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Convergence of a continuous Galerkin method for the Biot-Allard poroelasticity system

Jakob S. Stokke, Markus Bause, Florin A. Radu

TL;DR

The paper develops a space-time continuous Galerkin discretization for the Biot-Allard poroelastic system with memory effects modeled by a convolution. By recasting memory via an auxiliary differential equation, the authors obtain an efficient ADE-based scheme using equal-order finite elements in space and continuous Galerkin in time, and they prove an optimal-energy-norm error estimate of order $O(\tau^{k+1}+h^{r})$. The analysis avoids reliance on inf-sup stability by employing an energy approach and carefully balancing coupling terms, with extensions to non-equal-order elements discussed. Numerical results corroborate the theoretical convergence rates and illustrate the practical robustness and accuracy of the method for dynamic poroelastic problems with memory effects.

Abstract

We study a space-time finite element method for a system of poromechanics with memory effects that are modeled by a convolution integral. In the literature, the system is referred to as the Biot-Allard model. We recast the model as a first-order system in time, where the memory effects are transformed into an auxiliary differential equation. This allows for a computationally efficient numerical scheme. The system is discretized by continuous Galerkin methods in time and equal-order finite element methods in space. An optimal order error estimate is proved for the norm of the first-order energy of the unknowns of the system. The estimate is confirmed by numerical experiments.

Convergence of a continuous Galerkin method for the Biot-Allard poroelasticity system

TL;DR

The paper develops a space-time continuous Galerkin discretization for the Biot-Allard poroelastic system with memory effects modeled by a convolution. By recasting memory via an auxiliary differential equation, the authors obtain an efficient ADE-based scheme using equal-order finite elements in space and continuous Galerkin in time, and they prove an optimal-energy-norm error estimate of order . The analysis avoids reliance on inf-sup stability by employing an energy approach and carefully balancing coupling terms, with extensions to non-equal-order elements discussed. Numerical results corroborate the theoretical convergence rates and illustrate the practical robustness and accuracy of the method for dynamic poroelastic problems with memory effects.

Abstract

We study a space-time finite element method for a system of poromechanics with memory effects that are modeled by a convolution integral. In the literature, the system is referred to as the Biot-Allard model. We recast the model as a first-order system in time, where the memory effects are transformed into an auxiliary differential equation. This allows for a computationally efficient numerical scheme. The system is discretized by continuous Galerkin methods in time and equal-order finite element methods in space. An optimal order error estimate is proved for the norm of the first-order energy of the unknowns of the system. The estimate is confirmed by numerical experiments.

Paper Structure

This paper contains 17 sections, 14 theorems, 120 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite element spaces
  4. Auxilliaries
  5. Discretization
  6. Continuous system and stability estimates
  7. Bilinear forms and discrete operators
  8. Continuous Galerkin discretization
  9. Error analysis
  10. Auxiliary results for the error analysis
  11. Splitting of the error
  12. Variational equations for TEXT and TEXT
  13. Estimation of the interpolation errors
  14. Estimation of TEXT and TEXT
  15. Main error estimate
  16. ...and 2 more sections

Key Result

Lemma 2.1

Consider the Gauss quadrature formula gaussquadratureInKthpoint. For all polynomials $w\in\mathbb{P}_{k}(I_n;L^{2})$ it holds that

Figures (1)

  • Figure 1: Schematic of proof strategy. Error estimation by bounding the discrete error $E_{\tau,h}$ and the interpolation error $\eta$, each associated with all unknowns of the system.

Theorems & Definitions (29)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Definition 4.1: Special approximation $(\bm{w}_{1},\bm{w}_{2})$ of $(\bm{u},\partial_t\bm{u})$
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • proof
  • Lemma 4.5: Variational equations for $\bm{E}_{\tau,h}^{1},\bm{E}_{\tau,h}^{2}, e_{\tau,h}$ and $\bm{E}_{\tau,h}^{3}$
  • proof
  • ...and 19 more