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Fully mixed virtual element schemes for steady-state poroelastic stress-assisted diffusion

Isaac Bermudez, Bryan Gomez-Vargas, Andres E. Rubiano, Ricardo Ruiz-Baier

TL;DR

This work develops a fully mixed virtual element method for steady-state poroelastic stress-assisted diffusion, coupling Biot elasticity with stress-modulated diffusion under a strong symmetry constraint. The analysis leverages a Banach-space perturbation framework and a fixed-point strategy to establish existence and uniqueness for both the continuous and discrete problems, plus optimal a priori error estimates that are robust with respect to poromechanical parameters. The VEM discretisation enforces symmetry strongly, uses projections and stabilisations that ensure computability on general polyhedral meshes, and yields a stable, locking-free scheme for the coupled Biot-diffusion system. Numerical tests in 2D and 3D confirm optimal convergence and robustness, and a brain-tissue application highlights the method’s potential for modeling stress-influenced diffusion in biomedical contexts. The combination of a novel abstract stability result, fully mixed VEM discretisation, and rigorous error analysis provides a practical and theoretically solid tool for stress-assisted diffusion in poroelastic media.

Abstract

We propose a fully mixed virtual element method for the numerical approximation of the coupling between stress-altered diffusion and linear elasticity equations with strong symmetry of total poroelastic stress (using the Hellinger--Reissner principle). A novelty of this work is that we introduce a less restrictive assumption on the stress-assisted diffusion coefficient, requiring an analysis of the perturbed diffusion equation using Banach spaces. The solvability of the continuous and discrete problems is established using a suitable modification of the abstract theory for perturbed saddle-point problems in Banach spaces (which is in itself a new result of independent interest). In addition, we establish optimal a priori error estimates. The method and its analysis are robust with respect to the poromechanical parameters. We also include a number of numerical examples that illustrate the properties of the proposed scheme.

Fully mixed virtual element schemes for steady-state poroelastic stress-assisted diffusion

TL;DR

This work develops a fully mixed virtual element method for steady-state poroelastic stress-assisted diffusion, coupling Biot elasticity with stress-modulated diffusion under a strong symmetry constraint. The analysis leverages a Banach-space perturbation framework and a fixed-point strategy to establish existence and uniqueness for both the continuous and discrete problems, plus optimal a priori error estimates that are robust with respect to poromechanical parameters. The VEM discretisation enforces symmetry strongly, uses projections and stabilisations that ensure computability on general polyhedral meshes, and yields a stable, locking-free scheme for the coupled Biot-diffusion system. Numerical tests in 2D and 3D confirm optimal convergence and robustness, and a brain-tissue application highlights the method’s potential for modeling stress-influenced diffusion in biomedical contexts. The combination of a novel abstract stability result, fully mixed VEM discretisation, and rigorous error analysis provides a practical and theoretically solid tool for stress-assisted diffusion in poroelastic media.

Abstract

We propose a fully mixed virtual element method for the numerical approximation of the coupling between stress-altered diffusion and linear elasticity equations with strong symmetry of total poroelastic stress (using the Hellinger--Reissner principle). A novelty of this work is that we introduce a less restrictive assumption on the stress-assisted diffusion coefficient, requiring an analysis of the perturbed diffusion equation using Banach spaces. The solvability of the continuous and discrete problems is established using a suitable modification of the abstract theory for perturbed saddle-point problems in Banach spaces (which is in itself a new result of independent interest). In addition, we establish optimal a priori error estimates. The method and its analysis are robust with respect to the poromechanical parameters. We also include a number of numerical examples that illustrate the properties of the proposed scheme.
Paper Structure (24 sections, 23 theorems, 146 equations, 6 figures, 3 tables)

This paper contains 24 sections, 23 theorems, 146 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Scope.
  3. Outline.
  4. Recurrent notation.
  5. Strong mixed form.
  6. Weak formulation and continuous well-posedness analysis
  7. Derivation and main properties
  8. Unique solvability of decoupled Biot equations and mixed diffusion equations
  9. Unique solvability of the coupled problem via fixed-point theory
  10. Virtual element discretisation
  11. Assumptions on the mesh.
  12. Polynomial spaces.
  13. VEM for Hellinger--Reissner linear elasticity
  14. VEM for perturbed mixed second-order elliptic problems
  15. Polynomial projection and interpolation operators
  16. ...and 9 more sections

Key Result

Lemma 2.1

The bilinear forms $A(\bullet,\bullet)$, $B(\bullet,\bullet)$, $C(\bullet,\bullet)$, $D(\bullet,\bullet)$, $a_{\widehat{\boldsymbol{\sigma}}}(\bullet,\bullet)$, $b(\bullet,\bullet)$, and $c(\bullet,\bullet)$ are bounded. That is: where the boundedness constants are given by and $C_{\mathrm{emb}}$ is the constant from the continuous embedding $\mathbf{L}^4(\Omega) \hookrightarrow \mathbf{L}^2(\Om

Figures (6)

  • Figure 6.1: Example 1. Variety of 2D meshes used in the uniform refinement convergence test.
  • Figure 6.2: Example 1. Snapshots of the variables of interest for the Hexahedral mesh in the last refinement step with $k=2$. The parameters are set to unity, except for $\eta_1 = 10^{-3}$.
  • Figure 6.3: Example 2. Cross-section of a variety of 3D meshes used in the uniform refinement convergence test.
  • Figure 6.4: Example 2. Snapshots of the variables of interest for the Voronoi mesh in the last refinement step with $k=1$. The modulation parameter is set to $\eta_1 = 10^{-5}$, while the remaining parameters are set to unity.
  • Figure 6.5: Example 3. Two-dimensional schematic illustration of molecular clearance in brain tissue of a fluorescent CSF tracer. The experimental setup is shown at the top middle. The MRI scans for the sleep and awake states are shown on the left and right pannels. The bottom middle panels show the expected CSF tracer concentration computational simulations in a polytopal mesh of a coronal slice of the brain with 1,999 Voronoi cells.
  • ...and 1 more figures

Theorems & Definitions (25)

  • Lemma 2.1: boundedness of the bilinear forms
  • Lemma 2.2: symmetry and positive semi-definiteness of diagonal forms
  • Lemma 2.3: coercivity for the main diagonal forms
  • Lemma 2.4: continuous inf-sup conditions
  • Remark 2.1
  • Theorem 2.5: Abstract result for Q-elliptic perturbed saddle-point problems
  • Theorem 2.6: well-posedness of the Biot equations
  • Theorem 2.7: well-posedness of the mixed perturbed diffusion equation
  • Lemma 2.8: ball mapping property
  • Lemma 2.9: Lipschitz continuity
  • ...and 15 more