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arxiv: 2605.05451 · v1 · submitted 2026-05-06 · 🧮 math.NA · cs.NA

Hybridizable discontinuous Galerkin methods for poroelastic wave propagation with symmetric stress approximation

Jeonghun J. Lee , Manuel A. Sanchez This is my paper

Pith reviewed 2026-05-08 15:38 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords hybridizable discontinuous Galerkinporoelastic wave equationssymmetric stress approximationnearly incompressible materialsstatic condensationerror analysisCrank-Nicolson scheme
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The pith

Hybridizable discontinuous Galerkin methods for poroelastic waves produce strongly symmetric stress approximations with robust optimal convergence after static condensation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops hybridizable discontinuous Galerkin methods for poroelastic wave equations by first rewriting the system as a first-order symmetric hyperbolic problem. It then discretizes this system by combining an HDG+ approach for the elasticity terms with an LDG-H approach for the diffusion terms, ensuring the numerical stress tensor remains strongly symmetric. After static condensation the global system involves only trace variables for solid displacement and fluid pressure, and the resulting errors converge at optimal rates that stay robust even when the material is nearly incompressible. Comprehensive a priori error analyses cover both the semidiscrete formulation and the fully discrete Crank-Nicolson time-stepping scheme, while numerical examples confirm the predicted rates and demonstrate realistic wave-propagation behavior.

Core claim

By recasting the poroelastic equations as a first-order symmetric hyperbolic system and applying a tailored combination of HDG+ and LDG-H discretizations, the method yields a strongly symmetric stress approximation whose errors remain robust with respect to incompressibility and achieve optimal rates after static condensation to displacement and pressure traces; the analysis holds for both semidiscrete and Crank-Nicolson schemes.

What carries the argument

The combined HDG+ and LDG-H discretization of the rewritten first-order symmetric hyperbolic poroelastic system, followed by static condensation that retains only displacement and pressure trace variables.

If this is right

  • Strong symmetry of the stress tensor is obtained directly from the discretization without post-processing or additional constraints.
  • Error estimates remain uniform as the material approaches the incompressible limit.
  • Only displacement and pressure traces survive static condensation, yielding a smaller global algebraic system.
  • Optimal convergence rates hold for both the semidiscrete problem and the Crank-Nicolson fully discrete scheme.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The trace-only formulation after condensation may simplify coupling to other surface or interface models in multiphysics simulations.
  • The symmetry-preserving rewrite could be adapted to related first-order hyperbolic systems such as viscoelastic or thermoelastic wave propagation.
  • Because the method avoids locking without extra stabilization, it may enable reliable long-time simulations in geophysical or biomedical porous-media applications where material incompressibility is common.

Load-bearing premise

The original poroelastic equations can be rewritten as a first-order symmetric hyperbolic system that preserves the essential physics so the chosen HDG discretizations remain stable and accurate without extra stabilization.

What would settle it

A numerical experiment on a nearly incompressible test case (Poisson ratio approaching 0.5) that shows either loss of strong stress symmetry or degradation of the predicted optimal convergence rates after condensation.

Figures

Figures reproduced from arXiv: 2605.05451 by Jeonghun J. Lee, Manuel A. Sanchez.

Figure 1
Figure 1. Figure 1: Example 2. Simulation of a Gaussian pulse on isotropic sandstone media. Left: Fluid pressure approximation ph. Right: First component of the solid velocity approximation (vs,h)1. are represented without non-physical spurious oscillation, a common challenge in poroelastic simulations view at source ↗
Figure 2
Figure 2. Figure 2: Example 2. Simulation of a Gaussian pulse on anisotropic epoxy glass media. Left: First component of the solid velocity approximation (vs,h)1. Right: Second component of the solid velocity approximation (vs,h)2. Example 3: Wave propagation on heterogeneous poroelastic media. In our final numerical experiment, we extend the application of our method to a heterogeneous poroelastic medium. Simulating wave pro… view at source ↗
Figure 3
Figure 3. Figure 3: Example 3. Simulation of a Gaussian pulse on het￾erogeneous isotropic media. Left: First component of the solid velocity approximation (vs,h)1. Right: Second component of the solid velocity approximation (vs,h)2. variable; however, we also observe optimal convergence rates for the solid displace￾ment in numerical experiments. Numerical experiments with more physically real￾istic settings are consistent wit… view at source ↗
read the original abstract

In this paper, we develop hybridized discontinuous Galerkin (HDG) methods for poroelastic wave equations. We first rewrite the governing equations to a first-order symmetric hyperbolic system in order to use dual mixed formulations for discretization. Subsequently, we combine two HDG approaches in the discretization of the system, the $\text{HDG}+$ method for the linear elasticity equations and the $\text{LDG-H}$ method for the diffusion equations, with adjustments for the poroelastic wave equations. In our proposed HDG methods, the numerical approximation of the stress tensor is strongly symmetric and the convergence of the errors are robust for nearly incompressible materials. Upon performing static condensation, the system retains numerical trace variables solely for the solid displacement and the fluid pressure. We provide comprehensive error analyses for both the semidiscrete formulation and the Crank--Nicolson time-stepping scheme. Finally, extensive numerical examples illustrate optimal convergence results and simulate different poroelastic wave propagation scenarios relevant in the literature.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops hybridized discontinuous Galerkin (HDG) methods for poroelastic wave equations. It first rewrites the governing equations as a first-order symmetric hyperbolic system to enable dual mixed formulations. The discretization then combines the HDG+ method for the linear elasticity component with the LDG-H method for the diffusion component (with adjustments for poroelasticity). This yields strongly symmetric stress approximations that remain robust as the Lamé parameter λ approaches infinity. Static condensation reduces the global system to trace variables for solid displacement and fluid pressure only. Comprehensive error analyses are given for the semidiscrete problem and the Crank–Nicolson time-stepping scheme, and numerical examples demonstrate optimal convergence rates together with simulations of standard poroelastic wave scenarios.

Significance. If the central claims hold, the work supplies a practical and theoretically supported discretization for poroelastic waves that simultaneously enforces strong symmetry of the stress tensor and avoids locking in the nearly incompressible regime. The reduction to displacement and pressure traces via static condensation is a clear computational advantage. The provision of error estimates for both the semidiscrete and fully discrete (Crank–Nicolson) schemes, together with extensive numerical validation, adds concrete value for applications in geophysics and biomechanics. The combination of HDG+ and LDG-H within a single symmetric-hyperbolic framework is a technical contribution worth disseminating once the reformulation step is fully substantiated.

major comments (2)
  1. [Abstract and §2] Abstract and the reformulation step (presumably §2): The claim that the original poroelastic system can be rewritten as a first-order symmetric hyperbolic system “in order to use dual mixed formulations” is load-bearing for every subsequent result. The abstract does not display the resulting first-order system, so it is impossible to verify that the coupling terms between displacement, stress, pressure, and Darcy velocity remain exactly symmetric and hyperbolic without auxiliary stabilization or loss of the original Biot structure. This must be shown explicitly (with the precise matrix form of the hyperbolic operator) before the error analyses and robustness statements can be accepted.
  2. [Error-analysis sections (§4–5)] Error-analysis sections (presumably §4–5): The abstract asserts optimal convergence rates that are robust as λ → ∞ for both the semidiscrete and Crank–Nicolson schemes. These rates rest on the assumption that the rewritten symmetric hyperbolic system admits the same stability and approximation properties used in the HDG+ and LDG-H analyses. Without the explicit rewritten equations and the corresponding energy estimates, it is unclear whether the proofs carry over directly or require additional hypotheses that are not stated.
minor comments (2)
  1. [Numerical examples] Numerical examples: While optimal rates are reported, a direct comparison against a standard (non-hybridized) mixed method or against existing HDG schemes for Biot poroelasticity would strengthen the practical assessment of the new approach.
  2. [Throughout] Notation: Upon first introduction of the first-order system, all new variables (especially the auxiliary stress and velocity fields) should be listed with their physical meaning and the precise boundary conditions they satisfy.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. We address each major comment below and will make the necessary revisions to strengthen the presentation of the reformulation and error analysis.

read point-by-point responses

  1. Referee: [Abstract and §2] Abstract and the reformulation step (presumably §2): The claim that the original poroelastic system can be rewritten as a first-order symmetric hyperbolic system “in order to use dual mixed formulations” is load-bearing for every subsequent result. The abstract does not display the resulting first-order system, so it is impossible to verify that the coupling terms between displacement, stress, pressure, and Darcy velocity remain exactly symmetric and hyperbolic without auxiliary stabilization or loss of the original Biot structure. This must be shown explicitly (with the precise matrix form of the hyperbolic operator) before the error analyses and robustness statements can be accepted.

    Authors: We agree that the explicit form of the first-order symmetric hyperbolic system is essential for verifying the symmetry and hyperbolicity. Although the reformulation is derived and presented in Section 2, including the governing equations and the resulting system, we acknowledge that the abstract lacks this detail. In the revised manuscript, we will include the precise matrix form of the hyperbolic operator in the abstract to allow immediate verification. The coupling terms are constructed to maintain symmetry and the original Biot structure without additional stabilization, as shown in the derivation. revision: yes

  2. Referee: [Error-analysis sections (§4–5)] Error-analysis sections (presumably §4–5): The abstract asserts optimal convergence rates that are robust as λ → ∞ for both the semidiscrete and Crank–Nicolson schemes. These rates rest on the assumption that the rewritten symmetric hyperbolic system admits the same stability and approximation properties used in the HDG+ and LDG-H analyses. Without the explicit rewritten equations and the corresponding energy estimates, it is unclear whether the proofs carry over directly or require additional hypotheses that are not stated.

    Authors: The error analyses in Sections 4 and 5 rely on the well-posedness and stability of the symmetric hyperbolic system, which we establish in Section 2 prior to discretization. The energy estimates for the continuous system are provided there, and the discrete analyses adapt the standard HDG+ and LDG-H techniques accordingly. To clarify, we will add a dedicated subsection in Section 2 summarizing the energy estimates for the rewritten system and explicitly state the hypotheses under which the convergence rates hold, ensuring robustness as λ → ∞. This will make the connection to the error proofs transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard reformulation and independent error analysis

full rationale

The paper rewrites the poroelastic equations as a first-order symmetric hyperbolic system to enable dual mixed HDG+ and LDG-H discretizations, then performs static condensation and derives error estimates for the semidiscrete and Crank-Nicolson schemes. This chain is self-contained: the rewrite is presented as a modeling step to fit existing HDG frameworks, the symmetry and robustness claims follow from the subsequent analysis rather than being presupposed by definition, and no fitted parameters are relabeled as predictions. Any self-citations to prior HDG work serve as background for the component methods but are not load-bearing for the central poroelastic extension or its convergence proofs. The derivation does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the domain assumption that the poroelastic model admits a symmetric hyperbolic reformulation suitable for dual mixed methods, plus standard mathematical assumptions for well-posedness of hyperbolic systems and mixed FEM.

axioms (1)
  • domain assumption Poroelastic wave equations can be rewritten as a first-order symmetric hyperbolic system without loss of essential physical properties.
    Explicitly stated as the initial step to enable dual mixed formulations and HDG discretization.

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