Coupling of conforming and mixed finite element methods for a model of wave propagation in thermo-poroelasticity in the frequency domain

TL;DR

This work analyzes wave propagation in fully dynamic thermo-poroelastic media in the frequency domain, formulating a coupled system with inertial and relaxation terms. It develops a stabilized coupling of conforming and mixed finite element spaces, proving well-posedness via Fredholm's alternative and $ mathsf{T}$-coercivity, and establishes error estimates and locking-free convergence. The discrete scheme uses Bernardi–Raugel for displacement, Raviart–Thomas for fluid displacement, and $P_0$/$P_1$ for pressure/temperature, augmented by stabilization and reduced integration to suppress pressure and temperature oscillations. Numerical experiments confirm accuracy, robustness to large $\lambda$ and degenerating coefficients, and effective suppression of oscillations across diverse frequency ranges and heterogeneous media. The results provide a practical, provably robust framework for frequency-domain thermo-poroelastic wave simulations in complex porous media.

Abstract

A dynamic linear thermo-poroelasticity model, containing inertial and relaxation terms with second-order time derivatives, is investigated in this paper. The mathematical and numerical analysis of this model is performed in the frequency domain. The variational formulation is analyzed within the framework of Fredholm's alternative and T-coercivity. Under appropriate assumptions on the coefficients, the well-posedness of the problem is proved. For its discretization, we propose a stabilized coupling of conforming and mixed finite element spaces, which are free of volumetric locking, and both, pressure as well as temperature oscillations. By incorporating projections in certain sesquilinear forms, the well-posedness of the finite element solution can be obtained through a similar reasoning as in the continuous case. Optimal error estimates are derived for all variables. Numerical studies validate the accuracy and robustness of the proposed method.

Figures (5)

• Figure 1: Analytic solution. Numerical errors for different values of $\omega$ and different mesh sizes $h$.
• Figure 2: Analytic solution. Numerical error $\|\nabla(T-T_h)\|_0$ for different values of $\theta$, different mesh sizes $h$, and different values of $\delta$.
• Figure 3: Cantilever bracket problem (left) and layered domain problem (right).
• Figure 4: Cantilever bracket problem. Numerical approximations for the real and imaginary part of the pressure.
• Figure 5: Layered domain problem. Numerical approximations for the magnitude of pressure.

Theorems & Definitions (22)

• Definition 3.1: $\mathsf{T}$-coercivity
• Lemma 3.2: Well-posedness of $\mathsf{T}$-coercive sesquilinear form
• Lemma 3.3: The Fredholm alternative of index zero
• Lemma 3.5: $\mathbb{T}$-Coercivity of $a_1(\cdot,\cdot)$
• Proof 1
• Lemma 3.6: Continuity
• Proof 2
• Lemma 3.7: Well-posedness of $\mathcal{A}_{\mathrm{sub}}$
• Proof 3
• Remark 3.8: On the assumptions for Lemma \ref{['lemma:bijective-Asub']}