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Polytopal discontinuous Galerkin methods for low-frequency poroelasticity coupled to unsteady Stokes flow

Michele Botti, Ivan Fumagalli, Ilario Mazzieri

TL;DR

The paper develops and analyzes a polygonal discontinuous Galerkin method for simulating fluid exchange between a deformable poroelastic body and an adjacent free-flow channel, coupling the low-frequency Biot model with an unsteady Stokes flow via transmission conditions on the interface. It employs a two-displacement Biot formulation in the poroelastic region and a stress formulation of the Stokes problem with weak symmetry in the fluid, using PolydG discretization on polytopic meshes and a comprehensive stability and hp-error analysis, including an Inf-Sup condition for the fluid coupling. Theoretical results establish energy stability and optimal convergence in the hp setting, while numerical tests in 2D confirm the predicted rates and demonstrate the method’s effectiveness in scenarios with complex interface dynamics and potential locking. The work provides a robust, high-order framework for multi-physics simulations involving fluid–poroelastic structure interaction, with implications for geosciences, biomedical engineering, and environmental modeling.

Abstract

We focus on the numerical analysis of a polygonal discontinuous Galerkin scheme for the simulation of the exchange of fluid between a deformable saturated poroelastic structure and an adjacent free-flow channel. We specifically address wave phenomena described by the low-frequency Biot model in the poroelastic region and unsteady Stokes flow in the open channel, possibly an isolated cavity or a connected fracture system. The coupling at the interface between the two regions is realized by means of transmission conditions expressing conservation laws. The spatial discretization hinges on the weak form of the two-displacement poroelasticity system and a stress formulation of the Stokes equation with weakly imposed symmetry. We present a complete stability analysis for the proposed semi-discrete formulation and derive a-priori hp-error estimates.

Polytopal discontinuous Galerkin methods for low-frequency poroelasticity coupled to unsteady Stokes flow

TL;DR

The paper develops and analyzes a polygonal discontinuous Galerkin method for simulating fluid exchange between a deformable poroelastic body and an adjacent free-flow channel, coupling the low-frequency Biot model with an unsteady Stokes flow via transmission conditions on the interface. It employs a two-displacement Biot formulation in the poroelastic region and a stress formulation of the Stokes problem with weak symmetry in the fluid, using PolydG discretization on polytopic meshes and a comprehensive stability and hp-error analysis, including an Inf-Sup condition for the fluid coupling. Theoretical results establish energy stability and optimal convergence in the hp setting, while numerical tests in 2D confirm the predicted rates and demonstrate the method’s effectiveness in scenarios with complex interface dynamics and potential locking. The work provides a robust, high-order framework for multi-physics simulations involving fluid–poroelastic structure interaction, with implications for geosciences, biomedical engineering, and environmental modeling.

Abstract

We focus on the numerical analysis of a polygonal discontinuous Galerkin scheme for the simulation of the exchange of fluid between a deformable saturated poroelastic structure and an adjacent free-flow channel. We specifically address wave phenomena described by the low-frequency Biot model in the poroelastic region and unsteady Stokes flow in the open channel, possibly an isolated cavity or a connected fracture system. The coupling at the interface between the two regions is realized by means of transmission conditions expressing conservation laws. The spatial discretization hinges on the weak form of the two-displacement poroelasticity system and a stress formulation of the Stokes equation with weakly imposed symmetry. We present a complete stability analysis for the proposed semi-discrete formulation and derive a-priori hp-error estimates.

Paper Structure

This paper contains 18 sections, 8 theorems, 89 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The physical model and governing equations
  3. Notation
  4. The poroelasto-fluid problem
  5. Weak Formulation
  6. Stability analysis
  7. Discontinuous Galerkin space discretization
  8. Preliminaries
  9. Semi-discrete PolydG formulation
  10. Stability and semi-discrete error analysis
  11. Error analysis
  12. Time integration
  13. Numerical results
  14. Test case 1
  15. Test case 2
  16. ...and 3 more sections

Key Result

Lemma 1

The bilinear forms $\mathcal{M}^p$, $\mathcal{A}^p$, and $\mathcal{D}^p$ defined in eq:bilinear_forms are such that for any $\bm u, \bm v \in \bm H^1_{0,\Gamma_p^D}(\Omega_p)$ and any $\bm w, \bm z \in \bm H_{0,\Gamma_p^D}({\rm div},\Omega_p)$ it holds

Figures (9)

  • Figure 1: Test Case 1 and 2: polygonal mesh with Dirichlet (blue), Neumann (red), and interface (yellow) boundaries.
  • Figure 2: Test case 1 and 2. Physical parameters.
  • Figure 3: Test Case 1. Left: log-log plot of the computed error $\| (\bm e^u, \bm e^w)\|_{\rm E_p}$ as a function of the mesh size $h_p$ for $p_p = 1,2$. Right: log-log plot of the computed error $\| \bm e^\Sigma\|_{\rm E_f}$ as a function of the mesh size $h_f$ for $p_f = 1,2$. Final time $T=0.1$ and $\Delta t = 0.001$.
  • Figure 4: Test Case 1. Semi-log plot of the computed errors $\| (\bm e^u, \bm e^w)\|_{\rm E_p}$ and $\| \bm e^\Sigma\|_{\rm E_f}$ as a function of the polynomial degree $p=p_p = p_f$ fixing the number of mesh element equal to $100$. Final time $T=0.1$ and time step $\Delta t = 0.001$ (left), $\Delta t = 0.0001$ (right).
  • Figure 5: Test Case 2. Left: log-log plot of the computed error $\| (\bm e^u, \bm e^w, \bm e^\Sigma)\|_{\rm E}$ as a function of the mesh size $h = \max(h_p,h_f)$ for $p = p_p = p_f = 1,2,3,4$. Right: semi-log plot of the computed error $\| (\bm e^u, \bm e^w, \bm e^\Sigma)\|_{\rm E}$ as a function of the polynomial degree $p=p_p = p_f$ fixing the number of mesh element equal to $100$. Final time $T=0.1$ and time step $\Delta t = 0.001$.
  • ...and 4 more figures

Theorems & Definitions (17)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Definition 1
  • Lemma 3
  • proof
  • Theorem 2
  • ...and 7 more