A stabilized total pressure-formulation of the Biot's poroelasticity equations in frequency domain: numerical analysis and applications

TL;DR

This work develops a stabilized equal-order finite element method for Biot poroelasticity in the frequency domain using a total-pressure formulation. It proves well-posedness in both continuous and discrete settings via a Fredholm-based approach and introduces a Brezzi-Pitkáranta-type stabilization to handle low permeability and coefficient discontinuities. The method achieves optimal convergence for linear elements and demonstrates robustness across a range of permeabilities, including a brain elastography geometry. The framework provides a reliable forward model for poroelastography and supports future inverse problem formulations in tissue imaging.

Abstract

This work focuses on the numerical solution of the dynamics of a poroelastic material in the frequency domain. We provide a detailed stability analysis based on the application of the Fredholm alternative in the continuous case, considering a total pressure formulation of the Biot's equations. In the discrete setting, we propose a stabilized equal order finite element method complemented by an additional pressure stabilization to enhance the robustness of the numerical scheme with respect to the fluid permeability. Utilizing the Fredholm alternative, we extend the well-posedness results to the discrete setting, obtaining theoretical optimal convergence for the case of linear finite elements. We present different numerical experiments to validate the proposed method. First, we consider model problems with known analytic solutions in two and three dimensions. As next, we show that the method is robust for a wide range of permeabilities, including the case of discontinuous coefficients. Lastly, we show the application for the simulation of brain elastography on a realistic brain geometry obtained from medical imaging.

Figures (13)

• Figure 1: Example 1 (2D): Visualization of the analytical solution.
• Figure 2: Example 1. Error for displacement (red), pressure (green), and total pressure (magenta) as a function of the mesh size in centimeters. Top row: two-dimensional problem, bottom row: three-dimensional problem. Left: linear finite elements, right: quadratic finite elements. The dashed lines in each plot show the convergence rate $O(h)$ and $O(h^2)$.
• Figure 3: Example 1 (2D). Numerical error in the norm \ref{['norm:cont']} for different values of the permeability, for decreasing mesh sizes (from left to right), and for $\delta_2 = 0$ (red), and $\delta_2 \in \{0.0001, 0.01, 0.1\}$ (lighter to darker blue). Top row: $\mathcal{P}_1$ elements, bottom row: $\mathcal{P}_2$ elements.
• Figure 4: Example 1 (2D): Numerical error in the norm \ref{['norm:cont']} for different values of the Poisson modulus $\nu$, for decreasing mesh sizes (from left to right), and for $\delta_2 = 0$ (red), and $\delta_2 \in \{0.0001, 0.01, 0.1\}$ (lighter to darker blue, dashed lines). Top row: $\mathcal{P}_1$ elements, bottom row: $\mathcal{P}_2$ elements.
• Figure 5: Example 2 (layered domain): Set-up of the computational domain with varying permeability (on a coarse mesh).

Theorems & Definitions (40)

• Lemma 1: Poincaré inequality
• Lemma 2: Trace inequality
• Lemma 3: Korn inequality
• Definition 1: $T$-coercivity
• Theorem 1
• Theorem 2
• proof
• Remark 1
• Lemma 4: Continuity
• proof