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

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.

💡 Research Summary

This paper introduces and rigorously analyzes a novel fully-mixed finite element method for the dynamic poroelasticity problem in the low-frequency regime, a model crucial for applications in geosciences such as induced seismicity, geothermal energy, and CO2 storage.

The core innovation lies in the problem’s reformulation as a first-order, four-field hyperbolic system, where the unknowns are the solid velocity (u), filtration velocity (w), stress tensor (σ), and pore pressure (p). This approach offers advantages like local conservation laws and better approximation of flux and stress fields. A central challenge in mixed methods for elasticity is handling the symmetry constraint of the stress tensor (σ = σ^T). Instead of imposing it strongly in the function space or weakly via a Lagrange multiplier (which leads to a saddle-point problem), the authors propose a skew-symmetry penalization strategy. This involves adding a perturbation term to the weak symmetry constraint, effectively penalizing the skew-symmetric part of the stress tensor (skw(σ)) divided by a small parameter ε. Mathematically, this is equivalent to adding a stabilizing term to the original saddle-point system, which simplifies the analysis and discretization.

The spatial discretization employs stable mixed finite element pairs (like Raviart-Thomas elements for stresses and discontinuous spaces for velocities/pressure) on the four-field formulation, coupled with implicit time-stepping schemes. The authors then perform a comprehensive stability analysis for both the continuous and discrete problems.

A standout feature of this analysis is its robustness with respect to degenerate model parameters. The derived stability estimates are proven to hold uniformly in several physically critical limits: 1) when the storage coefficient (s0) approaches zero, modeling incompressible grains; 2) when the Lamé parameter (λ) becomes very large, avoiding volumetric locking; and 3) when the density parameters (ρ_s, ρ_f) vanish, causing the model to degenerate towards the quasi-static case. This robustness surpasses previous works that required strict positivity of these parameters. The analysis also accommodates mixed and non-homogeneous boundary conditions, enhancing practical applicability.

The theoretical findings are validated through extensive numerical tests. Convergence studies confirm optimal rates of convergence for all fields. Robustness tests demonstrate the method’s stability and accuracy across the full spectrum of degenerate parameter values, from the nearly incompressible to the quasi-static limit. Finally, a physically consistent wave propagation simulation in a poroelastic medium showcases the method’s capability to model relevant dynamic phenomena. In summary, this work presents a robust, stable, and computationally feasible mixed finite element framework for low-frequency dynamic poroelasticity, backed by a rigorous analysis that ensures reliability in challenging physical regimes.