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An $hp$ Error Analysis of HDG for Dynamic Poroelasticity

Salim Meddahi

TL;DR

The paper develops a four-field HDG method for dynamic Biot poroelasticity at low frequencies, establishing well-posedness and energy stability for the continuous model and its hp-discrete HDG approximation. It proves convergence: semi-discrete hp estimates are quasi-optimal in mesh size $h$ with stress/pressure converging as $O(h^{k+1})$ and velocities as $O(h^{k+2})$, while $k$-rates are suboptimal by a factor of $k^{1/2}$ and velocity errors are slightly degraded in $h$. The fully discrete scheme uses Crank–Nicolson in time, achieving second-order temporal accuracy and compatible spatial hp convergence, with rigorous stability and error bounds. Numerical experiments confirm the predicted rates and demonstrate the method’s effectiveness in capturing poroelastic wave dynamics, including viscous effects and boundary handling, using a 2D/3D HDG framework and hp adaptivity.

Abstract

This study introduces a hybridizable discontinuous Galerkin (HDG) method for simulating low-frequency wave propagation in poroelastic media. We present a novel four-field variational formulation and establish its well-posedness and energy stability. Our \(hp\)-convergence analysis of the HDG method for spatial discretization is complemented by a Crank-Nicolson scheme for temporal discretization. Numerical experiments validate the theoretical convergence rates and demonstrate the effectiveness of the method in accurately capturing poroelastic dynamics.

An $hp$ Error Analysis of HDG for Dynamic Poroelasticity

TL;DR

The paper develops a four-field HDG method for dynamic Biot poroelasticity at low frequencies, establishing well-posedness and energy stability for the continuous model and its hp-discrete HDG approximation. It proves convergence: semi-discrete hp estimates are quasi-optimal in mesh size with stress/pressure converging as and velocities as , while -rates are suboptimal by a factor of and velocity errors are slightly degraded in . The fully discrete scheme uses Crank–Nicolson in time, achieving second-order temporal accuracy and compatible spatial hp convergence, with rigorous stability and error bounds. Numerical experiments confirm the predicted rates and demonstrate the method’s effectiveness in capturing poroelastic wave dynamics, including viscous effects and boundary handling, using a 2D/3D HDG framework and hp adaptivity.

Abstract

This study introduces a hybridizable discontinuous Galerkin (HDG) method for simulating low-frequency wave propagation in poroelastic media. We present a novel four-field variational formulation and establish its well-posedness and energy stability. Our -convergence analysis of the HDG method for spatial discretization is complemented by a Crank-Nicolson scheme for temporal discretization. Numerical experiments validate the theoretical convergence rates and demonstrate the effectiveness of the method in accurately capturing poroelastic dynamics.

Paper Structure

This paper contains 14 sections, 14 theorems, 117 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notations and Sobolev spaces
  3. Four field formulation of a dynamic and linear poroelastic problem
  4. Well-posedness of the model problem
  5. Functional setting
  6. Application of the Hille-Yosida theory
  7. Finite element spaces and approximation properties
  8. The HDG semi-discrete method
  9. Convergence analysis of the HDG method
  10. The fully discrete scheme and its convergence analysis
  11. Numerical results
  12. Example 1: Validation of the convergence rates
  13. Example 2: Wave propagation in a poroelastic medium
  14. Conclusion

Key Result

Proposition 1

The linear space $\mathcal{X}_1$ endowed with the inner product $(\cdot, \cdot)_{\mathcal{X}_1}$ is a Hilbert space.

Figures (4)

  • Figure 1: 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']}. The exact solution is provided by \ref{['exactSol']}.
  • Figure 2: The figure illustrates transient behaviour at the source receiver $\boldsymbol{x}_r = (2000, 2200)\, \unit{m}$, displaying the $y$-components of solid velocity (left panel), the solid pressure (center panel) and fluid pressure (right panel). Two cases are compared: a purely inviscid fluid ($\eta = 0$) shown with dashed lines and a viscous fluid ($\eta = 0.0015$) represented by continuous lines. Problem \ref{['4field-a']}-\ref{['4field-d']} is solved using the source terms \ref{['sourceW']}, boundary conditions \ref{['bcW']}, and parameter set \ref{['coeffs']}, with $h=100$, $k=5$, and $\Delta t = 0.005$.
  • Figure 3: Snapshots of the $y$-component of the solid velocity $\boldsymbol{u}_s$ at times 0.7 s, 0.9 s, and 1.1 s (left to right panels). Problem \ref{['4field-a']}-\ref{['4field-d']} is solved using the source terms \ref{['sourceW']}, boundary conditions \ref{['bcW']}, and parameter set \ref{['coeffs']}, with $h=100$, $k=5$, and $\Delta t = 0.005$. The top row shows results for an inviscid fluid ($\eta = 0$), while the bottom row corresponds to a viscous fluid ($\eta = 0.0015$).
  • Figure 4: Snapshots of the von Mises stress, computed from the tensor $\boldsymbol{\sigma} - \alpha p \mathrm{I}_d$, at times 0.7 s, 0.9 s, and 1.1 s (left to right panels). Problem \ref{['4field-a']}-\ref{['4field-d']} is solved using the source terms \ref{['sourceW']}, boundary conditions \ref{['bcW']}, and parameter set \ref{['coeffs']}, with $h=100$, $k=5$, and $\Delta t = 0.005$. The top row shows results for an inviscid fluid ($\eta = 0$), while the bottom row corresponds to a viscous fluid ($\eta = 0.0015$).

Theorems & Definitions (31)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • ...and 21 more