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A priori and a posteriori error bounds for the fully mixed FEM formulation of poroelasticity with stress-dependent permeability

Arbaz Khan, Bishnu P. Lamichhane, Ricardo Ruiz-Baier, Segundo Villa-Fuentes

TL;DR

This work develops a fully mixed finite element framework for nonlinear poroelasticity with stress-dependent permeability, leveraging a $Hellinger\text{--}Reissner$-type formulation to achieve robustness in near-incompressible regimes and mass conservation via $PEERS_k$ and $RT_k$ spaces. A Banach fixed-point approach decouples the nonlinear permeability from the linear saddle-point structures, yielding existence and uniqueness under small-data assumptions, and enabling both a priori and computable a posteriori error analysis. The authors derive Céa-type estimates and optimal convergence rates, and design a residual-based a posteriori estimator that is reliable and efficient, driving adaptive mesh refinement in 2D and 3D. Numerical results validate the theory, showing optimal rates, conservation properties, and parameter-robust performance, including adaptive refinements near singularities and material interfaces. The framework is well-suited for soft-tissue and geomechanics applications where permeability depends on deformation, with potential extensions to transient, multiphysics, and thermo-poroelastic settings.

Abstract

We develop a family of mixed finite element methods for a model of nonlinear poroelasticity where, thanks to a rewriting of the constitutive equations, the permeability depends on the total poroelastic stress and on the fluid pressure and therefore we can use the Hellinger--Reissner principle with weakly imposed stress symmetry for Biot's equations. The problem is adequately structured into a coupled system consisting of one saddle-point formulation, one linearised perturbed saddle-point formulation, and two off-diagonal perturbations. This system's unique solvability requires assumptions on regularity and Lipschitz continuity of the inverse permeability, and the analysis follows fixed-point arguments and the Babuška--Brezzi theory. The discrete problem is shown uniquely solvable by applying similar fixed-point and saddle-point techniques as for the continuous case. The method is based on the classical PEERS$_k$ elements, it is exactly momentum and mass conservative, and it is robust with respect to the nearly incompressible as well as vanishing storativity limits. We derive a priori error estimates, we also propose fully computable residual-based a posteriori error indicators, and show that they are reliable and efficient with respect to the natural norms, and robust in the limit of near incompressibility. These a posteriori error estimates are used to drive adaptive mesh refinement. The theoretical analysis is supported and illustrated by several numerical examples in 2D and 3D.

A priori and a posteriori error bounds for the fully mixed FEM formulation of poroelasticity with stress-dependent permeability

TL;DR

This work develops a fully mixed finite element framework for nonlinear poroelasticity with stress-dependent permeability, leveraging a -type formulation to achieve robustness in near-incompressible regimes and mass conservation via and spaces. A Banach fixed-point approach decouples the nonlinear permeability from the linear saddle-point structures, yielding existence and uniqueness under small-data assumptions, and enabling both a priori and computable a posteriori error analysis. The authors derive Céa-type estimates and optimal convergence rates, and design a residual-based a posteriori estimator that is reliable and efficient, driving adaptive mesh refinement in 2D and 3D. Numerical results validate the theory, showing optimal rates, conservation properties, and parameter-robust performance, including adaptive refinements near singularities and material interfaces. The framework is well-suited for soft-tissue and geomechanics applications where permeability depends on deformation, with potential extensions to transient, multiphysics, and thermo-poroelastic settings.

Abstract

We develop a family of mixed finite element methods for a model of nonlinear poroelasticity where, thanks to a rewriting of the constitutive equations, the permeability depends on the total poroelastic stress and on the fluid pressure and therefore we can use the Hellinger--Reissner principle with weakly imposed stress symmetry for Biot's equations. The problem is adequately structured into a coupled system consisting of one saddle-point formulation, one linearised perturbed saddle-point formulation, and two off-diagonal perturbations. This system's unique solvability requires assumptions on regularity and Lipschitz continuity of the inverse permeability, and the analysis follows fixed-point arguments and the Babuška--Brezzi theory. The discrete problem is shown uniquely solvable by applying similar fixed-point and saddle-point techniques as for the continuous case. The method is based on the classical PEERS elements, it is exactly momentum and mass conservative, and it is robust with respect to the nearly incompressible as well as vanishing storativity limits. We derive a priori error estimates, we also propose fully computable residual-based a posteriori error indicators, and show that they are reliable and efficient with respect to the natural norms, and robust in the limit of near incompressibility. These a posteriori error estimates are used to drive adaptive mesh refinement. The theoretical analysis is supported and illustrated by several numerical examples in 2D and 3D.
Paper Structure (24 sections, 23 theorems, 147 equations, 4 figures, 4 tables)

This paper contains 24 sections, 23 theorems, 147 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Weak formulation and preliminary properties
  3. Derivation of weak forms
  4. Stability properties and suitable inf-sup conditions
  5. Analysis of the coupled problem
  6. A fixed‑point operator
  7. Well‑definedness of Lg
  8. Existence and uniqueness of weak solution
  9. Finite element discretisation
  10. Finite element spaces and Galerkin scheme
  11. Analysis of the discrete problem
  12. A priori error estimates
  13. Preliminaries
  14. Derivation of Céa estimates
  15. Rates of convergence
  16. ...and 9 more sections

Key Result

Lemma 3.1

Let $\widehat{p} \in \mathrm{W}$ (cf. eq:set-W). Then, there exists a unique $(\boldsymbol{\sigma},({\boldsymbol{u}},\boldsymbol{\rho})) \in \mathbb{H}_\mathrm{N}(\mathbf{div};\Omega)\times \mathbf{L}^2(\Omega)\times\mathbb{L}_{\mathrm{skew}}^2(\Omega)$ solution to eq:weak1. In addition, there exist

Figures (4)

  • Figure 7.1: Example 1. Sample of approximate solutions (stress magnitude, displacement magnitude, non-zero entry of rotation, flux magnitude, and fluid pressure) computed with the second-order scheme and plotted on the deformed domain (for reference we also show the contour of the undeformed domain).
  • Figure 7.2: Example 2. Sample of approximate solutions (stress magnitude, displacement magnitude, rotation magnitude, flux magnitude, and fluid pressure) computed with the first-order scheme and plotted on the deformed domain (for reference we also show the outline of the undeformed domain).
  • Figure 7.3: Example 3. Approximate primal variable solutions (solid displacement and fluid pressure) computed with the first-order scheme, and meshes generated after two, three, and four adaptive refinement steps.
  • Figure 7.4: Example 4. Cross sectional area of cervical spinal cord segmented from sheep imaging data in stover16, initial coarse mesh indicating subdomains with distinct material properties (outer pia mater in red, mid white matter in grey, inner grey matter in green), final adapted mesh after six refinement steps, and sample of stress, displacement, fluid flux and fluid pressure at the indentation test (bottom row figures are obtained with the lowest-order scheme and rendered on the deformed configuration).

Theorems & Definitions (53)

  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Remark 3.5
  • ...and 43 more