paper, we present and analyze a new mixed finite element formulation of a general family of quasi-static multiple-network poroelasticity (MPET) equations. The MPET equations describe flow and deformation in an elastic porous medium that is permeated by multiple fluid networks of differing characteristics. As such, the MPET equations represent a generalization of Biot’s equations, and numerical discretizations of the MPET equations face similar challenges. Here, we focus on the nearly incompressible case for which standard mixed finite element discretizations of the MPET equations perform poorly. Instead, we propose a new mixed finite element formulation based on introducing an additional total pressure variable. By presenting energy estimates for the continuous solutions and a priori error estimates for a family of compatible semidiscretizations, we show that this formulation is robust for nearly incompressible materials, small storage coefficients, and small or vanishing transfer between networks. These theoretical results are corroborated by numerical experiments. Our primary interest in the MPET equations stems from the use of these equations in modeling interactions between biological fluids and tissues in physiological settings. So, we additionally present physiologically realistic numerical results for blood and interstitial fluid flow interactions.

References in zbMATH (referenced in 12 articles , 1 standard article )

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  1. Boon, Wietse M.; Kuchta, Miroslav; Mardal, Kent-Andre; Ruiz-Baier, Ricardo: Robust preconditioners for perturbed saddle-point problems and conservative discretizations of Biot’S equations utilizing total pressure (2021)
  2. Borregales Reverón, Manuel Antonio; Kumar, Kundan; Nordbotten, Jan Martin; Radu, Florin Adrian: Iterative solvers for Biot model under small and large deformations (2021)
  3. Bürger, Raimund; Kumar, Sarvesh; Mora, David; Ruiz-Baier, Ricardo; Verma, Nitesh: Virtual element methods for the three-field formulation of time-dependent linear poroelasticity (2021)
  4. Khan, Arbaz; Powell, Catherine E.: Parameter-robust stochastic Galerkin mixed approximation for linear poroelasticity with uncertain inputs (2021)
  5. Kraus, Johannes; Lederer, Philip L.; Lymbery, Maria; Schöberl, Joachim: Uniformly well-posed hybridized discontinuous Galerkin/hybrid mixed discretizations for Biot’s consolidation model (2021)
  6. Mardal, Kent-Andre; Rognes, Marie E.; Thompson, Travis B.: Accurate discretization of poroelasticity without Darcy stability (2021)
  7. Oyarzúa, Ricardo; Rhebergen, Sander; Solano, Manuel; Zúñiga, Paulo: Error analysis of a conforming and locking-free four-field formulation for the stationary Biot’s model (2021)
  8. Piersanti, E.; Lee, J. J.; Thompson, T.; Mardal, K.-A.; Rognes, M. E.: Parameter robust preconditioning by congruence for multiple-network poroelasticity (2021)
  9. Qi, Wenya; Seshaiyer, Padmanabhan; Wang, Junping: A four-field mixed finite element method for Biot’s consolidation problems (2021)
  10. Hong, Qingguo; Kraus, Johannes; Lymbery, Maria; Philo, Fadi: Parameter-robust Uzawa-type iterative methods for double saddle point problems arising in Biot’s consolidation and multiple-network poroelasticity models (2020)
  11. Hong, Qingguo; Kraus, Johannes; Lymbery, Maria; Wheeler, Mary F.: Parameter-robust convergence analysis of fixed-stress split iterative method for multiple-permeability poroelasticity systems (2020)
  12. Lee, J. J.; Piersanti, E.; Mardal, K.-A.; Rognes, M. E.: A mixed finite element method for nearly incompressible multiple-network poroelasticity (2019)