Hybridizable Discontinuous Galerkin Methods for Coupled Poro-Viscoelastic and Thermo-Viscoelastic Systems

TL;DR

This work develops a single, first-order-in-time evolution framework that unifies poro-viscoelastic and thermo-viscoelastic dynamics via a Zener rheology with $\boldsymbol{\sigma}=\boldsymbol{\sigma}_E+\omega\boldsymbol{\sigma}_V$ and a Biot/Maxwell-Cattaneo coupling to a secondary field. An energy-stable, high-order hybridizable discontinuous Galerkin (HDG) method is formulated on general simplicial meshes, using hybrid traces for solid velocity and transport gradients to achieve a reduced globally coupled system and static condensation. Well-posedness is established through semigroup theory, and hp-error estimates are derived for the semi-discrete scheme, with numerical tests confirming optimal convergence rates and accurate wave-type propagation ($P$-, $S$-, and $T$-waves) in heterogeneous media. The results demonstrate robustness across parameter regimes, including nearly incompressible limits and strongly heterogeneous subdomains, and provide a solid foundation for adaptive hp refinement and nonlinear rheologies in multiphysics geomechanics and biomechanics applications.

Abstract

This article presents a unified mathematical framework for modeling coupled poro-viscoelastic and thermo-viscoelastic phenomena, formulated as a system of first-order in time partial differential equations. The model describes the evolution of solid velocity, elastic and viscous stress tensors, and additional fields related to either fluid pressure or temperature, depending on the physical context. We develop a hybridizable discontinuous Galerkin method for the numerical approximation of this coupled system, providing a high-order, stable discretization that efficiently handles the multiphysics nature of the problem. We establish stability analysis and derive optimal $hp$-error estimates for the semi-discrete formulation. The theoretical convergence rates are validated through comprehensive numerical experiments, demonstrating the method's accuracy and robustness across various test cases, including wave propagation in heterogeneous media with mixed viscoelastic properties.

Figures (7)

• Figure 1: The errors \ref{['Errors1']} are plotted against the mesh size h for various polynomial degrees k, using temporal over-refinements. Problem \ref{['eq:poro_thermo_visco']} is defined with coefficients \ref{['L1']} and the exact solution \ref{['exactSol']}.
• Figure 2: The errors \ref{['Errors1']} are plotted against the mesh size h for various polynomial degrees k, using temporal over-refinements. Problem \ref{['eq:poro_thermo_visco']} is defined with coefficients \ref{['L2']} and the exact solution \ref{['exactSol']}.
• Figure 3: Computed errors versus the polynomial degree $k$ with $h=1/4$ and $\Delta t = 10^{-6}$. The errors are measured at $t=0.3$, by employing the coefficients \ref{['L1']}.
• Figure 4: Computed errors for a sequence of uniform refinements in time with $h=1/16$ and $k=3$. The errors are measured at $t=0.5$, with the coefficients \ref{['L1']}.
• Figure 5: Snapshots of the temperature $\psi$ (top row) and the module of the solid velocity $\boldsymbol{u}$ (bottom row) at times 0.1 s, 0.3 s, and 0.5 s (left to right panels). Problem \ref{['eq:poro_thermo_visco']} is solved using the source terms \ref{['sourceTE']}, with vanishing initial and Dirichlet boundary conditions, and parameter set \ref{['L3']}, with $h=50$, $k=5$, and $\Delta t = 10^{-5}$.

Theorems & Definitions (19)

• Theorem 1
• proof
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Lemma 1
• proof