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Analysis of a family of time-continuous strongly conservative space-time finite element methods for the dynamic Biot model

Johannes Kraus, Maria Lymbery, Kevin Osthues, Fadi Philo

TL;DR

This work develops and analyzes a time‑continuous space–time Galerkin discretization for the dynamic Biot poroelastic model in a three‑field formulation $(\bm{u},\bm{w},p)$. By combining a continuous‑in‑time Petrov Galerkin method of degree $k$ with an $H(\mathrm{div})$‑conforming discontinuous Galerkin discretization for the displacement and a mixed flux–pressure approximation, the method achieves pointwise mass conservation and stability. The authors establish rigorous a priori error estimates in a combined energy norm and $L^2$ norms, proving optimal convergence rates in time and space (with improved displacement regularity on convex domains). This provides a theoretically sound, mass‑conserving framework for simulating dynamic poroelastic wave phenomena that couples solid deformation and fluid flow robustly across parameter regimes.

Abstract

We consider the dynamic Biot model describing the interaction between fluid flow and solid deformation including wave propagation phenomena in both the liquid and solid phases of a saturated porous medium. The model couples a hyperbolic equation for momentum balance to a second-order in time dynamic Darcy law and a parabolic equation for the balance of mass and is here considered in three-field formulation with the displacement of the elastic matrix, the fluid velocity, and the fluid pressure being the physical fields of interest. A family of variational space-time finite element methods is proposed that combines a continuous-in-time Galerkin ansatz of arbitrary polynomial degree with inf-sup stable $H(\rm{div})$-conforming approximations of discontinuous Galerkin (DG) type in case of the displacement and a mixed approximation of the flux, its time derivative and the pressure field. We prove error estimates in a combined energy norm as well as $L^2$~error estimates in space for the individual fields for both maximum and $L^2$ norm in time which are optimal for the displacement and pressure approximations.

Analysis of a family of time-continuous strongly conservative space-time finite element methods for the dynamic Biot model

TL;DR

This work develops and analyzes a time‑continuous space–time Galerkin discretization for the dynamic Biot poroelastic model in a three‑field formulation . By combining a continuous‑in‑time Petrov Galerkin method of degree with an ‑conforming discontinuous Galerkin discretization for the displacement and a mixed flux–pressure approximation, the method achieves pointwise mass conservation and stability. The authors establish rigorous a priori error estimates in a combined energy norm and norms, proving optimal convergence rates in time and space (with improved displacement regularity on convex domains). This provides a theoretically sound, mass‑conserving framework for simulating dynamic poroelastic wave phenomena that couples solid deformation and fluid flow robustly across parameter regimes.

Abstract

We consider the dynamic Biot model describing the interaction between fluid flow and solid deformation including wave propagation phenomena in both the liquid and solid phases of a saturated porous medium. The model couples a hyperbolic equation for momentum balance to a second-order in time dynamic Darcy law and a parabolic equation for the balance of mass and is here considered in three-field formulation with the displacement of the elastic matrix, the fluid velocity, and the fluid pressure being the physical fields of interest. A family of variational space-time finite element methods is proposed that combines a continuous-in-time Galerkin ansatz of arbitrary polynomial degree with inf-sup stable -conforming approximations of discontinuous Galerkin (DG) type in case of the displacement and a mixed approximation of the flux, its time derivative and the pressure field. We prove error estimates in a combined energy norm as well as ~error estimates in space for the individual fields for both maximum and norm in time which are optimal for the displacement and pressure approximations.
Paper Structure (13 sections, 13 theorems, 130 equations)

This paper contains 13 sections, 13 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Preliminaries
  4. Function spaces
  5. Quadrature formulas and interpolation operators
  6. Auxiliary results
  7. Variational formulations
  8. Space-time variational problem
  9. Discontinuous Galerkin method
  10. Error analysis
  11. Proof of Lemma \ref{['lem:dg_auxiliary_I']}
  12. Proof of Lemma \ref{['lem:dg_auxiliary_II']}
  13. Proof of Lemma \ref{['lem:dg_auxiliary_III']}

Key Result

Lemma 3.1

Let Then the following statements are satisfied, where $\beta_{i} \in \mathbb{R}$ for $i = 0, \ldots, k$.

Theorems & Definitions (25)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 5.1: Projection errors
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • ...and 15 more