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A robust fully-mixed finite element method with skew-symmetry penalization for low-frequency poroelasticity

Stefano Bonetti, Michele Botti, Patrick Vega

TL;DR

The paper develops a robust fully-mixed finite element method for dynamic, low-frequency poroelasticity by reformulating Biot's equations as a four-field system with skew-symmetry penalization. It introduces stable mixed discretizations for solid-fluid coupling and proves continuous and discrete stability that remains uniform under degenerate coefficients, including vanishing storativity and densities. A rigorous a priori analysis accompanies a fully discrete scheme with backward Euler time stepping, and numerical experiments validate convergence, robustness, and realistic wave-propagation behavior in poroelastic media. The approach yields locally conservative, well-posed simulations with nontrivial boundary conditions applicable to geophysical and engineering contexts.

Abstract

In this work, we present and analyze a fully-mixed finite element scheme for the dynamic poroelasticity problem in the low-frequency regime. We write the problem as a four-field, first-order, hyperbolic system of equations where the symmetry constraint on the stress field is imposed via penalization. This strategy is equivalent to adding a perturbation to the saddle point system arising when the stress symmetry is weakly-imposed. The coupling of solid and fluid phases is discretized by means of stable mixed elements in space and implicit time advancing schemes. The presented stability analysis is fully robust with respect to meaningful cases of degenerate model parameters. Numerical tests validate the convergence and robustness and assess the performances of the method for the simulation of wave propagation phenomena in porous materials.

A robust fully-mixed finite element method with skew-symmetry penalization for low-frequency poroelasticity

TL;DR

The paper develops a robust fully-mixed finite element method for dynamic, low-frequency poroelasticity by reformulating Biot's equations as a four-field system with skew-symmetry penalization. It introduces stable mixed discretizations for solid-fluid coupling and proves continuous and discrete stability that remains uniform under degenerate coefficients, including vanishing storativity and densities. A rigorous a priori analysis accompanies a fully discrete scheme with backward Euler time stepping, and numerical experiments validate convergence, robustness, and realistic wave-propagation behavior in poroelastic media. The approach yields locally conservative, well-posed simulations with nontrivial boundary conditions applicable to geophysical and engineering contexts.

Abstract

In this work, we present and analyze a fully-mixed finite element scheme for the dynamic poroelasticity problem in the low-frequency regime. We write the problem as a four-field, first-order, hyperbolic system of equations where the symmetry constraint on the stress field is imposed via penalization. This strategy is equivalent to adding a perturbation to the saddle point system arising when the stress symmetry is weakly-imposed. The coupling of solid and fluid phases is discretized by means of stable mixed elements in space and implicit time advancing schemes. The presented stability analysis is fully robust with respect to meaningful cases of degenerate model parameters. Numerical tests validate the convergence and robustness and assess the performances of the method for the simulation of wave propagation phenomena in porous materials.

Paper Structure

This paper contains 15 sections, 5 theorems, 50 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Dynamic poroelasticity
  3. Model problem
  4. First-order hyperbolic system
  5. Variational formulation
  6. Notation for functional spaces
  7. Weak-symmetry and skew-symmetry penalization
  8. Well-posedness
  9. Discrete setting
  10. Space and time meshes and discrete spaces
  11. Fully-discrete problem
  12. Numerical results
  13. Convergence test
  14. Robustness test
  15. Wave propagation in a poroleastic medium

Key Result

Lemma 2

There exist positive constants $\beta_{m}$ (only depending on $\Omega$ and $\Gamma_{\sigma}$) and $\beta_{f}$ (only depending on $\Omega$ and $\Gamma_{w}$) such that

Figures (6)

  • Figure 1: Convergence test of Section \ref{['sec:conv_test']}: computed errors in $L^2$-norm (left) and $\text{H}({\rm div})$-norm (right) versus $1/h$ (log-log scale). The errors are computed at the final time $T_f$. The polynomial degree of approximation is set to be $l = 0$.
  • Figure 2: Robustness test of Section \ref{['sec:rob_test']} ($s_0 = 0$, $\lambda = 10^6$): computed errors in $L^2$-norm (left) and $\text{H}({\rm div})$-norm (right) versus $1/h$ (log-log scale). The errors are computed at the final time $T_f$. The polynomial degree of approximation is set to be $l = 0$.
  • Figure 3: Robustness test of Section \ref{['sec:rob_test']} ($\rho_f = \rho_w = 0$): computed errors in $L^2$-norm (left) and $\text{H}({\rm div})$-norm (right) versus $1/h$ (log-log scale). The errors are computed at the final time $T_f$. The polynomial degree of approximation is set to be $l = 0$.
  • Figure 4: Wave propagation in poroelastic medium: modulus of the computed velocity field $|\boldsymbol{u}_{h}|$ at the time instants $t=0.8 s$ (left), $t=0.9 s$ (center), $t=1 s$ (right).
  • Figure 5: Wave propagation in poroelastic medium: computed vertical component of the velocity $u_{h,y}$ at the time instants $t=0.8 s$ (left), $t=0.9 s$ (center), $t=1 s$ (right).
  • ...and 1 more figures

Theorems & Definitions (11)

  • Remark 1
  • Lemma 2: Inf-sup conditions
  • proof
  • Lemma 3: $\boldsymbol{{\rm dev}-{\rm div}}$ and trace inequalities
  • Lemma 4: Bounds for $\mathcal{L}_{\boldsymbol{\Sigma}}$ and $\mathcal{L}_{\boldsymbol{W}}$
  • proof
  • Theorem 5: Stability
  • Remark 6
  • proof
  • Theorem 7
  • ...and 1 more